# Motion Switching and Chaos of a Particle in a Generalized Fermi-Acceleration Oscillator

Authors: A. C. J. Luo; Y. Guo
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Dynamic behaviors of a particle (or a bouncing ball) in a generalized Fermi-acceleration oscillator are investigated. The motion switching of a particle in the Fermi-oscillator causes the complexity and unpredictability of motion. Thus, the mechanism of motion switching of a particle in such a generalized Fermi-oscillator is studied through the theory of discontinuous dynamical systems, and the corresponding analytical conditions for the motion switching are developed. From solutions of linear systems in subdomains, four generic mappings are introduced, and mapping structures for periodic motions can be constructed. Thus, periodic motions in the Fermi-acceleration oscillator are predicted analytically, and the corresponding local stability and bifurcations are also discussed. From the analytical prediction, parameter maps of periodic and chaotic motions are achieved for a global view of motion behaviors in the Fermi-acceleration oscillator. Numerical simulations are carried out for illustrations of periodic and chaotic motions in such an oscillator. In existing results, motion switching in the Fermi-acceleration oscillator is not considered. The motion switching for many motion states of the Fermi-acceleration oscillator is presented for the first time. This methodology will provide a useful way to determine dynamical behaviors in the Fermi-acceleration oscillator.


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## Body

## 1. Introduction

The  Fermi-acceleration oscillator was first presented [1] by Fermi in 1949, which was used to explain the very
high energy of cosmic ray. Since then, such an oscillator has been extensively
investigated to interpret many physical and mechanical phenomena. In 1961, Ulam
[2] pointed out the statistical properties of a particle
in the Fermi-oscillator. In 1964, Zaslavskii and Chirikov [3] gave a comprehensive study of the Fermi-acceleration mechanism
in the one-dimensional case. In 1972, Lieberman and Lichtenberg [4] discussed the stochastic and adiabatic behaviors of
particles accelerated by periodic forcing, but such analysis was based on the
model presented by Zaslavskii and and Chirikov [3]. The corresponding stability of periodic motion was
discussed. Such results can be found from  Lichtenberg and Lieberman  [5].  In 1978,  Pustyl'nikov  [6] discussed the reducibility of non-autonomous systems
to normal forms in the neighborhood of an equilibrium point, and in 1995 he gave
a detailed description about the Fermi-acceleration mechanism [7]. In 1986, Jose and Cordery [8] studied a quantum Fermi-accelerator consisting of a
particle moving between a fixed wall and a periodic oscillator. In 1987,
Celaschi and Zimmerman [9] gave an experimental investigation to observe the
period-doubling route to chaos for a 1D system with two parameters. In 1988,
Kowalik et al. [10] completed an experimental investigation on chaotic behaviors
of a ball in the bouncing ball system, and used the Zaslavski's mapping to analyze
such chaotic behaviors. In 1989, Luna-Acosta [11] investigated the dynamics of the Fermi-accelerator
subject to a viscous friction. In 1996, Warr et al. [12] determined experimentally a probability distribution
function for a single-particle vibrating in one dimension. In 1998, Saif et al. [13] studied the dynamics of both classical and quantum
Fermi-acceleration oscillators and determined the dynamical localization of position
and momentum for a modulation amplitude. Lopac and Dananić [14] investigated chaotic dynamics and energy conservation
in a gravitationally driven Fermi-accelerator. In 2004, Bouchet et al. [15] presented a simple stochastic system to generate
anomalous diffusion of both position and velocity for the Fermi-accelerator. Leonel et al. [16] completed a scaling analysis of chaotic motion in the
Fermi-Ulam-simplified accelerator, and Leonel et al. [17] investigated the effect of a time-dependent perturbation
on a Fermi-accelerator model using discrete dynamical systems. In 2006, Leonel
and McClintock [18] discussed the influence of dissipation on a
simplified Fermi-Ulam accelerator model. In 2007, Leonel and de Carvalho [19] presented the Fermi-accelerator model with inelastic
collisions through a two-dimensional nonlinear area-contracting map. In 2007,
Karlis et al. [20] investigated the Fermi-acceleration of an ensemble of
noninteracted particles in two stochastic two-moving walls in the Fermi-Ulam
model. Leonel and Silva [21] discussed the dynamical properties of a prototype for
the Fermi-acceleration modeled by two nonlinear terms. Kamphorst et al. [22] numerically studied the energy change of a particle
bouncing in a time-varying billiard. In 2008, Leonel and Livorati [23] presented the average velocity behavior on a dissipative
Fermi-acceleration model with a scaling approach. The scaling characteristics
of a breathing circular billiard were investigated by Ladeira and da Silva [24]. All the
aforesaid investigations were based on the one motion state with impact. The motion
switching was not considered. In this paper, the motion switching from a motion
state into another one will be investigated.The similar studies of impacting systems have also
been carried out in mechanical engineering because the impact is an important
phenomenon in mechanical engineering. For instance, in  1982, Holmes [25] investigated the dynamics of repeated impacts of a
ball with a sinusoidal vibrating table. It was assumed that the mass of the
ball is much smaller than the mass of the table, and the ball and table always
impact at the same position. In  1983, Bapat et al. [26] investigated the asymptotically stable periodic
motions of a ball in an impact-pair. Shaw and Holmes [27] studied harmonic, subharmonic, and chaotic motions of
a single-degree-of-freedom nonlinear oscillator and analyzed the corresponding
bifurcations. In 1987, Whiston [28] studied the steady-state, vibroimpacting responses
of one-dimensional, harmonically excited, linear oscillator. In 1988, C. N. Bapat  and C. Bapat [29] used Fourier series and perturbation method to
determine the stability regions of two equispaced impact motion of an
impact-pair under a prescribed periodic displacement. In 1991, Nordmark [30]  studied the singularity caused by grazing impact in a
single degree of freedom impact oscillator. In 1992, Foale and Bishop [31] investigated a periodically forced linear oscillator
with instantaneous impacts at one or two stops. In 1994, Budd and Dux [32] investigated the chattering behavior of a
periodically forced, single-degree-of-freedom impact oscillator with a
restitution law for each impact. Foale [33] tried to determine bifurcations in a periodically
driven impact oscillator analytically, and the impact wall is fixed. In 1995,
Bapat [34] studied the motion of an inclined impact damper with
friction and collision on both sides. In order to understand dynamical
behaviors of a bouncing ball on a vibrating table, in 1996, Luo and Han [35] presented a reasonable model and analytically predicted
the periodic motions. In 2002, Luo [36] investigated the stability and bifurcation of
unsymmetrical periodic motions in a horizontal impact oscillator with a
periodic excitation. In 2004, Giusepponi et al. [37] discussed the randomness of motion of the bouncing
ball on a sinusoidally vibrating table. Luo [38] discussed the stability, saddle-node and
period-doubling bifurcations for the LR model motion in a horizontal impact
oscillator.In 2005, Luo [39] presented the mapping dynamics of periodic motions in
a nonsmooth piecewise system. To understand the complexity in
discontinuous dynamical systems, Luo [40, 41] developed a theory of the non-smooth dynamical
systems on connectable and accessible subdomains. Luo and Chen [42] applied such a theory to investigate the
grazing bifurcation and periodic motions in an idealized gear-transmission
system with impacts. In 2006, Luo and Gegg [43] used such a theory to develop the force criteria of
stick and nonstick motion in harmonically forced, friction-induced
oscillators. In 2007, Luo [44] discussed switching bifurcations of a flow to the
separation boundary. Luo and Rapp [45] studied the switching dynamics of a flow from one
domain into another adjacent domain in a periodically driven, discontinuous
dynamical system. Luo and O’Connor [46, 47] discussed the dynamics mechanism of impact chatters
and possible stick motions in a gear transmission system, and the moving
boundaries are controlled by other dynamical systems. In the gear model, the
two boundaries are movable. However, the Fermi-acceleration oscillator possesses both static and
time-varying boundaries in phase space for impacts and motion switching. In
existing investigations on the Fermi-acceleration oscillator or impact oscillators,
the dynamical systems are not switched except for impacts. Thus, in this paper,
the impacting and sticking of two systems on the boundaries will be considered,
which will provide a practical model to predict the motions of the
Fermi-acceleration oscillator.This paper will investigate
dynamics of a generalized Fermi-acceleration oscillator. The domains and
boundaries for such a problem will be introduced because of the discontinuity, and
the analytical conditions of stick and grazing motions will be developed at the
boundaries. The mapping technique will be used for the analytical prediction of
periodic motions of the extended Fermi-acceleration oscillator. The local
stability and bifurcation of periodic motions will be discussed using the eigenvalue
analysis. Bifurcation scenario and analytical prediction of motions will be
presented, and numerical simulations of periodic and chaotic motions will be carried
out. In addition, the Poincare mapping sections will be presented for
illustrations of chaotic motions, and the parameter maps will be presented as
well.


## 2. Physical Problem Statement

A generalized Fermi-acceleration oscillator consists of a particle moving vertically between a
fixed wall and the moving piston in a vibrating oscillator. The piston of massm2 is connected with a spring of constant k and a damper with coefficient c, and they are set on a periodically oscillating
base, as shown in Figure 1. The mass of particle is m1 and the restitution coefficients of impact are e1ande2 for the wall and piston, respectively. The gap
between the fixed wall and the equilibrium position of piston is h. The displacement of the oscillating base is D(t)=QcosΩt.Figure 1
Mechanical model.If the particle does not stay on
the piston, the corresponding motion of such a system is called thenonstick motion. For this case, the
equation of motion for such a system is given from the Newton's law, that is, (2.1)x¨(1)=−g,x¨(2)+2d(2)x˙(2)+(ω(2))2x(2)=(ω(2))2QcosΩt−2d(2)QΩsinΩt,  where d(2)=c/2m2,andω(2)=k/m2.x¨(i),x˙(i) and x(i) (i=1,2) are acceleration, velocity, and displacement,
respectively. g is the gravitational acceleration. The superscripts i=1 and 2 represent the particle and piston,
accordingly.If the particle stays on the piston and they move
together, such a motion is calledthe
stick motion. For the stick motion, there is a relation (x(1)=x(2) and x˙(1)=x˙(2)) and the corresponding equation of motion
becomes (2.2)x¨(i)+2d(0)x˙(i)+(ω(0))2x(i)=(ω(0))2QcosΩt−2d(0)QΩsinΩt,  where d(0)=c/2(m1+m2),ω(0)=k/(m1+m2). The superscripts i=1,2 represent the particle and piston, respectively.For nonstick motion, the impact relation between the particle
and the fixed wall is(2.3)x+(1)=x−(1)=h,x˙+(1)=−e1x˙−(1),x+(2)=x−(2),x˙+(2)=x˙−(2),  where (+) and (−) represent the after and before impact. The impact relations between the particle and the vibrating piston are (2.4)x+(1)=x+(2)=x−(1)=x−(2),x˙+(1)=m1x˙−(1)+m2x˙−(2)−m2e2(x˙−(1)−x˙−(2))m1+m2,x˙+(2)=m1x˙−(1)+m2x˙−(2)+m1e2(x˙−(1)−x˙−(2))m1+m2.


## 3. Analytical Conditions

In this section, the domains and boundaries of the extended Fermi-oscillator will be introduced in the absolute and relative coordinates. From such domains and boundaries, the analytical conditions of stick and grazing motions to each switching boundary will be developed from the theory of discontinuous dynamical systems.


### 3.1. Domains and Boundaries for Absolute Motions

To analyze the motion discontinuity in the Fermi-acceleration oscillator, the origin of the absolute coordinates is set at the equilibrium of piston. The domains for the particle and piston without stick in the absolute coordinates are defined as(3.1)Ω1(1)={(x(1),x˙(1))∣x(1)∈(x(2),h)},Ω1(2)={(x(2),x˙(2))∣x(2)∈(−∞,x(1))}.  The corresponding boundaries are
defined as (3.2)∂Ω1(+∞)(i)={(x(i),x˙(i))|φ1(+∞)(1)≡x(1)−h=0,x˙(1)≠0},∂Ω1(−∞)(i)={(x(i),x˙(i))|φ1(−∞)(i)≡x(i)−x(i¯)=0,x˙(i)≠x˙(i¯)},   where i¯=1and2 are for i=2and1,
accordingly. The subscripts “±∞” represent the permanent boundaries at x(1)=h and x(i)=x(i¯). If a flow in a subdomain of a dynamical system cannot pass over a switching boundary
into another subdomain without any transport law, such a boundary is called
the permanent boundary (e.g., Luo [48]). For this problem, the impact law, as a transport
law, should be applied for such permanent boundaries. In other words, the
impact relations in (2.3)
and (2.4) will be applied for
such permanent boundaries. The subdomains and boundaries for motions of particle
and piston in phase space are sketched in Figure 2. The subdomains are filled with diagonal lines. The
boundary (∂Ω1(+∞)(1)) for the particle is a dashed line with x(1)=h and the boundaries (∂Ω1(−∞)(1) and ∂Ω1(−∞)(2)) with x(1)=x(2) for the particle and piston are two dashed
curves.Absolute domains and boundaries without stick: (a) particle and (b) piston.
(a)(b)For this system, there is a stick motion of the particle and piston. Thus, the stick motion will appear and vanish under certain conditions. Such
onset and disappearance of the sticking motion will generate new boundaries and
domains. The domainsΩ0(i) and Ω1(i) (i=1,2) for the particle and piston with stick in the absolute coordinates are defined
as (3.3)Ω0(1)={(x(1),x˙(1))∣x(1)∈(−∞,xcr(2)),x˙(1)=x˙(2)},Ω0(2)={(x(2),x˙(2))∣x(2)∈(x(1),h),x˙(1)=x˙(2)},Ω1(1)={(x(1),x˙(1))∣x(1)∈(xcr(2),h),x˙(1)≠x˙(2)},Ω1(2)={(x(2),x˙(2))∣x(2)∈(−∞,xcr(1)),x˙(1)≠x˙(2)}.   The corresponding boundaries are defined as (3.4)∂Ω1(+∞)(i)={(x(i),x˙(i))|φ1(+∞)(1)≡x(1)−h=0,x˙(1)≠0},∂Ω10(i)={(x(i),x˙(i))|φ10(i)≡x(i)−xcr(i¯)=0,x˙(i)=x˙cr(i¯)},   where xcr(i) is an instantaneous value for the appearance or
vanishing of stick motion with x˙cr(1)=x˙cr(2) and xcr(1)=xcr(2). The instantaneous values are dependent on time, which cannot give specified values. The domains (Ω1(i) and Ω0(i)) describe the domains for the nonstick and
stick motions of the particle and piston, respectively. ∂Ω1(−∞)(1) is the impact chatter boundary for the particle
impacting with the piston, and ∂Ω10(1)and∂Ω01(1) are the stick onset and vanishing boundaries
for the particle sticking with the piston. Similarly, ∂Ω1(−∞)(2) is the impact chatter boundary for the piston
impacting with the particle, and ∂Ω10(2) and ∂Ω01(2) are the stick onset and vanishing boundaries
for the piston sticking with particle. In Figure 3, the nonstick and stick domains (Ω1(i) and Ω0(i)) are presented through the filled and shaded
regions. The boundaries (∂Ω10(i) and ∂Ω01(i), i=1,2) for the particle and piston are given by the
dashed curves in the absolute frame. The nonstick, permanent boundary (∂Ω1(+∞)(1)) for the particle is a dashed line, and such
a boundary is also applied for the piston as a permanent boundary.Absolute domains and boundaries with stick: (a) particle and (b) piston.
(a)(b)From the aforedefined domains, the vectors for absolute
motions are defined as(3.5)xλ(i)=(xλ(i),x˙λ(i))T,fλ(i)=(x˙λ(i),Fλ(i))Tfori=1,2,λ=0,1,   where the subscript (λ=0,1) represents the corresponding domain. The
equation of absolute motion of the generalized Fermi-oscillator is rewritten in
a state vector form of (3.6)x˙λ(i)=fλ(i)(xλ(i),t)fori=1,2,λ=0,1,   and one obtains (3.7)F1(1)(x1(1),t)=−g,F1(2)(x1(2),t)=−2dx˙1(2)−ω2x1(2)+ω2QcosΩt−2dQΩsinΩt   for the nonstick motion, and (3.8)F1(i)(x0(i),t)=−2d(0)x˙0(i)−(ω(0))2x0(i)+(ω(0))2QcosΩt−2d(0)QΩsinΩt   for the stick motion.


### 3.2. Domains and Boundaries for Relative Motions

Because the switching boundary varies with time, it is very
difficult to develop the switching conditions. Thus, the relative coordinates
for such a Fermi-oscillator are adopted herein. The relative displacement,
relative velocity, and relative acceleration between the particle and the piston arez(i)=x(i)−x(i¯),z˙(i)=x˙(i)−x˙(i¯), and z¨(i)=x¨(i)−x¨(i¯), where the index i=1,2 (with i¯=2,1) represents the particle and piston, respectively. The stick domain and
boundaries in the relative phase space become points, as shown in Figures 4(a) and 4(c). Because the
piston and particle have the same displacement and velocity for the entire
stick motion, the boundary and domains in the relative phase plane become zero. However, the relative interaction force between the particle and position are
different before or after the stick. Thus, the plane of
relative velocity and acceleration should be considered, and the stick
domains and boundaries in the relative velocity and acceleration (i.e., (z˙(i),z¨(i))) are presented in Figures 4(b) and 4(d). The filled domains are for stick motion. The shaded regions are for nonstick motion. The domains Ω0(i) and Ω1(i) for the relative motions of the particle and
piston are defined as (3.9)Ω0(i)={(z(i),z˙(i))∣z˙(i)=0,z(i)=0}fori=1,2,Ω1(1)={(z(1),z˙(1))∣z(1)∈(0,h−x(2))},Ω1(2)={(z(2),z˙(2))∣z(2)∈(−∞,0)}.   The boundaries ∂Ω1(+∞)(i),∂Ω1(−∞)(i),∂Ω10(i),and∂Ω01(i) for the relative motions of the particle and
piston are defined as (3.10)∂Ω1(+∞)(i)={(z(i),z˙(i))|φ1(+∞)(1)≡z(1)−h+x(2)=0},∂Ω1(−∞)(i)={(z(i),z˙(i))|φ1(−∞)(i)≡z(i)=0},∂Ω10(i)=∂Ω01(i)={(z(i),z˙(i))|φ10(i)≡z˙cr(i)=0,zcr(i)=0},   where ∂Ω1(−∞)(1) is the impact chatter boundary for the particle,
and two stick boundaries for the particle are ∂Ω10(1)and∂Ω01(1). Similarly, ∂Ω1(−∞)(2) is the impact chatter boundary for the piston,
and ∂Ω10(2)and∂Ω01(2) are two stick boundaries for the piston. ∂Ω1(+∞)(i) is the impact boundary for the particle and
the fixed wall.Relative domains and boundaries definition: (a)(z,z˙)-plane for particle, (b) (z˙,z¨)-plane for particle, (c) (z,z˙)-plane for piston, (d) (z˙,z¨)-plane for piston.
(a)(b)(c)(d)The vectors for relative motions are(3.11)zλ(i)=(zλ(i),z˙λ(i))T,gλ(i)=z˙λ(i)=(z˙λ(i),gλ(i))T,   where the superscript (i=1,2) represents the particle and piston, respectively. The subscript (λ=0,1) represents the corresponding domains. The
equation of relative motion is rewritten in a vector form of (3.12)z˙λ(i)=gλ(i)(zλ(i),xλ(i¯),t),x˙λ(i¯)=fλ(i¯)(xλ(i¯),t),   where (3.13)g1(1)(z1(1),x1(2),t)=−g+2d(2)x˙1(2)+(ω(2))2x1(2)−(ω(2))2QcosΩt+2d(2)QΩsinΩt,g1(2)(z1(2),x1(1),t)=−2d(2)z˙1(2)−(ω(2))2z1(2)+(ω(2))2QcosΩt−2d(2)QΩsinΩt−2d(2)x˙1(1)−(ω(2))2x1(1)+g,   for nonstick motion (3.14)z˙0(i)=0,g0(i)(z0(i),x0(i¯),t)=0,fori=1,2withi¯=2,1,   for stick motion.


### 3.3. Analytical Conditions

The analytical conditions of stick and grazing motion will
be developed from the theory of non-smooth dynamical systems in Luo   [40, 41, 44, 49, 50]. The grazing motion is a motion tangential to the switching
boundary in phase space. In the
Fermi-acceleration oscillator, once the ball meets
with the piston or the wall, there is a motion tangential to the switching boundary
in phase space. Such a motion is
called the grazing motion. To develop the analytical conditions of the grazing
and stick motion to the switching boundaries, the normal vectors of such boundaries
should be determined, that is, (3.15)n∂Ωαβ(i)=∇φαβ(i)=(∂φαβ(i)∂z(i),∂φαβ(i)∂z˙(i))Tfor  the  relative  frame,n∂Ωαβ=∇φαβ(i)=(∂φαβ(i)∂x(i),∂φαβ(i)∂x˙(i))Tfor  the  absolute  frame.   The normal vectors to the stick boundaries (n∂Ω10(1),n∂Ω01(1)) and impact boundary (n∂Ω1(+∞)(1),n∂Ω1(−∞)(1)) are (3.16)n∂Ω10(1)=n∂Ω01(1)=(0,1)T,n∂Ω1(−∞)(1)=(1,0)Tin  the  relative  frame,n∂Ω1(+∞)(1)=(1,0)Tin  the  absolute  frame.   Before the analytical conditions are developed, the zero-order and first-order G-functions in the relative
frame will be presented. For the
straight-line boundary, the zero-order and first-order G-functions are given
from Luo [49, 50], that is, (3.17)G∂Ω01(i)(0,0)(z0(i),tm±)=n∂Ω01(1)T⋅g(i)(z0(i),x0(i¯),tm±)=g0(i)(z0(i),x0(i¯),tm±)=z¨0(i)(tm±),G∂Ω10(i)(0,1)(z1(i),tm±)=n∂Ω10(1)T⋅g(i)(z1(i),x1(i¯),tm±)=g1(i)(z1(i),x1(i¯),tm±)=z¨1(i)(tm±),G∂Ω01(i)(1,0)(z0(i),tm±)=n∂Ω01(1)T⋅Dg(i)(z0(i),x0(i¯),tm±)=ddtg0(i)(z0(i),x0(i¯),t)|t=tm±=z⃛0(i)(tm±),G∂Ω10(i)(1,1)(z1(i),tm±)=n∂Ω10(1)T⋅Dg(i)(z1(i),x1(i¯),tm±)=ddtg1(i)(z1(i),x1(i¯),t)|t=tm±=z⃛1(i)(tm±).   Notice that the switching time tm is the motion at the corresponding switching
boundary, and tm±=tm±0 represents the motion in domains rather than
on the boundary. In G∂𝒟1(i)(k,λ),
two superscripts λ (λ=0,1) and k (k=0,1) represent the λ-subdomain and the kth order of G-functions, respectively. Again i=1and2 are for the particle and piston. The zero-order
G-function (k=0) is the normal component of the vector field
in the normal direction of boundary, and the first-order G-function (k=1) is the time-change rate of the normal
component of vector fields. The detailed discussion of G-functions can be
referred to Luo [48–50]. Similarly, the G-functions for the impact boundaries
are given by (3.18)G∂Ω1(+∞)(1)(0,1)(x1(1),tm±)=n∂Ω1(+∞)(1)T⋅f(1)(x1(1),tm±)=f(1)(x1(1),tm±)=x¨1(1)=−g,G∂Ω1(+∞)(1)(1,1)(x1(1),tm±)=n∂Ω1(+∞)(1)T⋅Df(1)(x1(1),tm±)=ddtf(1)(x1(1),t)|t=tm±=x⃛1(1)=0for∂Ω1(+∞)(1)in  the  absolute  frame,G∂Ω1(−∞)(i)(0,1)(z1(i),tm±)=n∂Ω1(−∞)(i)T⋅g(i)(z1(i),x1(i¯),tm±)=g(i)(z1(i),x1(i¯),tm±)=z¨1(i)(tm±),G∂Ω1(−∞)(i)(1,1)(z1(i),tm±)=n∂Ω1(−∞)(i)T⋅Dg(i)(z1(i),x1(i¯),tm±)=ddtg(i)(z1(i),x1(i¯),t)|t=tm±=z⃛1(i)(tm±)for∂Ω1(−∞)(i)in  the  relative  frame.   Based on the normal vectors and G-functions, the analytical
condition for stick motion is obtained for the condition for a passable flow
from domain Ω1(i) to Ω0(i) in Luo [49] or [50], that is, (3.19)(−1)iG∂Ω10(i)(0,1)(z1(i),tm−)>0,(−1)iG∂Ω10(i)(0,1)(z0(i),tm+)>0.   Therefore, one obtains (3.20)(−1)ig1(i)(z1(i),x1(i¯),tm−)>0,(−1)ig0(i)(z0(i),x0(i¯),tm+)>0.   With the relative force function per unit mass (i.e.,
relative acceleration z¨(i)=x¨(i)−x¨(i¯)), equations of    (3.20) gives (3.21)x¨(2)(tm±)>x¨(1)(tm±)=−g.
From (3.21), the stick motion
requires that the acceleration of the piston x¨(2)(tm±) is greater than x¨(1)(tm±)=−g. In a similar fashion, the condition for vanishing of the stick motion at ∂Ω01(i) in the relative frame is given by (3.22)G∂Ω01(i)(0,0)(z0(i),tm−)=0,G∂Ω01(i)(0,1)(z1(i),tm+)=0,(−1)iG∂Ω01(i)(1,0)(z0(i),tm−)<0,(−1)iG∂Ω01(i)(1,1)(z1(i),tm+)<0.   From equations of  (3.22), one obtains the relative force
relations or ∂Ω01(i) as (3.23)g0(i)(z0(i),x0(i¯),tm−)=0,g1(i)(z1(i),x1(i¯),tm+)=0,(−1)iddtg0(i)(z0(i),x0(i¯),tm−)<0,(−1)iddtg1(i)(z1(i),x1(i¯),tm+)<0.   With the relative acceleration (z¨(i)=x¨(i)−x¨(i¯)) and relative jerk (z⃛(i)=x⃛(i)−x⃛(i¯)), equations of  (3.22) can be rewritten as
(3.24)x¨(2)(tm±)=x¨(1)(tm±)=−g,x⃛m±(2)<x⃛m±(1)=0for∂Ω01(i).  The stick motion vanishing at the switching boundary
requires that the piston’s jerk is less than zero as both accelerations are the
gravitational acceleration.Based on theG-functions of a flow
to the impacting boundaries, the conditions of grazing motions are also
developed, that is, (3.25)G∂Ω1(+∞)(1)(0,1)(x1(1),tm±)=0,G∂Ω1(+∞)(1)(1,1)(x1(1),tm±)<0for∂Ω1(+∞)(i),(−1)iG∂Ω1(−∞)(1)(0,1)(z1(i),tm±)=0,(−1)iG∂Ω1(−∞)(1)(0,1)(z1(i),tm±)<0for∂Ω1(−∞)(i).   The grazing conditions for the boundaries of nonstick
motions reduce to (3.26)x˙(1)=0,x¨(1)=−g<0for∂Ω1(+∞)(1),x˙(1)=x˙(2),x¨(2)<x¨(1)=−gfor∂Ω1(−∞)(i).   From the first equation of (3.26),
the ball grazing at the fixed wall requires that the velocity of ball be zero
because the acceleration of ball (x¨(1)=−g) is always zero. However, from the second
equation of (3.26),
for the ball grazing at the impacting boundary with piston, the piston
acceleration should be less than the ball acceleration (i.e., x¨(1)=−g). In other words, this condition is also for
the piston motion grazing at this impacting boundary with the ball.In a similar fashion, the grazing
conditions for the boundary of the stick motion are(3.27)G∂Ω10(i)(0,1)(z1(i),tm±)=0,(−1)iG∂Ω10(i)(1,1)(z1(i),tm±)<0for∂Ω10(i),G∂Ω01(i)(0,0)(z0(i),tm±)=0,(−1)iG∂Ω01(i)(1,0)(z0(i),tm±)>0for∂Ω01(i).   Again, with the relative acceleration (z¨(i)=x¨(i)−x¨(i¯)) and relative jerk (z⃛(i)=x⃛(i)−x⃛(i¯)), the grazing conditions are given by the
accelerations and jerks of piston and ball, that is, (3.28)x¨(2)=x¨(1)=−g,x⃛(2)<x⃛(1)=0for∂Ω10(i),x¨(2)=x¨(1)=−g,x⃛(2)>x⃛(1)=0for∂Ω01(i).To geometrically explain the above
analytical conditions, consider a flow of the motion approaching the stick
boundary∂Ω10(1) at (z1(1),tm−). In order for such a flow to pass the boundary, the normal vector fields of G∂Ω10(i)(0,1)andG∂Ω01(i)(0,0) have to point toward the stick domain Ω1(i) (i.e., (−1)iG∂Ω10(i)(0,1)>0, (−1)iG∂Ω01(i)(0,0)>0) because the normal vector points to the
domain Ω1(i). Such conditions are given in (3.19)
(or (3.21))
and the flow of motion will pass over the boundary and gets into the stick
domain Ω0(i). Such a motion is a passable motion to the boundary ∂Ω10(1),
as shown in Figure 5(a). For this case, the stick motion is formed. Similarly,
for a flow from a stick domain to the nonstick domain, a flow of motion needs
to satisfy the conditions in (3.22)
(or (3.23)). Such a passable flow is called the vanishing of stick motion. When a flow of
motion approaches the impact chatter boundary ∂Ω1(−∞)(1) at (z1(1),tm−),
if the normal vector field G∂Ω1(−∞)(1)(0,1) be equal to zero and the total change rate of
the normal vector field G∂Ω1(−∞)(1)(1,1) is greater than zero in domain Ω1(1) (i.e., G∂Ω1(−∞)(1)(0,1)=0 and G∂Ω1(−∞)(1)(1,1)>0), the flow of motion will leave the boundary
and stay in domain Ω1(1),
as sketched in Figure
5(b). Such a motion is called a grazing motion to the
boundary ∂Ω1(−∞)(1).  The corresponding condition for grazing
at ∂Ω1(−∞)(1) is presented in (3.25). Similarly, for the stick boundary ∂Ω01(1),
the grazing condition in (3.27) can be geometrically
explained.(a) Passable motion and (b) grazing motion.
(a)(b)


## 3.1. Domains and Boundaries for Absolute Motions

To analyze the motion discontinuity in the Fermi-acceleration oscillator, the origin of the absolute coordinates is set at the equilibrium of piston. The domains for the particle and piston without stick in the absolute coordinates are defined as(3.1)Ω1(1)={(x(1),x˙(1))∣x(1)∈(x(2),h)},Ω1(2)={(x(2),x˙(2))∣x(2)∈(−∞,x(1))}.  The corresponding boundaries are
defined as (3.2)∂Ω1(+∞)(i)={(x(i),x˙(i))|φ1(+∞)(1)≡x(1)−h=0,x˙(1)≠0},∂Ω1(−∞)(i)={(x(i),x˙(i))|φ1(−∞)(i)≡x(i)−x(i¯)=0,x˙(i)≠x˙(i¯)},   where i¯=1and2 are for i=2and1,
accordingly. The subscripts “±∞” represent the permanent boundaries at x(1)=h and x(i)=x(i¯). If a flow in a subdomain of a dynamical system cannot pass over a switching boundary
into another subdomain without any transport law, such a boundary is called
the permanent boundary (e.g., Luo [48]). For this problem, the impact law, as a transport
law, should be applied for such permanent boundaries. In other words, the
impact relations in (2.3)
and (2.4) will be applied for
such permanent boundaries. The subdomains and boundaries for motions of particle
and piston in phase space are sketched in Figure 2. The subdomains are filled with diagonal lines. The
boundary (∂Ω1(+∞)(1)) for the particle is a dashed line with x(1)=h and the boundaries (∂Ω1(−∞)(1) and ∂Ω1(−∞)(2)) with x(1)=x(2) for the particle and piston are two dashed
curves.Absolute domains and boundaries without stick: (a) particle and (b) piston.
(a)(b)For this system, there is a stick motion of the particle and piston. Thus, the stick motion will appear and vanish under certain conditions. Such
onset and disappearance of the sticking motion will generate new boundaries and
domains. The domainsΩ0(i) and Ω1(i) (i=1,2) for the particle and piston with stick in the absolute coordinates are defined
as (3.3)Ω0(1)={(x(1),x˙(1))∣x(1)∈(−∞,xcr(2)),x˙(1)=x˙(2)},Ω0(2)={(x(2),x˙(2))∣x(2)∈(x(1),h),x˙(1)=x˙(2)},Ω1(1)={(x(1),x˙(1))∣x(1)∈(xcr(2),h),x˙(1)≠x˙(2)},Ω1(2)={(x(2),x˙(2))∣x(2)∈(−∞,xcr(1)),x˙(1)≠x˙(2)}.   The corresponding boundaries are defined as (3.4)∂Ω1(+∞)(i)={(x(i),x˙(i))|φ1(+∞)(1)≡x(1)−h=0,x˙(1)≠0},∂Ω10(i)={(x(i),x˙(i))|φ10(i)≡x(i)−xcr(i¯)=0,x˙(i)=x˙cr(i¯)},   where xcr(i) is an instantaneous value for the appearance or
vanishing of stick motion with x˙cr(1)=x˙cr(2) and xcr(1)=xcr(2). The instantaneous values are dependent on time, which cannot give specified values. The domains (Ω1(i) and Ω0(i)) describe the domains for the nonstick and
stick motions of the particle and piston, respectively. ∂Ω1(−∞)(1) is the impact chatter boundary for the particle
impacting with the piston, and ∂Ω10(1)and∂Ω01(1) are the stick onset and vanishing boundaries
for the particle sticking with the piston. Similarly, ∂Ω1(−∞)(2) is the impact chatter boundary for the piston
impacting with the particle, and ∂Ω10(2) and ∂Ω01(2) are the stick onset and vanishing boundaries
for the piston sticking with particle. In Figure 3, the nonstick and stick domains (Ω1(i) and Ω0(i)) are presented through the filled and shaded
regions. The boundaries (∂Ω10(i) and ∂Ω01(i), i=1,2) for the particle and piston are given by the
dashed curves in the absolute frame. The nonstick, permanent boundary (∂Ω1(+∞)(1)) for the particle is a dashed line, and such
a boundary is also applied for the piston as a permanent boundary.Absolute domains and boundaries with stick: (a) particle and (b) piston.
(a)(b)From the aforedefined domains, the vectors for absolute
motions are defined as(3.5)xλ(i)=(xλ(i),x˙λ(i))T,fλ(i)=(x˙λ(i),Fλ(i))Tfori=1,2,λ=0,1,   where the subscript (λ=0,1) represents the corresponding domain. The
equation of absolute motion of the generalized Fermi-oscillator is rewritten in
a state vector form of (3.6)x˙λ(i)=fλ(i)(xλ(i),t)fori=1,2,λ=0,1,   and one obtains (3.7)F1(1)(x1(1),t)=−g,F1(2)(x1(2),t)=−2dx˙1(2)−ω2x1(2)+ω2QcosΩt−2dQΩsinΩt   for the nonstick motion, and (3.8)F1(i)(x0(i),t)=−2d(0)x˙0(i)−(ω(0))2x0(i)+(ω(0))2QcosΩt−2d(0)QΩsinΩt   for the stick motion.


## 3.2. Domains and Boundaries for Relative Motions

Because the switching boundary varies with time, it is very
difficult to develop the switching conditions. Thus, the relative coordinates
for such a Fermi-oscillator are adopted herein. The relative displacement,
relative velocity, and relative acceleration between the particle and the piston arez(i)=x(i)−x(i¯),z˙(i)=x˙(i)−x˙(i¯), and z¨(i)=x¨(i)−x¨(i¯), where the index i=1,2 (with i¯=2,1) represents the particle and piston, respectively. The stick domain and
boundaries in the relative phase space become points, as shown in Figures 4(a) and 4(c). Because the
piston and particle have the same displacement and velocity for the entire
stick motion, the boundary and domains in the relative phase plane become zero. However, the relative interaction force between the particle and position are
different before or after the stick. Thus, the plane of
relative velocity and acceleration should be considered, and the stick
domains and boundaries in the relative velocity and acceleration (i.e., (z˙(i),z¨(i))) are presented in Figures 4(b) and 4(d). The filled domains are for stick motion. The shaded regions are for nonstick motion. The domains Ω0(i) and Ω1(i) for the relative motions of the particle and
piston are defined as (3.9)Ω0(i)={(z(i),z˙(i))∣z˙(i)=0,z(i)=0}fori=1,2,Ω1(1)={(z(1),z˙(1))∣z(1)∈(0,h−x(2))},Ω1(2)={(z(2),z˙(2))∣z(2)∈(−∞,0)}.   The boundaries ∂Ω1(+∞)(i),∂Ω1(−∞)(i),∂Ω10(i),and∂Ω01(i) for the relative motions of the particle and
piston are defined as (3.10)∂Ω1(+∞)(i)={(z(i),z˙(i))|φ1(+∞)(1)≡z(1)−h+x(2)=0},∂Ω1(−∞)(i)={(z(i),z˙(i))|φ1(−∞)(i)≡z(i)=0},∂Ω10(i)=∂Ω01(i)={(z(i),z˙(i))|φ10(i)≡z˙cr(i)=0,zcr(i)=0},   where ∂Ω1(−∞)(1) is the impact chatter boundary for the particle,
and two stick boundaries for the particle are ∂Ω10(1)and∂Ω01(1). Similarly, ∂Ω1(−∞)(2) is the impact chatter boundary for the piston,
and ∂Ω10(2)and∂Ω01(2) are two stick boundaries for the piston. ∂Ω1(+∞)(i) is the impact boundary for the particle and
the fixed wall.Relative domains and boundaries definition: (a)(z,z˙)-plane for particle, (b) (z˙,z¨)-plane for particle, (c) (z,z˙)-plane for piston, (d) (z˙,z¨)-plane for piston.
(a)(b)(c)(d)The vectors for relative motions are(3.11)zλ(i)=(zλ(i),z˙λ(i))T,gλ(i)=z˙λ(i)=(z˙λ(i),gλ(i))T,   where the superscript (i=1,2) represents the particle and piston, respectively. The subscript (λ=0,1) represents the corresponding domains. The
equation of relative motion is rewritten in a vector form of (3.12)z˙λ(i)=gλ(i)(zλ(i),xλ(i¯),t),x˙λ(i¯)=fλ(i¯)(xλ(i¯),t),   where (3.13)g1(1)(z1(1),x1(2),t)=−g+2d(2)x˙1(2)+(ω(2))2x1(2)−(ω(2))2QcosΩt+2d(2)QΩsinΩt,g1(2)(z1(2),x1(1),t)=−2d(2)z˙1(2)−(ω(2))2z1(2)+(ω(2))2QcosΩt−2d(2)QΩsinΩt−2d(2)x˙1(1)−(ω(2))2x1(1)+g,   for nonstick motion (3.14)z˙0(i)=0,g0(i)(z0(i),x0(i¯),t)=0,fori=1,2withi¯=2,1,   for stick motion.


## 3.3. Analytical Conditions

The analytical conditions of stick and grazing motion will
be developed from the theory of non-smooth dynamical systems in Luo   [40, 41, 44, 49, 50]. The grazing motion is a motion tangential to the switching
boundary in phase space. In the
Fermi-acceleration oscillator, once the ball meets
with the piston or the wall, there is a motion tangential to the switching boundary
in phase space. Such a motion is
called the grazing motion. To develop the analytical conditions of the grazing
and stick motion to the switching boundaries, the normal vectors of such boundaries
should be determined, that is, (3.15)n∂Ωαβ(i)=∇φαβ(i)=(∂φαβ(i)∂z(i),∂φαβ(i)∂z˙(i))Tfor  the  relative  frame,n∂Ωαβ=∇φαβ(i)=(∂φαβ(i)∂x(i),∂φαβ(i)∂x˙(i))Tfor  the  absolute  frame.   The normal vectors to the stick boundaries (n∂Ω10(1),n∂Ω01(1)) and impact boundary (n∂Ω1(+∞)(1),n∂Ω1(−∞)(1)) are (3.16)n∂Ω10(1)=n∂Ω01(1)=(0,1)T,n∂Ω1(−∞)(1)=(1,0)Tin  the  relative  frame,n∂Ω1(+∞)(1)=(1,0)Tin  the  absolute  frame.   Before the analytical conditions are developed, the zero-order and first-order G-functions in the relative
frame will be presented. For the
straight-line boundary, the zero-order and first-order G-functions are given
from Luo [49, 50], that is, (3.17)G∂Ω01(i)(0,0)(z0(i),tm±)=n∂Ω01(1)T⋅g(i)(z0(i),x0(i¯),tm±)=g0(i)(z0(i),x0(i¯),tm±)=z¨0(i)(tm±),G∂Ω10(i)(0,1)(z1(i),tm±)=n∂Ω10(1)T⋅g(i)(z1(i),x1(i¯),tm±)=g1(i)(z1(i),x1(i¯),tm±)=z¨1(i)(tm±),G∂Ω01(i)(1,0)(z0(i),tm±)=n∂Ω01(1)T⋅Dg(i)(z0(i),x0(i¯),tm±)=ddtg0(i)(z0(i),x0(i¯),t)|t=tm±=z⃛0(i)(tm±),G∂Ω10(i)(1,1)(z1(i),tm±)=n∂Ω10(1)T⋅Dg(i)(z1(i),x1(i¯),tm±)=ddtg1(i)(z1(i),x1(i¯),t)|t=tm±=z⃛1(i)(tm±).   Notice that the switching time tm is the motion at the corresponding switching
boundary, and tm±=tm±0 represents the motion in domains rather than
on the boundary. In G∂𝒟1(i)(k,λ),
two superscripts λ (λ=0,1) and k (k=0,1) represent the λ-subdomain and the kth order of G-functions, respectively. Again i=1and2 are for the particle and piston. The zero-order
G-function (k=0) is the normal component of the vector field
in the normal direction of boundary, and the first-order G-function (k=1) is the time-change rate of the normal
component of vector fields. The detailed discussion of G-functions can be
referred to Luo [48–50]. Similarly, the G-functions for the impact boundaries
are given by (3.18)G∂Ω1(+∞)(1)(0,1)(x1(1),tm±)=n∂Ω1(+∞)(1)T⋅f(1)(x1(1),tm±)=f(1)(x1(1),tm±)=x¨1(1)=−g,G∂Ω1(+∞)(1)(1,1)(x1(1),tm±)=n∂Ω1(+∞)(1)T⋅Df(1)(x1(1),tm±)=ddtf(1)(x1(1),t)|t=tm±=x⃛1(1)=0for∂Ω1(+∞)(1)in  the  absolute  frame,G∂Ω1(−∞)(i)(0,1)(z1(i),tm±)=n∂Ω1(−∞)(i)T⋅g(i)(z1(i),x1(i¯),tm±)=g(i)(z1(i),x1(i¯),tm±)=z¨1(i)(tm±),G∂Ω1(−∞)(i)(1,1)(z1(i),tm±)=n∂Ω1(−∞)(i)T⋅Dg(i)(z1(i),x1(i¯),tm±)=ddtg(i)(z1(i),x1(i¯),t)|t=tm±=z⃛1(i)(tm±)for∂Ω1(−∞)(i)in  the  relative  frame.   Based on the normal vectors and G-functions, the analytical
condition for stick motion is obtained for the condition for a passable flow
from domain Ω1(i) to Ω0(i) in Luo [49] or [50], that is, (3.19)(−1)iG∂Ω10(i)(0,1)(z1(i),tm−)>0,(−1)iG∂Ω10(i)(0,1)(z0(i),tm+)>0.   Therefore, one obtains (3.20)(−1)ig1(i)(z1(i),x1(i¯),tm−)>0,(−1)ig0(i)(z0(i),x0(i¯),tm+)>0.   With the relative force function per unit mass (i.e.,
relative acceleration z¨(i)=x¨(i)−x¨(i¯)), equations of    (3.20) gives (3.21)x¨(2)(tm±)>x¨(1)(tm±)=−g.
From (3.21), the stick motion
requires that the acceleration of the piston x¨(2)(tm±) is greater than x¨(1)(tm±)=−g. In a similar fashion, the condition for vanishing of the stick motion at ∂Ω01(i) in the relative frame is given by (3.22)G∂Ω01(i)(0,0)(z0(i),tm−)=0,G∂Ω01(i)(0,1)(z1(i),tm+)=0,(−1)iG∂Ω01(i)(1,0)(z0(i),tm−)<0,(−1)iG∂Ω01(i)(1,1)(z1(i),tm+)<0.   From equations of  (3.22), one obtains the relative force
relations or ∂Ω01(i) as (3.23)g0(i)(z0(i),x0(i¯),tm−)=0,g1(i)(z1(i),x1(i¯),tm+)=0,(−1)iddtg0(i)(z0(i),x0(i¯),tm−)<0,(−1)iddtg1(i)(z1(i),x1(i¯),tm+)<0.   With the relative acceleration (z¨(i)=x¨(i)−x¨(i¯)) and relative jerk (z⃛(i)=x⃛(i)−x⃛(i¯)), equations of  (3.22) can be rewritten as
(3.24)x¨(2)(tm±)=x¨(1)(tm±)=−g,x⃛m±(2)<x⃛m±(1)=0for∂Ω01(i).  The stick motion vanishing at the switching boundary
requires that the piston’s jerk is less than zero as both accelerations are the
gravitational acceleration.Based on theG-functions of a flow
to the impacting boundaries, the conditions of grazing motions are also
developed, that is, (3.25)G∂Ω1(+∞)(1)(0,1)(x1(1),tm±)=0,G∂Ω1(+∞)(1)(1,1)(x1(1),tm±)<0for∂Ω1(+∞)(i),(−1)iG∂Ω1(−∞)(1)(0,1)(z1(i),tm±)=0,(−1)iG∂Ω1(−∞)(1)(0,1)(z1(i),tm±)<0for∂Ω1(−∞)(i).   The grazing conditions for the boundaries of nonstick
motions reduce to (3.26)x˙(1)=0,x¨(1)=−g<0for∂Ω1(+∞)(1),x˙(1)=x˙(2),x¨(2)<x¨(1)=−gfor∂Ω1(−∞)(i).   From the first equation of (3.26),
the ball grazing at the fixed wall requires that the velocity of ball be zero
because the acceleration of ball (x¨(1)=−g) is always zero. However, from the second
equation of (3.26),
for the ball grazing at the impacting boundary with piston, the piston
acceleration should be less than the ball acceleration (i.e., x¨(1)=−g). In other words, this condition is also for
the piston motion grazing at this impacting boundary with the ball.In a similar fashion, the grazing
conditions for the boundary of the stick motion are(3.27)G∂Ω10(i)(0,1)(z1(i),tm±)=0,(−1)iG∂Ω10(i)(1,1)(z1(i),tm±)<0for∂Ω10(i),G∂Ω01(i)(0,0)(z0(i),tm±)=0,(−1)iG∂Ω01(i)(1,0)(z0(i),tm±)>0for∂Ω01(i).   Again, with the relative acceleration (z¨(i)=x¨(i)−x¨(i¯)) and relative jerk (z⃛(i)=x⃛(i)−x⃛(i¯)), the grazing conditions are given by the
accelerations and jerks of piston and ball, that is, (3.28)x¨(2)=x¨(1)=−g,x⃛(2)<x⃛(1)=0for∂Ω10(i),x¨(2)=x¨(1)=−g,x⃛(2)>x⃛(1)=0for∂Ω01(i).To geometrically explain the above
analytical conditions, consider a flow of the motion approaching the stick
boundary∂Ω10(1) at (z1(1),tm−). In order for such a flow to pass the boundary, the normal vector fields of G∂Ω10(i)(0,1)andG∂Ω01(i)(0,0) have to point toward the stick domain Ω1(i) (i.e., (−1)iG∂Ω10(i)(0,1)>0, (−1)iG∂Ω01(i)(0,0)>0) because the normal vector points to the
domain Ω1(i). Such conditions are given in (3.19)
(or (3.21))
and the flow of motion will pass over the boundary and gets into the stick
domain Ω0(i). Such a motion is a passable motion to the boundary ∂Ω10(1),
as shown in Figure 5(a). For this case, the stick motion is formed. Similarly,
for a flow from a stick domain to the nonstick domain, a flow of motion needs
to satisfy the conditions in (3.22)
(or (3.23)). Such a passable flow is called the vanishing of stick motion. When a flow of
motion approaches the impact chatter boundary ∂Ω1(−∞)(1) at (z1(1),tm−),
if the normal vector field G∂Ω1(−∞)(1)(0,1) be equal to zero and the total change rate of
the normal vector field G∂Ω1(−∞)(1)(1,1) is greater than zero in domain Ω1(1) (i.e., G∂Ω1(−∞)(1)(0,1)=0 and G∂Ω1(−∞)(1)(1,1)>0), the flow of motion will leave the boundary
and stay in domain Ω1(1),
as sketched in Figure
5(b). Such a motion is called a grazing motion to the
boundary ∂Ω1(−∞)(1).  The corresponding condition for grazing
at ∂Ω1(−∞)(1) is presented in (3.25). Similarly, for the stick boundary ∂Ω01(1),
the grazing condition in (3.27) can be geometrically
explained.(a) Passable motion and (b) grazing motion.
(a)(b)


## 4. Mapping Structures and Motions

In this section, the switching sets and mapping structures will be introduced to symbolically describe motions in the generalized Fermi-acceleration oscillator. The switching sets will be defined from the switching boundaries. From the switching sets, the generic mappings in domains will be introduced. The mapping structure for periodic and chaotic motions will be constructed from such generic mappings. The stability and bifurcation conditions will also be discussed via the eigenvalue analysis.


### 4.1. Switching Sets and Generic Mappings

Based on the switching boundaries in (3.2), the switching sets of the generalized Fermi-acceleration oscillator without stick are defined as(4.1)Σ1(−∞)=Σ1(−∞)(1)⊗Σ1(−∞)(2)={(xi(1),x˙i(1),xi(2),x˙i(2),ti)∣xi(1)=xi+(2),x˙i(1)≠x˙i(2)},Σ1(+∞)=Σ1(+∞)(1)⊗Σ1(+∞)(2)={(xi(1),x˙i(1),xi(2),x˙i(2),ti)∣xi(1)=h,x˙i(1)≠0}, where the symbol ⊗ represents the direct product of switching
subsets, and the switching subsets (Σ1(−∞)(i) and Σ1(+∞)(i)) on the boundaries (∂Ω1(−∞)(i) and ∂Ω1(+∞)(i)) are defined as  (4.2)Σ1(−∞)(i)={(xk(i),x˙k(i),tk)∣xk(i)=xk(i¯),x˙k(i)≠x˙k(i¯)}⊂∂Ω1(−∞)(i),Σ1(+∞)(i)={(xk(i),x˙k(i),tk)∣xk(1)=h,x˙k(1)≠0}⊂∂Ω1(+∞)(i).  Thus, the generic mappings for motions without stick are
defined as (4.3)P1:Σ1(−∞)→Σ1(+∞),P2:Σ1(+∞)→Σ1(−∞),P3:Σ1(−∞)→Σ1(−∞),  where the mappings P1 and P2 are the global mappings and the mapping P3 is a local mapping. No any mapping exists from Σ1(+∞) to Σ1(+∞) because x¨(1)=−g. The global mapping will map one switching set into another one. However, the
local mapping will map a switching set into itself. Since the resultant
switching set is the direct product of the two switching subsets, a resultant
mapping should have two components, that is, (4.4)P1=(P1(1),P1(2)),P2=(P2(1),P1(2)),P3=(P3(1),P1(2)),P1(i):Σ1(−∞)(i)→Σ1(+∞)(i),P2(i):Σ1(+∞)(i)→Σ1(−∞)(i),P3(i):Σ1(−∞)(i)→Σ1(−∞)(i).   From the above definitions,
the switching subsets and submappings are sketched in Figures 6(a) and 6(b)
for the particle and piston. The switching points in the switching sets are
presented by the circular symbols, and the mappings are depicted by the curves with
arrows.Switching sets and generic mappings for nonstick motion (in
absolute coordinates): (a) particle and
(b) piston.
(a)(b)Similarly, from the switching boundaries in (3.2)
and (3.4),
the switching sets of the generalized Fermi-acceleration oscillator with stick
are defined as (4.5)Σ10=Σ10(1)⊗Σ10(2)={(xk(1),x˙k(1),xk(2),x˙k(2),tk)∣xk(1)=xk(2),x˙k(1)=x˙k(2)},Σ1(+∞)=Σ1(+∞)(1)⊗Σ1(+∞)(2)={(xk(1),x˙k(1),xk(2),x˙k(2),tk)∣xk(1)=h,x˙k(1)≠0},    where the switching set Σ10 is defined on the boundary ∂Ω10(4.6)Σ10(i)={(xk(i),x˙k(i),tk)∣xk(i)=xk(i¯),x˙k(i)=x˙k(i¯)}⊂∂Ω10(i),Σ1(+∞)(i)={(xk(i),x˙k(i),tk)∣xk(1)=h,x˙k(1)≠0}⊂∂Ω1(+∞)(i).   Thus, the generic mappings are
defined as (4.7)P0:Σ10→Σ10,P1:Σ10→Σ1(+∞),P2:Σ1(+∞)→Σ10,P3:Σ1(−∞)→Σ10.   Similarly, a resultant mapping should have two components, so one
obtains (4.8)P0=(P0(1),P0(2)),P1=(P1(1),P1(2)),P2=(P2(1),P2(2)),P3=(P3(1),P3(2)),P0(i):Σ10(i)→Σ10(i),P1(i):Σ1(−∞)(i)→Σ1(+∞)(i),P2(i):Σ1(+∞)(i)→Σ1(−∞)(i),P3(i):Σ1(−∞)(i)→Σ1(−∞)(i),   where P1 and P2 are the global mappings  and P0 and P3 are the local mappings. The mappings in  (4.8),  are
sketched in Figure 7.Switching sets and generic mappings for stick motion: (a) particle and (b) piston.
(a)(b)From the above definitions, a mapping will
map a switching set into another switching set (or itself) through the corresponding
dynamical system in a specified domain. In such a domain, the dynamical system
given in (2.1) or (2.2) is continuous. The
solutions for such a continuous system are listed in the appendix. From the
displacement and velocity solutions of the particle and piston in the appendix,
with switching sets, four algebraic governing equations for a generic mapping Pj (j=1,2,3) can be expressed in a form of (4.9)Pj:{f1(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,f2(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,f3(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,f4(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,   with (4.10)xk(1)=h,xk+1(1)=xk+1(2)forP1,xk(1)=xk(2),xk+1(1)=hforP2,xk(1)=xk(1),xk+1(1)=xk+1(2)forP3.  The governing equations for the stick mapping P0 can be expressed by (4.11)P0:{f1(0)(xk(i),x˙k(i),tk,xk+1(i),x˙k+1(i),tk+1)=0,f2(0)(xk(i),x˙k(i),tk,xk+1(i),x˙k+1(i),tk+1)=0,f4(0)(xk+1(i),x˙k+1(i),tk+1)=g1(i)(0,xk+1(i),tk+1)=0,   with (4.12)xk(i)=xk(i¯),x˙k(i)=x˙k(i¯),xk+1(i)=xk+1(i¯),x˙k+1(i)=x˙k+1(i¯),x¨k(2)>−g,x¨k+1(2)<−g,for  mappingP0.


### 4.2. Motions with Mapping Structures

The notation for mapping action is introduced as(4.13)Pjkjk−1⋯j1=Pjk∘Pjk−1∘⋯∘Pj1,   where jm∈{0,1,2,3} with m=1,2,…,k. For a motion with n-time repeats of
mapping structure of Pj1j2⋯jk,
the mapping structure is expressed as (4.14)Pjkjk−1⋯j1(m)=(Pjk∘Pjk−1∘⋯∘Pj1)∘⋯∘(Pjk∘Pjk−1∘⋯∘Pj1)︸n=P(jkjk−1⋯j1)n.   Any periodic motion can be described using the above notations. Consider a mapping structure for the generalized Fermi-acceleration oscillator
as (4.15)P=P3n∘P0k3∘P3m∘P2k2∘P1k1=P3n0k33m2k21k1,   where kj∈{0,1} and m,n∈{0,ℕ}. From the mapping of  (4.15), a periodic motion can be expressed. By period-doubling,
the corresponding mapping structures for motion relative to the foregoing
mapping structure P3n0k33m2k21k1 of  (4.15) are (4.16)P=P3n0k33m2k21k1∘P3n0k33m2k21k1=P(3n0k33m2k21k1)2,⋮P=P(3n0k33m2k21k1)2l−1∘P(3n0k33m2k21k1)2l−1=P(3n0k33m2k21k1)2l.   For l→∞,
the chaotic motion relative to P3n0k33m2k21k1 can be obtained.Once the
grazing bifurcation of the periodic motion ofP3n0k33m2k21k1 occurs, the corresponding mapping structure will
change dramatically. For instance, the changed mapping structure can be expressed
by two parts  of mapping structures, that is, (4.17)P=P3n20k323m22k221k12∘P3n10k313m12k211k11=P(3n20k323m22k221k12)(3n10k313m12k211k11),   where kjs∈{0,1} and ms,ns∈ℕ (s=1,2). For a generalized case, the mapping
structure is  (4.18)P=P3nl0k3l3ml2k2l1k1l∘⋯∘P3n10k313m12k211k11︸l-terms=P(3nl0k3l3ml2k2l1k1l)⋯(3n10k313m12k211k11)︸l-terms,  where kjs∈{0,1} and ms,ns∈ℕ (s=1,2,…,l). Such mapping structure can be used to express
all possible motions in the generalized Fermi-acceleration oscillator. In
addition, the grazing bifurcation of chaotic motions will cause the strange
attractor fragmentation (see Luo [41]).Without stick,
there are two types of motions: (i) impact only at the boundary∂Ω1(−∞) and (ii) impacts at both of the boundaries ∂Ω1(−∞) and ∂Ω1(+∞). For the first case, the motion interacts with the boundary ∂Ω1(−∞) (k1s=0, k2s=0, and k3s=0, s∈ℕ) only. This motion can be described by the generic
mapping P3. The motion pattern relative to P3m,
caused by the period-doubling bifurcation, is given by (4.19)P=P3m,P=P3m∘P3m=P32m,⋮P=P32l−1m∘P32l−1m=P32lm.   The motion pattern relative to P3m can be generated by the grazing bifurcation,
and the corresponding mapping structures are (4.20)P=P3m,P=P3m1∘P3∘P3m2=P3m+1,⋮P=P3m∘P3n=P3m+n.
The second case is that the motion interacts with boundaries ∂Ω1(−∞) and ∂Ω1(+∞) (k3s=0, s∈ℕ, m,n∈{0,ℕ}). Consider a mapping structure of P3m2k21k1. The motion pattern generated by the period-doubling is expressed by (4.21)P=P3m2k21k1∘P3m2k21k1=P(3m2k21k1)2,…,P=P(3m2k21k1)2l−1∘P(3m2k21k1)2l−1=P(3m2k21k1)2l. Similarly, the motion pattern generated
by the grazing bifurcation can be discussed. To illustrate such mapping
structures, the relation between the mappings and motions in phase plane are
sketched in Figure 8. The mappings and motions for the piston can be
similarly presented. A motion pertaining to mapping P3m21 is sketched in Figure 8(a). In such a motion, no any stick motions exists. However, a motion with stick is associated with mapping P3n03m21 in Figure
8(b). The mappings are depicted through thick curves, and
the impacts are represented by the thin arrows. The stick motion is in the
shaded area with mapping P0. In a similar fashion, any motion with a
specific mapping structure can be illustrated.Mapping structure for motions: (a)P13m2 and  (b) P13n03m2.
(a)(b)


### 4.3. Periodic Motions and Local Stability

Consider a mapPj(j=0,1,2,3) mapping from the initial state (xk(1),x˙k(1),xk(2),x˙k(2),tk) to the final state (xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1):(4.22)Pj:(xk(1),x˙k(1),xk(2),x˙k(2),tk)→(xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1).   From (4.9),
the mapping in (4.22)  gives four algebraic equations.  With boundary constraints, the
five variables for the final state can be determined.  Without loss of
generality, substitution of the boundary constraint into the four algebraic governing
equations for Pj (j=0,1,2,3) yields (4.23)f(j)(Xk,Yk+1)=0,
where Xk=(Xk1,Xk2,Xk3,Xk4)T with Xkr∈{xk(1),x˙k(1),xk(2),x˙k(2),tk} and Yk=(Yk1,Yk2,Yk3,Yk4)T with Ykr∈{xk(1),x˙k(1),xk(2),x˙k(2),tk} and f(j)=(f1(j),f2(j),f3(j),f4(j))T. Taking the total derivative of (4.23) with
respect to Xk leads to
(4.24)[∂f(j)∂Xk]+[∂f(j)∂Yk+1][∂Yk+1∂Xk]=0or∂fI(j)∂XkJ+∑R=14∂fI(j)∂Y(k+1)R∂Y(k+1)R∂XkJ=0forI,J=1,2,3,4.   From the equation of (4.24), one obtains (4.25)DPj=[∂Yk+1∂Xk]=−[∂f(j)∂Yk+1]−1[∂f(j)∂Xk],   where (4.26)[∂f(j)∂Yk+1]=[∂fI(j)∂Y(k+1)J]4×4forI,J=1,2,3,4;[∂f(j)∂Xk]=[∂fI(j)∂XkJ]4×4forI,J=1,2,3,4;DPj=[∂Yk+1∂Xk]=[∂Y(k+1)I∂XkJ]4×4forI,J=1,2,3,4.   Consider a simple periodic motion of the mapping structure
of P21=P2∘P1 as an example, sketched in Figure 9. With the mapping structure P21=P2∘P1, one obtains
(4.27)P1:(tk,xk(2),x˙k(1),x˙k(2))→(tk+1,xk+1(2),x˙k+1(1),x˙k+1(2)),P2:(tk+1,xk+1(2),x˙k+1(1),x˙k+1(2))→(tk+2,xk+2(2),x˙k+2(1),x˙k+2(2)).   Let Xk1=xk(2)=xk(1),Xk2=x˙k(1),Xk3=x˙k(2),Xk4=tk because the impact occurs at xk(1)=xk(2). However, Yk1=xk(2),Yk2=x˙k(1),Yk3=x˙k(2),Yk4=tk because the impact occurs at xk(1)=h. The corresponding governing equations for such a periodic motion are (4.28)f(1)(Xk,Yk+1)=0forP1,f(2)(Yk+1,Xk+2)=0forP2.   For a period-N motion, the periodicity requires (4.29)Xk*=Xk+2*,
or (4.30)xk+2∗(i)=xk∗(i),x˙k+2∗(i)=x˙k∗(i),tk+2*=tk*+2NπΩ,N=1,2,…,fori=1,2.   Solving (4.28) and (4.29)
produces the switching sets of the periodic motion. The Jacobian matrix for
each mapping is developed from (4.28)
for j=1,2(4.31)DPj=[∂(tk+j,xk+j(2),x˙k+j(1),x˙k+j(2))∂(tk+j−1,xk+j−1(2),x˙k+j−1(1),x˙k+j−1(2))]4×4.   The total Jacobian matrix of the periodic motion can be
computed by (4.32)DP=DP2⋅DP1.
The Jacobian matrix of a periodic motion with a generalized
mapping structure can be developed in a similar fashion. For a stable periodic
motion, the magnitudes of all the eigenvalues should be inside a unit circle, that is, (4.33)|λi|<1,(i=1,2,3,4).   If (4.33) cannot
be satisfied, then the periodic motion is unstable. Consider a pair of complex eigenvalues
(λ1,2) and the others (λ3,4). If |λ1,2|=1 and |λ3,4|<1,
the Neirmark bifurcation of the periodic motion takes place. Consider two real
eigenvalues (λ1,2) and the others (λ3,4). If λ1=−1 and |λ2,3,4|<1,
the period doubling bifurcation of the periodic motion occurs. If λ1=1 and |λ2,3,4|<1,
the saddle-node bifurcation of the periodic motion occurs. However, the
eigenvalue analysis cannot be used to predict stick motion and grazing
bifurcation. Both of them should be determined through the normal vector
fields, and the stick motion is determined by (3.19)
and the grazing bifurcation is determined by (3.25).Figure 9
The
mapping structure of a simple periodic motion ofP21.


## 4.1. Switching Sets and Generic Mappings

Based on the switching boundaries in (3.2), the switching sets of the generalized Fermi-acceleration oscillator without stick are defined as(4.1)Σ1(−∞)=Σ1(−∞)(1)⊗Σ1(−∞)(2)={(xi(1),x˙i(1),xi(2),x˙i(2),ti)∣xi(1)=xi+(2),x˙i(1)≠x˙i(2)},Σ1(+∞)=Σ1(+∞)(1)⊗Σ1(+∞)(2)={(xi(1),x˙i(1),xi(2),x˙i(2),ti)∣xi(1)=h,x˙i(1)≠0}, where the symbol ⊗ represents the direct product of switching
subsets, and the switching subsets (Σ1(−∞)(i) and Σ1(+∞)(i)) on the boundaries (∂Ω1(−∞)(i) and ∂Ω1(+∞)(i)) are defined as  (4.2)Σ1(−∞)(i)={(xk(i),x˙k(i),tk)∣xk(i)=xk(i¯),x˙k(i)≠x˙k(i¯)}⊂∂Ω1(−∞)(i),Σ1(+∞)(i)={(xk(i),x˙k(i),tk)∣xk(1)=h,x˙k(1)≠0}⊂∂Ω1(+∞)(i).  Thus, the generic mappings for motions without stick are
defined as (4.3)P1:Σ1(−∞)→Σ1(+∞),P2:Σ1(+∞)→Σ1(−∞),P3:Σ1(−∞)→Σ1(−∞),  where the mappings P1 and P2 are the global mappings and the mapping P3 is a local mapping. No any mapping exists from Σ1(+∞) to Σ1(+∞) because x¨(1)=−g. The global mapping will map one switching set into another one. However, the
local mapping will map a switching set into itself. Since the resultant
switching set is the direct product of the two switching subsets, a resultant
mapping should have two components, that is, (4.4)P1=(P1(1),P1(2)),P2=(P2(1),P1(2)),P3=(P3(1),P1(2)),P1(i):Σ1(−∞)(i)→Σ1(+∞)(i),P2(i):Σ1(+∞)(i)→Σ1(−∞)(i),P3(i):Σ1(−∞)(i)→Σ1(−∞)(i).   From the above definitions,
the switching subsets and submappings are sketched in Figures 6(a) and 6(b)
for the particle and piston. The switching points in the switching sets are
presented by the circular symbols, and the mappings are depicted by the curves with
arrows.Switching sets and generic mappings for nonstick motion (in
absolute coordinates): (a) particle and
(b) piston.
(a)(b)Similarly, from the switching boundaries in (3.2)
and (3.4),
the switching sets of the generalized Fermi-acceleration oscillator with stick
are defined as (4.5)Σ10=Σ10(1)⊗Σ10(2)={(xk(1),x˙k(1),xk(2),x˙k(2),tk)∣xk(1)=xk(2),x˙k(1)=x˙k(2)},Σ1(+∞)=Σ1(+∞)(1)⊗Σ1(+∞)(2)={(xk(1),x˙k(1),xk(2),x˙k(2),tk)∣xk(1)=h,x˙k(1)≠0},    where the switching set Σ10 is defined on the boundary ∂Ω10(4.6)Σ10(i)={(xk(i),x˙k(i),tk)∣xk(i)=xk(i¯),x˙k(i)=x˙k(i¯)}⊂∂Ω10(i),Σ1(+∞)(i)={(xk(i),x˙k(i),tk)∣xk(1)=h,x˙k(1)≠0}⊂∂Ω1(+∞)(i).   Thus, the generic mappings are
defined as (4.7)P0:Σ10→Σ10,P1:Σ10→Σ1(+∞),P2:Σ1(+∞)→Σ10,P3:Σ1(−∞)→Σ10.   Similarly, a resultant mapping should have two components, so one
obtains (4.8)P0=(P0(1),P0(2)),P1=(P1(1),P1(2)),P2=(P2(1),P2(2)),P3=(P3(1),P3(2)),P0(i):Σ10(i)→Σ10(i),P1(i):Σ1(−∞)(i)→Σ1(+∞)(i),P2(i):Σ1(+∞)(i)→Σ1(−∞)(i),P3(i):Σ1(−∞)(i)→Σ1(−∞)(i),   where P1 and P2 are the global mappings  and P0 and P3 are the local mappings. The mappings in  (4.8),  are
sketched in Figure 7.Switching sets and generic mappings for stick motion: (a) particle and (b) piston.
(a)(b)From the above definitions, a mapping will
map a switching set into another switching set (or itself) through the corresponding
dynamical system in a specified domain. In such a domain, the dynamical system
given in (2.1) or (2.2) is continuous. The
solutions for such a continuous system are listed in the appendix. From the
displacement and velocity solutions of the particle and piston in the appendix,
with switching sets, four algebraic governing equations for a generic mapping Pj (j=1,2,3) can be expressed in a form of (4.9)Pj:{f1(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,f2(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,f3(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,f4(j)(xk(1),x˙k(1),xk(2),x˙k(2),tk,xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1)=0,   with (4.10)xk(1)=h,xk+1(1)=xk+1(2)forP1,xk(1)=xk(2),xk+1(1)=hforP2,xk(1)=xk(1),xk+1(1)=xk+1(2)forP3.  The governing equations for the stick mapping P0 can be expressed by (4.11)P0:{f1(0)(xk(i),x˙k(i),tk,xk+1(i),x˙k+1(i),tk+1)=0,f2(0)(xk(i),x˙k(i),tk,xk+1(i),x˙k+1(i),tk+1)=0,f4(0)(xk+1(i),x˙k+1(i),tk+1)=g1(i)(0,xk+1(i),tk+1)=0,   with (4.12)xk(i)=xk(i¯),x˙k(i)=x˙k(i¯),xk+1(i)=xk+1(i¯),x˙k+1(i)=x˙k+1(i¯),x¨k(2)>−g,x¨k+1(2)<−g,for  mappingP0.


## 4.2. Motions with Mapping Structures

The notation for mapping action is introduced as(4.13)Pjkjk−1⋯j1=Pjk∘Pjk−1∘⋯∘Pj1,   where jm∈{0,1,2,3} with m=1,2,…,k. For a motion with n-time repeats of
mapping structure of Pj1j2⋯jk,
the mapping structure is expressed as (4.14)Pjkjk−1⋯j1(m)=(Pjk∘Pjk−1∘⋯∘Pj1)∘⋯∘(Pjk∘Pjk−1∘⋯∘Pj1)︸n=P(jkjk−1⋯j1)n.   Any periodic motion can be described using the above notations. Consider a mapping structure for the generalized Fermi-acceleration oscillator
as (4.15)P=P3n∘P0k3∘P3m∘P2k2∘P1k1=P3n0k33m2k21k1,   where kj∈{0,1} and m,n∈{0,ℕ}. From the mapping of  (4.15), a periodic motion can be expressed. By period-doubling,
the corresponding mapping structures for motion relative to the foregoing
mapping structure P3n0k33m2k21k1 of  (4.15) are (4.16)P=P3n0k33m2k21k1∘P3n0k33m2k21k1=P(3n0k33m2k21k1)2,⋮P=P(3n0k33m2k21k1)2l−1∘P(3n0k33m2k21k1)2l−1=P(3n0k33m2k21k1)2l.   For l→∞,
the chaotic motion relative to P3n0k33m2k21k1 can be obtained.Once the
grazing bifurcation of the periodic motion ofP3n0k33m2k21k1 occurs, the corresponding mapping structure will
change dramatically. For instance, the changed mapping structure can be expressed
by two parts  of mapping structures, that is, (4.17)P=P3n20k323m22k221k12∘P3n10k313m12k211k11=P(3n20k323m22k221k12)(3n10k313m12k211k11),   where kjs∈{0,1} and ms,ns∈ℕ (s=1,2). For a generalized case, the mapping
structure is  (4.18)P=P3nl0k3l3ml2k2l1k1l∘⋯∘P3n10k313m12k211k11︸l-terms=P(3nl0k3l3ml2k2l1k1l)⋯(3n10k313m12k211k11)︸l-terms,  where kjs∈{0,1} and ms,ns∈ℕ (s=1,2,…,l). Such mapping structure can be used to express
all possible motions in the generalized Fermi-acceleration oscillator. In
addition, the grazing bifurcation of chaotic motions will cause the strange
attractor fragmentation (see Luo [41]).Without stick,
there are two types of motions: (i) impact only at the boundary∂Ω1(−∞) and (ii) impacts at both of the boundaries ∂Ω1(−∞) and ∂Ω1(+∞). For the first case, the motion interacts with the boundary ∂Ω1(−∞) (k1s=0, k2s=0, and k3s=0, s∈ℕ) only. This motion can be described by the generic
mapping P3. The motion pattern relative to P3m,
caused by the period-doubling bifurcation, is given by (4.19)P=P3m,P=P3m∘P3m=P32m,⋮P=P32l−1m∘P32l−1m=P32lm.   The motion pattern relative to P3m can be generated by the grazing bifurcation,
and the corresponding mapping structures are (4.20)P=P3m,P=P3m1∘P3∘P3m2=P3m+1,⋮P=P3m∘P3n=P3m+n.
The second case is that the motion interacts with boundaries ∂Ω1(−∞) and ∂Ω1(+∞) (k3s=0, s∈ℕ, m,n∈{0,ℕ}). Consider a mapping structure of P3m2k21k1. The motion pattern generated by the period-doubling is expressed by (4.21)P=P3m2k21k1∘P3m2k21k1=P(3m2k21k1)2,…,P=P(3m2k21k1)2l−1∘P(3m2k21k1)2l−1=P(3m2k21k1)2l. Similarly, the motion pattern generated
by the grazing bifurcation can be discussed. To illustrate such mapping
structures, the relation between the mappings and motions in phase plane are
sketched in Figure 8. The mappings and motions for the piston can be
similarly presented. A motion pertaining to mapping P3m21 is sketched in Figure 8(a). In such a motion, no any stick motions exists. However, a motion with stick is associated with mapping P3n03m21 in Figure
8(b). The mappings are depicted through thick curves, and
the impacts are represented by the thin arrows. The stick motion is in the
shaded area with mapping P0. In a similar fashion, any motion with a
specific mapping structure can be illustrated.Mapping structure for motions: (a)P13m2 and  (b) P13n03m2.
(a)(b)


## 4.3. Periodic Motions and Local Stability

Consider a mapPj(j=0,1,2,3) mapping from the initial state (xk(1),x˙k(1),xk(2),x˙k(2),tk) to the final state (xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1):(4.22)Pj:(xk(1),x˙k(1),xk(2),x˙k(2),tk)→(xk+1(1),x˙k+1(1),xk+1(2),x˙k+1(2),tk+1).   From (4.9),
the mapping in (4.22)  gives four algebraic equations.  With boundary constraints, the
five variables for the final state can be determined.  Without loss of
generality, substitution of the boundary constraint into the four algebraic governing
equations for Pj (j=0,1,2,3) yields (4.23)f(j)(Xk,Yk+1)=0,
where Xk=(Xk1,Xk2,Xk3,Xk4)T with Xkr∈{xk(1),x˙k(1),xk(2),x˙k(2),tk} and Yk=(Yk1,Yk2,Yk3,Yk4)T with Ykr∈{xk(1),x˙k(1),xk(2),x˙k(2),tk} and f(j)=(f1(j),f2(j),f3(j),f4(j))T. Taking the total derivative of (4.23) with
respect to Xk leads to
(4.24)[∂f(j)∂Xk]+[∂f(j)∂Yk+1][∂Yk+1∂Xk]=0or∂fI(j)∂XkJ+∑R=14∂fI(j)∂Y(k+1)R∂Y(k+1)R∂XkJ=0forI,J=1,2,3,4.   From the equation of (4.24), one obtains (4.25)DPj=[∂Yk+1∂Xk]=−[∂f(j)∂Yk+1]−1[∂f(j)∂Xk],   where (4.26)[∂f(j)∂Yk+1]=[∂fI(j)∂Y(k+1)J]4×4forI,J=1,2,3,4;[∂f(j)∂Xk]=[∂fI(j)∂XkJ]4×4forI,J=1,2,3,4;DPj=[∂Yk+1∂Xk]=[∂Y(k+1)I∂XkJ]4×4forI,J=1,2,3,4.   Consider a simple periodic motion of the mapping structure
of P21=P2∘P1 as an example, sketched in Figure 9. With the mapping structure P21=P2∘P1, one obtains
(4.27)P1:(tk,xk(2),x˙k(1),x˙k(2))→(tk+1,xk+1(2),x˙k+1(1),x˙k+1(2)),P2:(tk+1,xk+1(2),x˙k+1(1),x˙k+1(2))→(tk+2,xk+2(2),x˙k+2(1),x˙k+2(2)).   Let Xk1=xk(2)=xk(1),Xk2=x˙k(1),Xk3=x˙k(2),Xk4=tk because the impact occurs at xk(1)=xk(2). However, Yk1=xk(2),Yk2=x˙k(1),Yk3=x˙k(2),Yk4=tk because the impact occurs at xk(1)=h. The corresponding governing equations for such a periodic motion are (4.28)f(1)(Xk,Yk+1)=0forP1,f(2)(Yk+1,Xk+2)=0forP2.   For a period-N motion, the periodicity requires (4.29)Xk*=Xk+2*,
or (4.30)xk+2∗(i)=xk∗(i),x˙k+2∗(i)=x˙k∗(i),tk+2*=tk*+2NπΩ,N=1,2,…,fori=1,2.   Solving (4.28) and (4.29)
produces the switching sets of the periodic motion. The Jacobian matrix for
each mapping is developed from (4.28)
for j=1,2(4.31)DPj=[∂(tk+j,xk+j(2),x˙k+j(1),x˙k+j(2))∂(tk+j−1,xk+j−1(2),x˙k+j−1(1),x˙k+j−1(2))]4×4.   The total Jacobian matrix of the periodic motion can be
computed by (4.32)DP=DP2⋅DP1.
The Jacobian matrix of a periodic motion with a generalized
mapping structure can be developed in a similar fashion. For a stable periodic
motion, the magnitudes of all the eigenvalues should be inside a unit circle, that is, (4.33)|λi|<1,(i=1,2,3,4).   If (4.33) cannot
be satisfied, then the periodic motion is unstable. Consider a pair of complex eigenvalues
(λ1,2) and the others (λ3,4). If |λ1,2|=1 and |λ3,4|<1,
the Neirmark bifurcation of the periodic motion takes place. Consider two real
eigenvalues (λ1,2) and the others (λ3,4). If λ1=−1 and |λ2,3,4|<1,
the period doubling bifurcation of the periodic motion occurs. If λ1=1 and |λ2,3,4|<1,
the saddle-node bifurcation of the periodic motion occurs. However, the
eigenvalue analysis cannot be used to predict stick motion and grazing
bifurcation. Both of them should be determined through the normal vector
fields, and the stick motion is determined by (3.19)
and the grazing bifurcation is determined by (3.25).Figure 9
The
mapping structure of a simple periodic motion ofP21.


## 5. Predictions and Simulations

In this section,
bifurcation scenario for the generalized Fermi-acceleration oscillator will be
presented first, and analytical predictions of periodic motions will be
completed through the mapping structures. Periodic and chaotic motions will be
presented for illustration of the analytical conditions. Poincare mapping
sections will be given for illustrations of chaotic motions in the
Fermi-acceleration oscillator. The parameter maps for certain parameters will
also be presented for an overview of dynamical behaviors of particle and piston
in the generalized Fermi-acceleration oscillator.


### 5.1. Bifurcation Scenario

To
obtain a bifurcation scenario in the generalized Fermi-acceleration oscillator,
the closed-form solutions of particle and piston in the appendix are used for numerical
simulations. To achieve a motion with stick, the stick conditions in (3.21)
(or x¨k(2)>−g) with xk(1)=xk(2) and x˙k(1)=x˙k(2) should be embedded in the computer program. Before stick, the particle and piston are separated. One should set a tolerance
of the velocity difference (i.e., |x˙k(1)−x˙k(2)|<ε). In computation, the tolerance ε<10−4 is used for the stick motion. In other words,
when |x˙k(1)−x˙k(2)|<10−4,
it was assumed x˙k(1)≈x˙k(2). If x¨k(2)>−g,
the stick motion of particle and piston is formed, and the two separated
dynamical systems in (2.1)
will be switched into one dynamical system in (2.2)
for the stick motion. Further, the solution of the stick motion in the appendix
will be employed for numerical simulations. For the vanishing of the stick
motion, the condition in (3.24)
(i.e., x¨(2)(tm±)=x¨(1)(tm±)=−g and x⃛m±(2)<x⃛m±(1)=0) will also be embedded in the computer
program. Once the stick motion disappears, the system in (2.2)
will become two separated dynamical systems for particle and piston in (2.1). Similarly, the grazing conditions in (3.26)
and (3.28)
at the corresponding boundaries were embedded in the computer program. The
switching points for the particle and piston with switching times are recorded
for bifurcation scenarios.Consider a set of
parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5). Switching points on the boundaries versus excitation
frequency in a bifurcation scenario are shown in Figure 10. The acronyms PD,
SN, and GB denote the period-doubling, saddle node, and grazing bifurcations,
respectively. In Figure 10(a), the switching displacement of particle versus
excitation frequency is presented. The corresponding bifurcations are marked by
dashed vertical lines and labeled by the corresponding acronyms (i.e., PD, SN,
and GB). The switching displacements for particle impacts for x(1)=h lie on a straight line. However, the
switching displacement for the particle impacting with the piston will be
changed. The region for the stick motion is labeled. It is observed that the
stick motion occurs at the low excitation frequency. Many windows of periodic
motions are also observed. The first obvious window of periodic motions is
relative to mapping P21 for Ω∈(21.2,35.1) with one excitation period (1T). However, the second obvious window of
periodic motions is relative to mapping P21 for Ω∈(59.3,70.2) with two excitation periods (2T). In Figure 10(b), shown are the switching
velocities of particle before impact.  So the upper branch of the
switching velocity of particle is relative to impacts at the fixed wall, and
the lower branch of the switching velocity shows the impacts between the
particle and piston. In Figure 10(c), the switching displacements of piston
are plotted. The switching displacements of impacts and stick between the piston
and particle are the same, which are presented by the lower branch of
switching displacement. The upper branch of the switching displacement of
piston is to record the displacement of piston at impacts between the particle
and the fixed wall. Before and after such impacts, the corresponding
switching velocity of the piston will not be changed. Therefore, the switching
velocities of piston before impacts
are plotted in Figure 10(d). In
addition, switching times for switching points are very important. Thus,
switching phases pertaining to the switching times are plotted in Figure
10(e).  From the bifurcation scenario, the
motion of P21 lies in the range of Ω∈(21.2,35.1) and Ω∈(59.3,70.2). The motion of P(21)2 lies in the range of Ω∈(19.4,21.2) and Ω∈(56.8,59.3). The motion of P(21)4 is in the range of Ω∈(19.1,19.4) and Ω∈(56.4,56.8). In the region of Ω∈(0,19.1),
the zone of complex motions exists, and in the region of Ω∈(35.1,56.4) and Ω∈(70.2,80), the zone of chaos exists. In the zone of complex motion, the majority of motions are observable
periodic motions with complicated mapping structure. In the zone of chaos,
the majority of motions is chaotic. Inside the range of Ω∈(0,6.3) there are some simple stick motions. The detailed
mapping structures for periodic motions are tabulated in Table 1.Table 1
Summary of driving frequency for periodic motions (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
MappingExcitationMappingExcitationStructureFrequencyStructure1P(321)2P(2T)(6.33, 6.56)16P(32121)2P(4T)(11.01, 11.05) (11.35, 11.40)2P321P(T)(6.56, 7.01)17P32121P(2T)(11.05, 11.16) (11.40, 11.52)3P(333321)2P(2T)(7.41, 7.43)18P(321)4P(8T)(11.75, 11.76)4P3(321)2P(3T)(7.60, 7.80) (7.91, 7.93)19P(321)2P(4T)(11.76, 11.77)5P(33(321)2)2P(6T)(7.87, 7.89)20P32P(4T)(11.77, 11.79)6P33(321)2P(3T)(8.05, 8.07) (8.18, 8.20)21P34P(8T)(11.79, 11.80)7P3(3(321)2)2P(6T)(8.27, 8.29)22P(21)4P(4T)(19.15, 19.40)8P(33321)2P(6T)(8.62, 8.64)23P(21)2P(2T)(19.40, 21.25)9P(3(3321)2)2P(8T)(8.76, 8.78)24P21P(T)(21.25, 35.20)10P3(3321)2P(4T)(8.94, 8.96)25P(21)4P(8T)(56.42, 56.92)11P(3321)4P(8T)(9.06, 9.07)26P(21)2P(4T)(56.92, 59.35)12P(3321)2P(4T)(9.07, 9.09) (9.81, 9.83)27P21P(2T)(59.35, 70.23)14P(321)233(21)2P(6T)(10.42, 10.44)28Chaos & Complex
Motion—(7.01, 19.15) (19.15, 35.20)
(70.23, 80.00)15P(32121)4P(8T)(11.00, 11.01) (11.34, 11.35)Bifurcation scenario varying with driving
frequency:  (a) switching displacement
and (b) switching velocity of the particle; (c) switching displacement and (d)
switching velocity of the piston; (e) Switching phase (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)(c)(d)(e)


### 5.2. Analytical Prediction

The analytical prediction of
periodic motions is based on the corresponding mapping structures presented in
Section4 to solve the switching points of the periodic motions. In other words, for a specified mapping structure, a set of nonlinear algebraic
equations can be developed, and such nonlinear algebraic equations are solved
by the Newton-Raphson method to obtain the switching points. The local stability
and bifurcation analysis of the periodic motion are also completed as presented
in Section 4.2. In this section, the analytical prediction of
periodic motions will be presented via two examples, and the corresponding
eigenvalue analysis will be given.Considerm1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5. The analytical prediction of the periodic
motions pertaining to mapping P(21)k (k=1,2,22⋯) is presented in Figure 11, and the
corresponding eigenvalues of the periodic motions are shown in Figure 12. The
solid curves represent stable solutions, which can be observed in the
bifurcation scenario. The dashed curves denote unstable solutions that may
exist in the region of chaotic motion in the bifurcation scenario. The
acronyms PD, SN, and GB represent the period-doubling, saddle-node, and grazing
bifurcations, respectively. From this
analytical prediction of periodic motions, the stable solution disappears
at the saddle-node bifurcation with Ω=35.20. The unstable solution of mapping structure P21 disappears at the grazing bifurcation with Ω=5.88. The unstable solutions of mapping structure P(21)2 and P(21)4 vanish at the grazing bifurcations with Ω=9.39. The saddle-node bifurcation of the
periodic motion of P21 occurs
at Ω=35.20. This is because one of the eigenvalues equals 1 and all the others are inside
a unit circle, as shown in Figure 12. At Ω=19.1,19.4,and21.2,
one of the eigenvalues is equal to −1 and all the others are inside a unit circle,
and the period-doubling bifurcations of the corresponding motion of P(21)4, P(21)2, and P21 occur. At Ω=5.88and9.39,
the grazing bifurcations of periodic motions of P21 and (P(21)4, P(21)2) occur,
and such periodic motions are unstable. Such a grazing bifurcation should be determined
from the normal vector fields and the corresponding grazing conditions are
given in (3.26)
and (3.28). The grazing bifurcations of unstable
solutions imply that the corresponding motion
relative to a specific mapping structure will disappear in the region of a chaotic
motion. The chaotic motion will be switched into another motion with a sudden
switching of motion. Both grazing
and stick bifurcations give a catastrophe mechanism of the motion switching in
the Fermi-acceleration oscillator.Analytical prediction of periodic motion forΩ∈(21,35):
(a) switching displacement and (b) switching velocity of the particle; (c) switching
displacement and (d) switching velocity of the piston; (e) switching phase. Solid
and dashed curves represent the stable and unstable solutions, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5.)
(a)(b)(c)(d)(e)Eigenvalues varying with excitation frequency forΩ∈(19.15,35.20):
(a) magnitude  and (b) real part. Solid
and dashed curves represent the stable and unstable solutions, respectively. (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)The methodology presented in Section4 can give the analytical prediction of any periodic
motions. For a further demonstration of this technique, consider the same set
of parameters except for the range of excitation frequency for the analytical
prediction of periodic motions relative to the mapping structures of P(132)k (k=1,2,22⋯).  With
varying excitation frequency, the switching displacements, switching velocities
and switching phases of periodic motions of particles, and piston are
analytically predicted and presented in Figure 13. The periodic motion of P132 is stable for Ω∈(3.24,4.38)∪(6.57,7.00) and unstable for Ω∈(4.38,6.57). The corresponding period-doubling bifurcations occur at Ω=4.38and6.57 and the saddle-node bifurcations occur at Ω=3.24and7.00. The periodic motion of P(132)2 is unstable for Ω∈(4.80,6.31) and stable for Ω∈(4.38,4.80)∪(6.31,6.57). The period-doubling bifurcations of the periodic motion of P(132)2 are at Ω=4.80and6.31,
and the corresponding saddle-node bifurcations are at Ω=4.38and6.57. In the vicinity of Ω=4.80and6.31,
there are many periodic motions relative to P(132)4, P(132)8, and chaos. Since the ranges of excitation frequency
for periodic motions are very small, herein such analytical results will not be
presented.  For this analytical
prediction of periodic motions relative to P132 and P(132)2,
the corresponding stability and bifurcations of the periodic motions are presented
through the magnitudes and real parts of eigenvalues, as shown in Figure 14. For such periodic motions, no grazing and stick bifurcation can be observed. The motion in the range of Ω∈(4.80,6.31) is complex periodic motions and chaotic
motions.Analytical prediction of periodic motion relative to mapping structureP123 for Ω∈(3,7.5):
(a) switching displacement and (b) switching velocity of the particle; (c) switching
displacement and (d) switching velocity of the piston; (e) switching phase. Solid
and dashed curves represent the stable and unstable solutions, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)(c)(d)(e)Eigenvalues varying with excitation frequency forΩ∈(19.15,35.20):
(a) magnitude  and (b) real part. Solid
and dashed curves represent the stable and unstable solutions, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)For a better
understanding of the switching dynamics of the Fermi-acceleration oscillator, a
parameter map for periodic motions and chaos should be developed from the
analytical prediction, and the corresponding local stability and bifurcation
conditions with the grazing and stick conditions will be adopted.  The parameter map about excitation amplitude
and frequency is presented in Figure15 for parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1).
The “forbidden area" in the parameter map exists because the displacement
of piston cannot pass over the location of the fixed wall. In Figure 15(a), the
parameter map of (Ω,Q) for Ω∈(0,200) and Q∈(0,35) is presented. Periodic motions of P(12)k (k=1,2,22,23,…), P(312), and P(312)2 with complex and chaotic motions are
presented. The small regions of periodic motions cannot be clearly
illustrated.  To illustrate such regions,
a zoomed view of the parameter map in such regions is given in Figure 15(b). The regions for periodic motion of P(21)4, P312, and P(312)2 are presented clearly.Parameter maps: (a) overall view, (b)
zoomed view, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1).
(a)(b)


### 5.3. Simulations

From the previous
section, periodic motions were analytically predicted. It is significant to
illustrate motions for a better understanding of the dynamics of the
Fermi-acceleration oscillator. To illustrate motions in the generalized Fermi-acceleration
oscillator, the closed-form solutions of particle and piston in the appendix are
used again for numerical simulations. As in the section of bifurcation
scenario, the corresponding conditions for motion switching should be embedded
in the computer program. The time histories of displacement and velocity of
both the particle and piston will be presented for periodic motions. In
addition, phase planes of both particle and piston will be illustrated. For
chaotic motions, the Poincare mapping sections consisting of the switching
points on the switching boundaries will be presented. In numerical
illustrations, the switching points are labeled by circular symbols. The large
and small circular symbols are for the particle and piston in the
Fermi-acceleration oscillator, respectively.Consider
a set of parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5) with Ω=6.7. From the analytical prediction, one obtains the initial condition (i.e., t0=0.0741633079,x0(1)=1.0,x˙0(1)=5.4104888431,x0(2)=0.3280350633,x˙0(2)=1.1504459376) for a periodic motion of P213.  Using such an initial condition, a numerical
simulation gives phase plane and the time histories of displacements and
velocities of particle and piston, as shown in Figure 16.  The initial conditions are marked by green
circular symbols. The arrows give the directions of motions for the particle
and piston. The switching points are marked by the hollow circular symbols. The
corresponding mappings are labeled by Pλ(1),Pλ(2)(λ=1,2,3). In Figure 16(a), the switching
displacements of particle and piston at the impacting points are of the same. However, the switching velocities for both of them are different because impacts
cause the velocity to be discontinuous, as shown in Figure 16(b). In Figures
16(a) and 16(b), the solid and dashed curves depict the motions of particle and
piston, respectively. When the particle impacts with the fixed wall, the piston
motion will not be influenced because the piston moves by itself. Thus, the piston velocity will not be changed. When the impact between the particle and
piston occurs, the velocity change of the piston is very small because the particle’s
mass is much smaller than the piston’s mass. If the particle’s mass is with the
same quantitative level of the piston’s mass, the velocity change before and after impacts can be clearly observed. In Figure 16(c), a trajectory
of particle for such a periodic motion of P213 in phase plane is plotted. The dashed curves
are the switching boundary for the particle motion, which is given by the displacement
of piston. The periodic motion of particle consists of three pieces of
trajectories. Similarly, a trajectory of piston with the corresponding switching
boundary for the periodic motion is plotted in Figure 16(d).Periodic motion ofP(213) (Ω=6.7 and e2=0.7):  (a)
displacement history, (b) velocity history, (c) phase plane of particle, and (d)
phase plane of piston. Solid and dashed curves indicate the motion of particle
and piston, respectively, (m1=0.01, m2=1, k=10, c=6, e1=0.9, h=1, Q=0.5,
t0=0.0741633079, x0(1)=1.0, x˙0(1)=5.4104888431, x0(2)=0.3280350633,  x˙0(2)=1.1504459376).
(a)(b)(c)(d)As discussed before, there are two
kinds of motions for motion switching: (i) stick motion and nonstick motion. In other words, the nonstick motion means that the particle does not stay on
the piston, and the stick motion means that the particle stays on the piston
and moves together. In Figure16, the impacting motion of the particle and
piston without stick are presented. The stick motion of particle and piston is
an important phenomenon. Thus, consider parameters (m1=0.001,m2=1,k=20,c=8,e1=0.9,e2=0.01,h=1,Q=0.5) with Ω=6.5 to demonstrate a periodic motion with stick. From the analytical prediction, one obtains an initial condition (t0=0.70442282,x0(1)=x0(2)=−0.38755043,x˙0(1)=x˙0(2)=1.97682140) for a periodic motion with stick, which is
relative to a mapping  structure of P032. When such  an initial condition is used for a numerical simulation,  the
corresponding results are presented in Figure 17. Before the stick motion is
formed, there are two times of the impact chattering between the particle and
piston. In numerical computations, the relative velocity tolerance of the
particle and piston at the impact boundary is set to be |x˙0(1)−x˙0(2)|<ε=10−4. Under such a condition, it is assumed that the velocities of particle and
piston are of the same. If the chosen impact restriction coefficient is
relatively large, this set of parameters is very difficult to observe the
periodic motion with stick. The detailed discussion of such impact chattering
can be referred to Luo and O'Connor [46, 47]. To observe the stick motion of particle and piston
in periodic motions, the acceleration and jerk of particle and piston are very
important from the analytical conditions for motion switching in Section 3.3. So the time histories of the displacements,
velocities, accelerations and jerks of particle and piston are presented in Figure
17, and the trajectories of particle and piston in phase plane are plotted in
Figure 18. For the nonstick part of motion, the discussion is the same
as in Figure 16.  For the part of stick
motion, the displacement and velocity for particle and piston should be x(1)=x(2) and x˙(1)=x˙(2),
as shown in Figures 17(a) and 17(b). Such a stick motion is labeled by (P0(1),P0(2)). The solid and dashed curves depict the motions of particle and piston, respectively. In addition, the condition of stick motion in (3.19)
should be satisfied, which is expressed by the relative acceleration (i.e., g(i)=x¨(i)−x¨(i¯), i∈{1,2}) with a(i)≡x¨(i),a(i¯)≡x¨(i¯).  The condition of stick motion (i.e., g(1)<0 or g(2)>0) gives the absolute acceleration relation
(i.e., −g=x¨(1)<x¨(2)) under which the stick motion will be formed. Such a condition of stick motion is observed in the shaded area in Figure 17(c).  The vanishing condition of stick motion is
given by g(i)=x¨(i)−x¨(i¯)=0 with the relative jerks ((d/st)g(i)=J(i)−J(i¯)) with J(i)≡x⃛(i),J(i¯)≡x⃛(i¯). In other words, (d/dt)g(1)=J(1)−J(2)>0 or (d/dt)g(2)=J(2)−J(1)<0. From such conditions, one obtains J(2)<J(1)=0 from which the stick motion will
disappear.  Such a condition can be observed
in Figures 17(c) and 17(d). For the nonstick motion, the acceleration and jerk
for particle are constants (i.e., a(1)=−g and J(1)=0). The dashed curves in Figures 17(c) and 17(d)
give the acceleration and jerks for the piston.
For a further observation of the analytical condition of stick, the
phase plane, acceleration, and jerk versus displacement for particle and piston
are presented in Figure 18.  In Figures
18(a) and 18(d), the trajectories of periodic motion with stick for particle and
piston are presented, and the dashed curves are the displacement boundaries for
motion switching. The arrows still show the directions of flows for such a
periodic motion. The stick motion portions are shaded by the gray color. It is
observed that the displacement and velocity for the stick motion are
overlapped. To look into the condition for stick motion, in Figures 18(b), 18(c), 18(e), and 18(f), the vertical arrows show the motion switching at the switching
boundaries. At the initial condition (or stick starting point), the
acceleration of particle is less than the piston’s acceleration (i.e., a(2)>a(1)=−9.8) with the same displacement and velocity. So
the stick motion will be formed, because the jerk of particle is zero, and the
jerk of piston is negative. With increasing time, the acceleration of piston
becomes small. Until a(2)=a(1)=−9.8,
the stick motion should still exist. However, for a(2)=a(1)=−9.8,  if  the
jerk of the piston is still negative, then the acceleration of piston will be
less than the particle’s acceleration. Thus, the stick motion will disappear
and the nonstick motion is formed. Such a switching mechanism of the stick motion
is clearly observed in Figures 18(b), 18(c), 18(e), and  18(f).  Finally, the trajectories in phase plane for
a set of periodic motions are presented in Figures 19(a)–19(f) with parameters (m1=0.001,m2=1,k=20,c=8,e1=0.9,h=1,Q=0.5). The mapping structures, other parameters,
and initial conditions are listed in Table 2 for periodic motions. The green
points are the starting points (initial points), and the solid and dashed
curves represent the motions of particle and piston, respectively. The other
periodic motions can be similarly illustrated.Table 2
Initial conditions for periodic motions  (m1=0.01,m2=1,k=10,c=6,e1=0.9,h=1,Q=0.5).
Mapping(Ω,e2)Initial timet0(x0(1),x˙0(1))(x0(2),x˙0(2))P320(6.5,0.01)0.70490614(−0.38755043,1.97682140)(−0.38755043,1.97682140)P32(11.8,0.01)0.51035890(0.01422235,0.87719897)(0.01422235,2.83304464)P(213)(6.7,0.7)0.07416331(1.0,5.41048884)(0.32803506,1.15044594)P(213)2(6.5,0.7)0.08347153(1.0,5.54878710)(0.34730036,0.98801503)P12(22.7,0.7)0.08699328(1.0,8.11298196)(0.04059727,−1.64929089)P(12)2(20.4,0.7)0.39975776(1.0,8.17571963)(0.07102098,−1.44790328)Periodic
motion ofP034 (Ω=6.5 and e2=0.01):  (a)
displacement time-history, (b) velocity time-history, (c) phase plane of
particle, and (d) phase plane of piston. Solid and dashed curves represents the
motions of particle and piston, respectively, (m1=0.01, m2=1, k=10, c=6, e1=0.9, h=1, Q=0.5) (t0=0.70442282, x0(1)=−0.38755043, x˙0(1)=1.90527812, x0(2)=−0.38755043, x˙0(2)=1.96895970).
(a)(b)(c)(d)Periodic
motion ofP33 (Ω=6.5 and e2=0.01). Particle: (a) phase plane, (b) acceleration
and (c) jerks versus displacement of particle. Piston: (d) phase plane, (e)
acceleration, and (f) jerks versus displacement of piston, (m1=0.01, m2=1,  k=10,  c=6, e1=0.9, h=1, Q=0.5 ) (t0=0.70442282, x0(1)=−0.38755043, x˙0(1)=1.90527812, x0(2)=−0.38755043,  x˙0(2)=1.96895970).
(a)(b)(c)(d)(e)(f)Phase
planes. Solid and dashed curves represent motions of particle and piston, respectively,
(m1=0.01,  m2=1,  k=10,  c=6, e1=0.9, Q=0.5):  (a) P032 (Ω=6.5, e2=0.01) (t0=0.70442282,  x0(1)=x0(2)=−0.38755043,  x˙0(1)=1.90527812,  x˙0(2)=1.96895970),  (b) P32 (Ω=11.8, e2=0.01) (t0=0.51035890,  x0(1)=x0(2)=0.01422235,  x˙0(1)=0.87719897,  x˙0(2)=2.83304464), (c) P(213) (Ω=6.7, e2=0.7) (t0=0.07416331,  x0(1)=1.0,  x˙0(1)=5.41048884,  x0(2)=0.32803506,  x˙0(2)=1.15044594),  (d) P(213)2 (Ω=6.5, e2=0.7) (t0=0.08347153,  x0(1)=1.0,  x˙0(1)=5.54878710,  x0(2)=0.34730036,  x˙0(2)=0.98801503),  (e) P12 (Ω=22.7, e2=0.7) (t0=0.08699328,  x0(1)=1.0,  x˙0(1)=8.11298196,  x0(2)=0.04059727,  x˙0(2)=−1.64929089),  and (f) P(12)2 (Ω=20.4, e2=0.7) (t0=0.39975776,  x0(1)=1.0,  x˙0(1)=8.17571963,  x0(2)=0.07102098,  x˙0(2)=−1.44790328).
(a)(b)(c)(d)(e)(f)Finally, a chaotic motion relative
toP⋅⋅⋅21212121333 is illustrated for parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5,Ω=12) and the initial conditions (t0=0.40024561,
x0(1)=x0(2)=−0.22472018,
x˙0(1)=2.58913681,x˙0(2)=1.19198278) are used. The time histories of displacement
and velocity of particles and piston are presented in Figures 20(a) and 20(b),
respectively. In addition, the trajectories of particle and piston in phase
plane are plotted in Figures 20(c) and 20(d), accordingly. The particle motion
is very chaotic and such a chaotic motion of particle takes a big region in phase
plane. However, the trajectory of piston is in a small region of ring in phase
plane. This is because the particle’s mass is much smaller than the piston’s
mass. The impact effect on the particle motion is much bigger than on the piston
motion. To further looking into the chaotic behavior of this motion, the
corresponding Poincare mapping sections for such chaotic motion are presented
in Figure 21. The Poincare mapping sections consist of the switching points
on the switching boundaries. In Figure 21(a) and 21(b), the switching
displacement (xk(1)) and velocity (x˙k(1)) versus switching phase (mod(Ωtk,2π)) for the particle are presented. The
switching displacement and velocity of piston versus switching phase (mod(Ωtk,2π)) are also illustrated in Figures 21(c) and
21(d), respectively. The Poincare mapping sections in phase space for the
particle and piston are shown in Figures 21(e) and 21(f), respectively. Again,
because the particle’s mass is chosen to be much smaller than the piston's mass,
the impact effect on the piston is not very obvious. Therefore, one often
assumed the piston have a fixed sinusoidal motion. Once the particle’s mass is
not much less than the piston’s mass, the impact effect on the piston motion
will become significant. The aforementioned assumption should not be adequate. For
a further investigation of chaos in such a Fermi-acceleration oscillator, the
Lyapunov exponent can be used, and the scaling properties of chaotic motions
can be studied. This paper mainly focuses on the motion switching of the
multimotion states. Therefore, the other investigations on nonlinear dynamics
of the generalized Fermi-acceleration oscillator can be performed in sequel.Chaotic motion (Ω=12): (a) displacement time-history, (b) velocity
time-history, (c) phase plane for particle, and (d) phase plane for piston. Solid
and dashed curves represent the motion of particle and piston, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5) and the initial condition (t0=0.40024561,  x0(1)=x0(2)=−0.22472018,  x˙0(1)=2.58913681,x˙0(2)=1.19198278).
(a)(b)(c)(d)Poincare mapping sections of chaotic motion (Ω=12). (a) Switching displacement of particle, (b)
switching velocity of particle, (c) switching displacement of piston, (d)
switching velocity of piston, (e) Poincare section of particle, and (f) Poincare
section of piston. (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5) and initial condition (t0=0.40024561,  x0(1)=x0(2)=−0.22472018,  x˙0(1)=2.58913681,x˙0(2)=1.19198278).
(a)(b)(c)(d)(e)(f)


## 5.1. Bifurcation Scenario

To
obtain a bifurcation scenario in the generalized Fermi-acceleration oscillator,
the closed-form solutions of particle and piston in the appendix are used for numerical
simulations. To achieve a motion with stick, the stick conditions in (3.21)
(or x¨k(2)>−g) with xk(1)=xk(2) and x˙k(1)=x˙k(2) should be embedded in the computer program. Before stick, the particle and piston are separated. One should set a tolerance
of the velocity difference (i.e., |x˙k(1)−x˙k(2)|<ε). In computation, the tolerance ε<10−4 is used for the stick motion. In other words,
when |x˙k(1)−x˙k(2)|<10−4,
it was assumed x˙k(1)≈x˙k(2). If x¨k(2)>−g,
the stick motion of particle and piston is formed, and the two separated
dynamical systems in (2.1)
will be switched into one dynamical system in (2.2)
for the stick motion. Further, the solution of the stick motion in the appendix
will be employed for numerical simulations. For the vanishing of the stick
motion, the condition in (3.24)
(i.e., x¨(2)(tm±)=x¨(1)(tm±)=−g and x⃛m±(2)<x⃛m±(1)=0) will also be embedded in the computer
program. Once the stick motion disappears, the system in (2.2)
will become two separated dynamical systems for particle and piston in (2.1). Similarly, the grazing conditions in (3.26)
and (3.28)
at the corresponding boundaries were embedded in the computer program. The
switching points for the particle and piston with switching times are recorded
for bifurcation scenarios.Consider a set of
parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5). Switching points on the boundaries versus excitation
frequency in a bifurcation scenario are shown in Figure 10. The acronyms PD,
SN, and GB denote the period-doubling, saddle node, and grazing bifurcations,
respectively. In Figure 10(a), the switching displacement of particle versus
excitation frequency is presented. The corresponding bifurcations are marked by
dashed vertical lines and labeled by the corresponding acronyms (i.e., PD, SN,
and GB). The switching displacements for particle impacts for x(1)=h lie on a straight line. However, the
switching displacement for the particle impacting with the piston will be
changed. The region for the stick motion is labeled. It is observed that the
stick motion occurs at the low excitation frequency. Many windows of periodic
motions are also observed. The first obvious window of periodic motions is
relative to mapping P21 for Ω∈(21.2,35.1) with one excitation period (1T). However, the second obvious window of
periodic motions is relative to mapping P21 for Ω∈(59.3,70.2) with two excitation periods (2T). In Figure 10(b), shown are the switching
velocities of particle before impact.  So the upper branch of the
switching velocity of particle is relative to impacts at the fixed wall, and
the lower branch of the switching velocity shows the impacts between the
particle and piston. In Figure 10(c), the switching displacements of piston
are plotted. The switching displacements of impacts and stick between the piston
and particle are the same, which are presented by the lower branch of
switching displacement. The upper branch of the switching displacement of
piston is to record the displacement of piston at impacts between the particle
and the fixed wall. Before and after such impacts, the corresponding
switching velocity of the piston will not be changed. Therefore, the switching
velocities of piston before impacts
are plotted in Figure 10(d). In
addition, switching times for switching points are very important. Thus,
switching phases pertaining to the switching times are plotted in Figure
10(e).  From the bifurcation scenario, the
motion of P21 lies in the range of Ω∈(21.2,35.1) and Ω∈(59.3,70.2). The motion of P(21)2 lies in the range of Ω∈(19.4,21.2) and Ω∈(56.8,59.3). The motion of P(21)4 is in the range of Ω∈(19.1,19.4) and Ω∈(56.4,56.8). In the region of Ω∈(0,19.1),
the zone of complex motions exists, and in the region of Ω∈(35.1,56.4) and Ω∈(70.2,80), the zone of chaos exists. In the zone of complex motion, the majority of motions are observable
periodic motions with complicated mapping structure. In the zone of chaos,
the majority of motions is chaotic. Inside the range of Ω∈(0,6.3) there are some simple stick motions. The detailed
mapping structures for periodic motions are tabulated in Table 1.Table 1
Summary of driving frequency for periodic motions (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
MappingExcitationMappingExcitationStructureFrequencyStructure1P(321)2P(2T)(6.33, 6.56)16P(32121)2P(4T)(11.01, 11.05) (11.35, 11.40)2P321P(T)(6.56, 7.01)17P32121P(2T)(11.05, 11.16) (11.40, 11.52)3P(333321)2P(2T)(7.41, 7.43)18P(321)4P(8T)(11.75, 11.76)4P3(321)2P(3T)(7.60, 7.80) (7.91, 7.93)19P(321)2P(4T)(11.76, 11.77)5P(33(321)2)2P(6T)(7.87, 7.89)20P32P(4T)(11.77, 11.79)6P33(321)2P(3T)(8.05, 8.07) (8.18, 8.20)21P34P(8T)(11.79, 11.80)7P3(3(321)2)2P(6T)(8.27, 8.29)22P(21)4P(4T)(19.15, 19.40)8P(33321)2P(6T)(8.62, 8.64)23P(21)2P(2T)(19.40, 21.25)9P(3(3321)2)2P(8T)(8.76, 8.78)24P21P(T)(21.25, 35.20)10P3(3321)2P(4T)(8.94, 8.96)25P(21)4P(8T)(56.42, 56.92)11P(3321)4P(8T)(9.06, 9.07)26P(21)2P(4T)(56.92, 59.35)12P(3321)2P(4T)(9.07, 9.09) (9.81, 9.83)27P21P(2T)(59.35, 70.23)14P(321)233(21)2P(6T)(10.42, 10.44)28Chaos & Complex
Motion—(7.01, 19.15) (19.15, 35.20)
(70.23, 80.00)15P(32121)4P(8T)(11.00, 11.01) (11.34, 11.35)Bifurcation scenario varying with driving
frequency:  (a) switching displacement
and (b) switching velocity of the particle; (c) switching displacement and (d)
switching velocity of the piston; (e) Switching phase (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)(c)(d)(e)


## 5.2. Analytical Prediction

The analytical prediction of
periodic motions is based on the corresponding mapping structures presented in
Section4 to solve the switching points of the periodic motions. In other words, for a specified mapping structure, a set of nonlinear algebraic
equations can be developed, and such nonlinear algebraic equations are solved
by the Newton-Raphson method to obtain the switching points. The local stability
and bifurcation analysis of the periodic motion are also completed as presented
in Section 4.2. In this section, the analytical prediction of
periodic motions will be presented via two examples, and the corresponding
eigenvalue analysis will be given.Considerm1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5. The analytical prediction of the periodic
motions pertaining to mapping P(21)k (k=1,2,22⋯) is presented in Figure 11, and the
corresponding eigenvalues of the periodic motions are shown in Figure 12. The
solid curves represent stable solutions, which can be observed in the
bifurcation scenario. The dashed curves denote unstable solutions that may
exist in the region of chaotic motion in the bifurcation scenario. The
acronyms PD, SN, and GB represent the period-doubling, saddle-node, and grazing
bifurcations, respectively. From this
analytical prediction of periodic motions, the stable solution disappears
at the saddle-node bifurcation with Ω=35.20. The unstable solution of mapping structure P21 disappears at the grazing bifurcation with Ω=5.88. The unstable solutions of mapping structure P(21)2 and P(21)4 vanish at the grazing bifurcations with Ω=9.39. The saddle-node bifurcation of the
periodic motion of P21 occurs
at Ω=35.20. This is because one of the eigenvalues equals 1 and all the others are inside
a unit circle, as shown in Figure 12. At Ω=19.1,19.4,and21.2,
one of the eigenvalues is equal to −1 and all the others are inside a unit circle,
and the period-doubling bifurcations of the corresponding motion of P(21)4, P(21)2, and P21 occur. At Ω=5.88and9.39,
the grazing bifurcations of periodic motions of P21 and (P(21)4, P(21)2) occur,
and such periodic motions are unstable. Such a grazing bifurcation should be determined
from the normal vector fields and the corresponding grazing conditions are
given in (3.26)
and (3.28). The grazing bifurcations of unstable
solutions imply that the corresponding motion
relative to a specific mapping structure will disappear in the region of a chaotic
motion. The chaotic motion will be switched into another motion with a sudden
switching of motion. Both grazing
and stick bifurcations give a catastrophe mechanism of the motion switching in
the Fermi-acceleration oscillator.Analytical prediction of periodic motion forΩ∈(21,35):
(a) switching displacement and (b) switching velocity of the particle; (c) switching
displacement and (d) switching velocity of the piston; (e) switching phase. Solid
and dashed curves represent the stable and unstable solutions, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5.)
(a)(b)(c)(d)(e)Eigenvalues varying with excitation frequency forΩ∈(19.15,35.20):
(a) magnitude  and (b) real part. Solid
and dashed curves represent the stable and unstable solutions, respectively. (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)The methodology presented in Section4 can give the analytical prediction of any periodic
motions. For a further demonstration of this technique, consider the same set
of parameters except for the range of excitation frequency for the analytical
prediction of periodic motions relative to the mapping structures of P(132)k (k=1,2,22⋯).  With
varying excitation frequency, the switching displacements, switching velocities
and switching phases of periodic motions of particles, and piston are
analytically predicted and presented in Figure 13. The periodic motion of P132 is stable for Ω∈(3.24,4.38)∪(6.57,7.00) and unstable for Ω∈(4.38,6.57). The corresponding period-doubling bifurcations occur at Ω=4.38and6.57 and the saddle-node bifurcations occur at Ω=3.24and7.00. The periodic motion of P(132)2 is unstable for Ω∈(4.80,6.31) and stable for Ω∈(4.38,4.80)∪(6.31,6.57). The period-doubling bifurcations of the periodic motion of P(132)2 are at Ω=4.80and6.31,
and the corresponding saddle-node bifurcations are at Ω=4.38and6.57. In the vicinity of Ω=4.80and6.31,
there are many periodic motions relative to P(132)4, P(132)8, and chaos. Since the ranges of excitation frequency
for periodic motions are very small, herein such analytical results will not be
presented.  For this analytical
prediction of periodic motions relative to P132 and P(132)2,
the corresponding stability and bifurcations of the periodic motions are presented
through the magnitudes and real parts of eigenvalues, as shown in Figure 14. For such periodic motions, no grazing and stick bifurcation can be observed. The motion in the range of Ω∈(4.80,6.31) is complex periodic motions and chaotic
motions.Analytical prediction of periodic motion relative to mapping structureP123 for Ω∈(3,7.5):
(a) switching displacement and (b) switching velocity of the particle; (c) switching
displacement and (d) switching velocity of the piston; (e) switching phase. Solid
and dashed curves represent the stable and unstable solutions, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)(c)(d)(e)Eigenvalues varying with excitation frequency forΩ∈(19.15,35.20):
(a) magnitude  and (b) real part. Solid
and dashed curves represent the stable and unstable solutions, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5).
(a)(b)For a better
understanding of the switching dynamics of the Fermi-acceleration oscillator, a
parameter map for periodic motions and chaos should be developed from the
analytical prediction, and the corresponding local stability and bifurcation
conditions with the grazing and stick conditions will be adopted.  The parameter map about excitation amplitude
and frequency is presented in Figure15 for parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1).
The “forbidden area" in the parameter map exists because the displacement
of piston cannot pass over the location of the fixed wall. In Figure 15(a), the
parameter map of (Ω,Q) for Ω∈(0,200) and Q∈(0,35) is presented. Periodic motions of P(12)k (k=1,2,22,23,…), P(312), and P(312)2 with complex and chaotic motions are
presented. The small regions of periodic motions cannot be clearly
illustrated.  To illustrate such regions,
a zoomed view of the parameter map in such regions is given in Figure 15(b). The regions for periodic motion of P(21)4, P312, and P(312)2 are presented clearly.Parameter maps: (a) overall view, (b)
zoomed view, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1).
(a)(b)


## 5.3. Simulations

From the previous
section, periodic motions were analytically predicted. It is significant to
illustrate motions for a better understanding of the dynamics of the
Fermi-acceleration oscillator. To illustrate motions in the generalized Fermi-acceleration
oscillator, the closed-form solutions of particle and piston in the appendix are
used again for numerical simulations. As in the section of bifurcation
scenario, the corresponding conditions for motion switching should be embedded
in the computer program. The time histories of displacement and velocity of
both the particle and piston will be presented for periodic motions. In
addition, phase planes of both particle and piston will be illustrated. For
chaotic motions, the Poincare mapping sections consisting of the switching
points on the switching boundaries will be presented. In numerical
illustrations, the switching points are labeled by circular symbols. The large
and small circular symbols are for the particle and piston in the
Fermi-acceleration oscillator, respectively.Consider
a set of parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5) with Ω=6.7. From the analytical prediction, one obtains the initial condition (i.e., t0=0.0741633079,x0(1)=1.0,x˙0(1)=5.4104888431,x0(2)=0.3280350633,x˙0(2)=1.1504459376) for a periodic motion of P213.  Using such an initial condition, a numerical
simulation gives phase plane and the time histories of displacements and
velocities of particle and piston, as shown in Figure 16.  The initial conditions are marked by green
circular symbols. The arrows give the directions of motions for the particle
and piston. The switching points are marked by the hollow circular symbols. The
corresponding mappings are labeled by Pλ(1),Pλ(2)(λ=1,2,3). In Figure 16(a), the switching
displacements of particle and piston at the impacting points are of the same. However, the switching velocities for both of them are different because impacts
cause the velocity to be discontinuous, as shown in Figure 16(b). In Figures
16(a) and 16(b), the solid and dashed curves depict the motions of particle and
piston, respectively. When the particle impacts with the fixed wall, the piston
motion will not be influenced because the piston moves by itself. Thus, the piston velocity will not be changed. When the impact between the particle and
piston occurs, the velocity change of the piston is very small because the particle’s
mass is much smaller than the piston’s mass. If the particle’s mass is with the
same quantitative level of the piston’s mass, the velocity change before and after impacts can be clearly observed. In Figure 16(c), a trajectory
of particle for such a periodic motion of P213 in phase plane is plotted. The dashed curves
are the switching boundary for the particle motion, which is given by the displacement
of piston. The periodic motion of particle consists of three pieces of
trajectories. Similarly, a trajectory of piston with the corresponding switching
boundary for the periodic motion is plotted in Figure 16(d).Periodic motion ofP(213) (Ω=6.7 and e2=0.7):  (a)
displacement history, (b) velocity history, (c) phase plane of particle, and (d)
phase plane of piston. Solid and dashed curves indicate the motion of particle
and piston, respectively, (m1=0.01, m2=1, k=10, c=6, e1=0.9, h=1, Q=0.5,
t0=0.0741633079, x0(1)=1.0, x˙0(1)=5.4104888431, x0(2)=0.3280350633,  x˙0(2)=1.1504459376).
(a)(b)(c)(d)As discussed before, there are two
kinds of motions for motion switching: (i) stick motion and nonstick motion. In other words, the nonstick motion means that the particle does not stay on
the piston, and the stick motion means that the particle stays on the piston
and moves together. In Figure16, the impacting motion of the particle and
piston without stick are presented. The stick motion of particle and piston is
an important phenomenon. Thus, consider parameters (m1=0.001,m2=1,k=20,c=8,e1=0.9,e2=0.01,h=1,Q=0.5) with Ω=6.5 to demonstrate a periodic motion with stick. From the analytical prediction, one obtains an initial condition (t0=0.70442282,x0(1)=x0(2)=−0.38755043,x˙0(1)=x˙0(2)=1.97682140) for a periodic motion with stick, which is
relative to a mapping  structure of P032. When such  an initial condition is used for a numerical simulation,  the
corresponding results are presented in Figure 17. Before the stick motion is
formed, there are two times of the impact chattering between the particle and
piston. In numerical computations, the relative velocity tolerance of the
particle and piston at the impact boundary is set to be |x˙0(1)−x˙0(2)|<ε=10−4. Under such a condition, it is assumed that the velocities of particle and
piston are of the same. If the chosen impact restriction coefficient is
relatively large, this set of parameters is very difficult to observe the
periodic motion with stick. The detailed discussion of such impact chattering
can be referred to Luo and O'Connor [46, 47]. To observe the stick motion of particle and piston
in periodic motions, the acceleration and jerk of particle and piston are very
important from the analytical conditions for motion switching in Section 3.3. So the time histories of the displacements,
velocities, accelerations and jerks of particle and piston are presented in Figure
17, and the trajectories of particle and piston in phase plane are plotted in
Figure 18. For the nonstick part of motion, the discussion is the same
as in Figure 16.  For the part of stick
motion, the displacement and velocity for particle and piston should be x(1)=x(2) and x˙(1)=x˙(2),
as shown in Figures 17(a) and 17(b). Such a stick motion is labeled by (P0(1),P0(2)). The solid and dashed curves depict the motions of particle and piston, respectively. In addition, the condition of stick motion in (3.19)
should be satisfied, which is expressed by the relative acceleration (i.e., g(i)=x¨(i)−x¨(i¯), i∈{1,2}) with a(i)≡x¨(i),a(i¯)≡x¨(i¯).  The condition of stick motion (i.e., g(1)<0 or g(2)>0) gives the absolute acceleration relation
(i.e., −g=x¨(1)<x¨(2)) under which the stick motion will be formed. Such a condition of stick motion is observed in the shaded area in Figure 17(c).  The vanishing condition of stick motion is
given by g(i)=x¨(i)−x¨(i¯)=0 with the relative jerks ((d/st)g(i)=J(i)−J(i¯)) with J(i)≡x⃛(i),J(i¯)≡x⃛(i¯). In other words, (d/dt)g(1)=J(1)−J(2)>0 or (d/dt)g(2)=J(2)−J(1)<0. From such conditions, one obtains J(2)<J(1)=0 from which the stick motion will
disappear.  Such a condition can be observed
in Figures 17(c) and 17(d). For the nonstick motion, the acceleration and jerk
for particle are constants (i.e., a(1)=−g and J(1)=0). The dashed curves in Figures 17(c) and 17(d)
give the acceleration and jerks for the piston.
For a further observation of the analytical condition of stick, the
phase plane, acceleration, and jerk versus displacement for particle and piston
are presented in Figure 18.  In Figures
18(a) and 18(d), the trajectories of periodic motion with stick for particle and
piston are presented, and the dashed curves are the displacement boundaries for
motion switching. The arrows still show the directions of flows for such a
periodic motion. The stick motion portions are shaded by the gray color. It is
observed that the displacement and velocity for the stick motion are
overlapped. To look into the condition for stick motion, in Figures 18(b), 18(c), 18(e), and 18(f), the vertical arrows show the motion switching at the switching
boundaries. At the initial condition (or stick starting point), the
acceleration of particle is less than the piston’s acceleration (i.e., a(2)>a(1)=−9.8) with the same displacement and velocity. So
the stick motion will be formed, because the jerk of particle is zero, and the
jerk of piston is negative. With increasing time, the acceleration of piston
becomes small. Until a(2)=a(1)=−9.8,
the stick motion should still exist. However, for a(2)=a(1)=−9.8,  if  the
jerk of the piston is still negative, then the acceleration of piston will be
less than the particle’s acceleration. Thus, the stick motion will disappear
and the nonstick motion is formed. Such a switching mechanism of the stick motion
is clearly observed in Figures 18(b), 18(c), 18(e), and  18(f).  Finally, the trajectories in phase plane for
a set of periodic motions are presented in Figures 19(a)–19(f) with parameters (m1=0.001,m2=1,k=20,c=8,e1=0.9,h=1,Q=0.5). The mapping structures, other parameters,
and initial conditions are listed in Table 2 for periodic motions. The green
points are the starting points (initial points), and the solid and dashed
curves represent the motions of particle and piston, respectively. The other
periodic motions can be similarly illustrated.Table 2
Initial conditions for periodic motions  (m1=0.01,m2=1,k=10,c=6,e1=0.9,h=1,Q=0.5).
Mapping(Ω,e2)Initial timet0(x0(1),x˙0(1))(x0(2),x˙0(2))P320(6.5,0.01)0.70490614(−0.38755043,1.97682140)(−0.38755043,1.97682140)P32(11.8,0.01)0.51035890(0.01422235,0.87719897)(0.01422235,2.83304464)P(213)(6.7,0.7)0.07416331(1.0,5.41048884)(0.32803506,1.15044594)P(213)2(6.5,0.7)0.08347153(1.0,5.54878710)(0.34730036,0.98801503)P12(22.7,0.7)0.08699328(1.0,8.11298196)(0.04059727,−1.64929089)P(12)2(20.4,0.7)0.39975776(1.0,8.17571963)(0.07102098,−1.44790328)Periodic
motion ofP034 (Ω=6.5 and e2=0.01):  (a)
displacement time-history, (b) velocity time-history, (c) phase plane of
particle, and (d) phase plane of piston. Solid and dashed curves represents the
motions of particle and piston, respectively, (m1=0.01, m2=1, k=10, c=6, e1=0.9, h=1, Q=0.5) (t0=0.70442282, x0(1)=−0.38755043, x˙0(1)=1.90527812, x0(2)=−0.38755043, x˙0(2)=1.96895970).
(a)(b)(c)(d)Periodic
motion ofP33 (Ω=6.5 and e2=0.01). Particle: (a) phase plane, (b) acceleration
and (c) jerks versus displacement of particle. Piston: (d) phase plane, (e)
acceleration, and (f) jerks versus displacement of piston, (m1=0.01, m2=1,  k=10,  c=6, e1=0.9, h=1, Q=0.5 ) (t0=0.70442282, x0(1)=−0.38755043, x˙0(1)=1.90527812, x0(2)=−0.38755043,  x˙0(2)=1.96895970).
(a)(b)(c)(d)(e)(f)Phase
planes. Solid and dashed curves represent motions of particle and piston, respectively,
(m1=0.01,  m2=1,  k=10,  c=6, e1=0.9, Q=0.5):  (a) P032 (Ω=6.5, e2=0.01) (t0=0.70442282,  x0(1)=x0(2)=−0.38755043,  x˙0(1)=1.90527812,  x˙0(2)=1.96895970),  (b) P32 (Ω=11.8, e2=0.01) (t0=0.51035890,  x0(1)=x0(2)=0.01422235,  x˙0(1)=0.87719897,  x˙0(2)=2.83304464), (c) P(213) (Ω=6.7, e2=0.7) (t0=0.07416331,  x0(1)=1.0,  x˙0(1)=5.41048884,  x0(2)=0.32803506,  x˙0(2)=1.15044594),  (d) P(213)2 (Ω=6.5, e2=0.7) (t0=0.08347153,  x0(1)=1.0,  x˙0(1)=5.54878710,  x0(2)=0.34730036,  x˙0(2)=0.98801503),  (e) P12 (Ω=22.7, e2=0.7) (t0=0.08699328,  x0(1)=1.0,  x˙0(1)=8.11298196,  x0(2)=0.04059727,  x˙0(2)=−1.64929089),  and (f) P(12)2 (Ω=20.4, e2=0.7) (t0=0.39975776,  x0(1)=1.0,  x˙0(1)=8.17571963,  x0(2)=0.07102098,  x˙0(2)=−1.44790328).
(a)(b)(c)(d)(e)(f)Finally, a chaotic motion relative
toP⋅⋅⋅21212121333 is illustrated for parameters (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5,Ω=12) and the initial conditions (t0=0.40024561,
x0(1)=x0(2)=−0.22472018,
x˙0(1)=2.58913681,x˙0(2)=1.19198278) are used. The time histories of displacement
and velocity of particles and piston are presented in Figures 20(a) and 20(b),
respectively. In addition, the trajectories of particle and piston in phase
plane are plotted in Figures 20(c) and 20(d), accordingly. The particle motion
is very chaotic and such a chaotic motion of particle takes a big region in phase
plane. However, the trajectory of piston is in a small region of ring in phase
plane. This is because the particle’s mass is much smaller than the piston’s
mass. The impact effect on the particle motion is much bigger than on the piston
motion. To further looking into the chaotic behavior of this motion, the
corresponding Poincare mapping sections for such chaotic motion are presented
in Figure 21. The Poincare mapping sections consist of the switching points
on the switching boundaries. In Figure 21(a) and 21(b), the switching
displacement (xk(1)) and velocity (x˙k(1)) versus switching phase (mod(Ωtk,2π)) for the particle are presented. The
switching displacement and velocity of piston versus switching phase (mod(Ωtk,2π)) are also illustrated in Figures 21(c) and
21(d), respectively. The Poincare mapping sections in phase space for the
particle and piston are shown in Figures 21(e) and 21(f), respectively. Again,
because the particle’s mass is chosen to be much smaller than the piston's mass,
the impact effect on the piston is not very obvious. Therefore, one often
assumed the piston have a fixed sinusoidal motion. Once the particle’s mass is
not much less than the piston’s mass, the impact effect on the piston motion
will become significant. The aforementioned assumption should not be adequate. For
a further investigation of chaos in such a Fermi-acceleration oscillator, the
Lyapunov exponent can be used, and the scaling properties of chaotic motions
can be studied. This paper mainly focuses on the motion switching of the
multimotion states. Therefore, the other investigations on nonlinear dynamics
of the generalized Fermi-acceleration oscillator can be performed in sequel.Chaotic motion (Ω=12): (a) displacement time-history, (b) velocity
time-history, (c) phase plane for particle, and (d) phase plane for piston. Solid
and dashed curves represent the motion of particle and piston, respectively, (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5) and the initial condition (t0=0.40024561,  x0(1)=x0(2)=−0.22472018,  x˙0(1)=2.58913681,x˙0(2)=1.19198278).
(a)(b)(c)(d)Poincare mapping sections of chaotic motion (Ω=12). (a) Switching displacement of particle, (b)
switching velocity of particle, (c) switching displacement of piston, (d)
switching velocity of piston, (e) Poincare section of particle, and (f) Poincare
section of piston. (m1=0.01,m2=1,k=10,c=6,e1=0.9,e2=0.7,h=1,Q=0.5) and initial condition (t0=0.40024561,  x0(1)=x0(2)=−0.22472018,  x˙0(1)=2.58913681,x˙0(2)=1.19198278).
(a)(b)(c)(d)(e)(f)


## 6. Conclusion

In this paper, the mechanism of motion switching of a
particle in such a generalized Fermi-oscillator was studied through the theory
of discontinuous dynamical systems, and the corresponding analytical conditions
for the motion switching were developed. From the solutions of linear systems
in each domain, the generic mappings
are introduced. Further, the mapping structures for periodic motions were
developed, and such periodic motions in the Fermi-acceleration oscillator
were predicted analytically. The corresponding local stability and bifurcation
are carried out. From the analytical prediction, parameter maps of regular and
chaotic motions were achieved for a global view of motions in the Fermi-acceleration oscillator. Illustrations of periodic and chaotic motions in such
an oscillator were done. This
methodology will provide a useful way to determine dynamical behaviors in the
Fermi-acceleration oscillator.



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*Source: 298906-2009-04-27.xml*