# Scaling Properties of a Hybrid Fermi-Ulam-Bouncer Model

Authors: Diego F. M. Oliveira; Rafael A. Bizão; Edson D. Leonel
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Some dynamical properties for a one-dimensional hybrid Fermi-Ulam-bouncer model are studied
under the framework of scaling description. The model is described by using a two-dimensional nonlinear area preserving mapping. Our results show that the chaotic regime below the lowest energy invariant spanning curve is scaling invariant and the obtained critical exponents are used to find a universal plot for the second momenta of the average velocity.


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## Body

## 1. Introduction

The investigation  of nonlinear dynamical systems has
awaken special interest  along last decades since they can explain and/or
predict some  phenomena until then incomprehensible. A special class of systems
that present nonlinear phenomena and that can be described via recursive
equations is the so-called classical nonlinear billiard problems [1, 2]. Generally, a billiard
problem consists of a system in which a point-like particle moves freely inside
a bounded region and suffers specular reflections with the boundaries. This
system has been extensively studied during several years considering both its
classical and quantized formulation.It is well known in the literature that the phase
space of billiard problems highly depends on the shape of the boundary. The
dynamics of the particle might generate phase spaces of different kinds that
can be settled in three different classes of universality including (i)
integrable, (ii) ergodic, and (iii) mixed. A typical example of case (i) is the
circle billiard, as the integrability of such a
case resembles the angular momentum conservation. Two examples of case (ii) are
the Bunimovich stadium [3] and the Sinai billiard [4]. In case (iii), the most
important property present in the phase space is that chaotic seas, generally
surrounding Kolmogorov-Arnold-Moser (KAM) islands, are confined by invariant
spanning curves [2]
sometimes also called as primary KAM tori. Particularly, such curves cross the
phase plane from one side to the other one thus separating different portions
of the phase space. The mixed structure in the phase portrait is such that
regions of regular motion and regions with chaotic behavior could coexist
together. This property is generic for nondegenerate Hamiltonian systems
[5]. Some examples of
systems with this particular structure are the one-dimensional Fermi-Ulam model
[6–12], the bouncer model
[13–19], time-dependent potentials well [20, 21], and many other different
problems considering different degrees of freedom.In this paper, we will consider a 1D model that is described using the formalism
of discrete mappings, the so-called hybrid Fermi-Ulam-bouncer model [22, 23]. We are seeking to understand
and describe a scaling property present in the chaotic sea of such a model. The
model was originally proposed to merge together two different nonlinear
problems commonly studied apart from each other, that is, the Fermi-Ulam model
and the bouncer model. The Fermi-Ulam model (FUM) consists of a classical
particle, in the total absence of any external field, which is confined to
bounce between two rigid walls, where one of them is fixed and the other one is
periodically time varying. The returning mechanism of the particle for a next
collision with the moving wall is due to a reflection with the fixed wall. On
the other hand, the bouncer model consists of a classical particle falling in a
constant gravitational field and that hits a periodically oscillating wall. The
returning mechanism of the bouncer model is related to the gravitational field
only. Despite the very similar models, the different returning mechanisms of
the two models cause profound consequences on the dynamics of the particle
[24]. Particularly,
depending on the combination of the control parameters and initial conditions,
the average velocity of the particle reaches a constant value for sufficient
long time in the FUM, while it diverges on the bouncer model. Such a divergency
is basically related to the phenomenon of Fermi acceleration. Therefore, Fermi
acceleration (FA) is a phenomenon in which a classical particle acquires
unbounded energy from collisions with a massive moving wall [25, 26]. Applications of FA have acquired a broad interest in
different fields of physics including plasma physics [27], astrophysics [28, 29], atomic physics [30], optics [31, 32], and even in the well-known time-dependent billiard
problems [33, 34]. The interesting posed question
that should be answered is whether the FA results from the nonlinear dynamics
itself considering the complete absence of any imposed random motion. An
one-dimensional model exhibiting such a phenomenon, which is modelled by a
nonlinear mapping, is the bouncer model. For two-dimensional time-dependent
billiards (billiards with moving boundaries), the answer for this question is
not unique and it depends on the kind of phase space for the corresponding
static version of the problem. Therefore, as conjectured by
Loskutov et al. [35]
(such a conjecture is known in the literature as the LRA conjecture), the
regular dynamics for a fixed boundary implies a bound to the energy gained by the
bouncing particle, but the chaotic dynamics of a billiard with a fixed boundary
is a sufficient condition for FA in the system when a boundary perturbation is
introduced. Such a conjecture was confirmed in different billiards [36–38]. It was, therefore, observed recently
FA  in a time
varying elliptical billiard [39]. The oval billiard, however, seems not to exhibit FA
for the breathing case [40].Thus, the hybrid Fermi-Ulam-bouncer model consists of
a classical particle which is confined in and bouncing elastically between two
rigid walls in the presence of a constant gravitational field. Thus, properties
that are individually observed in the FUM and bouncer model come together and
coalesce in the hybrid Fermi-Ulam-bouncer model [22]. As we will show, the phase
space of this model is of mixed kind. Our main goal is to characterize some
properties on the chaotic regime for the region below the first invariant
spanning curve, particularly the behavior of the average velocity and the
deviation of the average velocity.This paper is organized as follows. In Section2, we
give all the details needed for the construction of the nonlinear mapping. We
illustrate some results for the phase space of the model considering both the
simplified and the complete versions of the Fermi-Ulam-bouncer model. The
scaling hypotheses are present and numerical results are discussed in Section 3. Finally, in Section 4, we draw our conclusions.


## 2. The Model and the Mapping

We discuss in this section all the details needed for
the mapping construction. We also present the phase space and obtain the
positive Lyapunov exponent for the low-energy chaotic sea. The one-dimensional
hybrid Fermi-Ulam-bouncer model thus consists of a classical particle confined
to bounce elastically between two rigid walls. One of the walls is assumed to
be fixed at the positiony=l, while the other one moves periodically in time
according to the equation yw(t)=εcos(wt).
Here, ε and w denote, respectively, the amplitude of
oscillation and the frequency of the moving wall. Additionally, the particle is
suffering the action of a constant gravitational field g′.
As it is so usual in the literature, the dynamics of the particle is described
in terms of a two-dimensional nonlinear area preserving map T that gives the velocity of the particle vn and the corresponding time tn at the instant of the nth
impact with the moving wall, that is, T(vn,tn)=(vn+1,tn+1).
Before we write the equations of the mapping, it is convenient to define
dimensionless and much more appropriated variables to describe the dynamics of
the particle. We define ϵ=ε/l, g=g′/(w2l), and Vn=vn/(wl).
Finally, we measure the time in terms of ϕn=wtn.
Incorporating this new set of variables in the dynamics, the map is written
as(2.1)T:{Vn+1=Vn*+gϕc−2ϵsin(ϕn+1),ϕn+1=(ϕn+ΔTn)mod(2π),where the expressions for both Vn* and ΔTn depend on the kind of collision occurs. There
are three different possible situations, namely (i) multiple collisions, (ii)
collisions without reflection in the upper wall, and (iii) collisions with
reflection in the upper wall. Considering the first case, where the particle
suffers an impact before leaving the collision zone, which is given by the
interval y∈[−ϵ,ϵ],
the expressions are Vn*=−Vn and ΔTn=ϕc,
where ϕc is obtained as the smallest solution of G(ϕc)=0 with ϕc∈(0,2π].
Such solution is equivalent to the position of the particle being the same as
the position of the moving wall. Thus, the transcendental equation G(ϕc) is given by(2.2)G(ϕc)=ϵcos(ϕn+ϕc)−ϵcos(ϕn)−Vnϕc+gϕc22.If G(ϕc) does not have a solution in the interval ϕc∈(0,2π],
the particle leaves the collision zone without suffering a successive hit.
Considering the case (ii), the needed condition for the particle not colliding
with the upper wall after leaving the collision zone is Vn≤2g[1−ϵcos(ϕn)].
In this case, Vn*=Vn2+2gϵ[cos(ϕn)−1] and ΔTn=ϕu+ϕd+ϕc,
where the auxiliary terms assume the following expressions:(2.3)ϕu=Vng,ϕd=Vn2+2gϵ[cos(ϕn)−1]g.The term ϕc is obtained by solution of F(ϕc)=0.
Again, considering the condition that matches the same position of the moving
wall and of the particle. Thus, F(ϕc) is given by(2.4)F(ϕc)=ϵcos(ϕn+ϕu+ϕd+ϕc)−ϵ−Vn*ϕc+gϕc22.Finally, for the case (iii), whereVn>2g[1−ϵcos(ϕn)],
the corresponding expressions for Vn* and ΔTn are the same as those of the case (ii).
However, the auxiliary terms, ϕu and ϕd, assume the following
expressions:(2.5)ϕu=Vn−Vn2−2g[1−ϵcos(ϕn)]g,ϕd=Vn2+2gϵ[ϵcos(ϕn)−1]−Vn2−2g[1−ϵcos(ϕn)]g.The value of ϕc is obtained from (2.4) using, however, the new
expressions for ϕu and ϕd.Figure1(a) shows the phase space for the complete
hybrid Fermi-Ulam-bouncer model generated from (2.1). The control parameters used
in Figure 1 are ϵ=10−3 and g=5.628×10−5.
It is easy to see the presence of KAM islands surrounded by a chaotic sea, that
is, limited by an invariant spanning curve. The presence of the invariant
spanning curves in the phase space implies that the particle cannot acquire
unlimited energy. To illustrate the behavior of the average energy, we
define(2.6)Ej=1n+1∑i=0nV2¯i,j,   and finally
obtain the average energy as  (2.7)E¯(n,ϵ)=1M∑j=1MEj,where M corresponds to an ensemble of different
initial conditions.(a) Phase space for the complete hybrid
Fermi-Ulam-bouncer model. (b) Behavior of the average energy,E¯,
as function of n.
The control parameters used in (a) and (b) were ϵ=10−3 and g=5.628×10−5.
(a)(b)Figure1(b) shows the behavior of the average energy, E¯,
as a function of n.
We can see that the energy grows for small n and then, after a changeover, it reaches a
regime of saturation for large n.
The initial velocity was considered fixed as V0=2.1ϵ, while we have considered an ensemble of M=5000 different initial phases ϕ0 uniformly distributed in the interval [0,2π).
Each initial condition was evolved up to 107 iterations. Since the instant of each impact
of the particle with the moving wall can only be obtained numerically, the
computational time for evaluation of (2.7) is very large. As an attempt to reduce
the time consuming in the numerical simulations, we will discuss in Section 2.1 a simplified version of the hybrid Fermi-Ulam-bouncer model.


### 2.1. A Simplified Hybrid Fermi-Ulam-Bouncer Model

In this section, we discuss a simplification used in
the model. For the complete model, the instant of the impact of the particle
with the moving wall is obtained via a solution of a transcendental equation,
which yields the simulations to be long-time consuming. However, instead of
considering solving transcendental equations, we will use a simplification in
the model which is commonly used in the literature [2]. We will suppose that both
walls are fixed (one of them is fixed at y=0 and the other one is fixed at y=l), but that, when the particle suffers a
collision with one of then (say, with the one at y=0), the particle exchanges momentum and energy
as if the wall was moving. This simplification carries a great advantage of
allowing us to speed up our numerical simulations when compared to the complete
version of the model. Such a simplification also retains the nonlinearity of
the problem. Considering the same dimensionless variables defined for the
complete model, the map for the simplified hybrid Fermi-Ulam-bouncer model can
be written as(2.8)T:{Vn+1=|Vn−2ϵsin(ϕn+1)|,ϕn+1=(ϕn+ΔTn)mod(2π).The modulus function used in the
equation of the velocity on the mapping (2.8) was introduced to preserve the
particle into the region between the walls. The expressions for ΔTn depend on the following conditions:(1)
collision without reflection in the upper
wall. Such a condition is verified ifVn≤2g(1−ϵcos(ϕn)).
In this case, the particle rises and decelerates due to the gravitational field
only until reaches instantaneously the rest. Then the particle is accelerated
downward until it collides with the wall at y=0.
Thus, ΔTn is given by(2.9)ΔTn=2Vng.(2)
Collision with reflection in the upper wall.
For such a collision, the condition that must be observed isVn>2g(1−ϵcos(ϕn)).
In this case, the particle goes in the upward direction and hits the upper
wall. It is then reflected downward and is also accelerated by the gravitational
field. The expression for ΔTn is written as(2.10)ΔTn=Vn−Vn2−2gg.
The mapping (2.8) preserves the phase space measure since that det(J)=±1.The phase space for the mapping (2.8) is shown in Figure 2(a) for the same control parameters used in Figure 1. We have also evaluated numerically the positive Lyapunov exponent for
the chaotic sea of Figure 2(a). It is well known that the Lyapunov exponent has
great applicability as a practical tool that can quantify the average expansion
or contraction rate for a small volume of initial conditions. As discussed in
[41], the Lyapunov
exponents are defined as(2.11)λj=limn→∞1nln|Λj|,j=1,2,where Λj are the eigenvalues of M=∏i=1nJi(Vi,ϕi) and Ji is the Jacobian matrix evaluated over the
orbit (Vi,ϕi).
However, a direct implementation of a computational algorithm to evaluate (2.11)
has a severe limitation to obtain M.
Even in the limit of short n,
the components of M can assume very different orders of magnitude
for chaotic orbits and periodic attractors, yielding impracticable the
implementation of the algorithm. In order to avoid such problem, we note that J can be written as J=ΘT, where Θ is an orthogonal matrix and T is a right triangular matrix. Thus, we rewrite M as M=JnJn−1,…,J2Θ1Θ1−1J1,
where T1=Θ1−1J1.
A product of J2Θ1 defines a new J2′.
In a next step, it is easy to show that M=JnJn−1,…,J3Θ2Θ2−1J2′T1.
The same procedure can be used to obtain T2=Θ2−1J2′, and so on. Using this procedure, the problem
is reduced to evaluate the diagonal elements of Ti: T11i,T22i.
Finally, the Lyapunov exponents are now given by(2.12)λj=limn→∞1n∑i=1nln|Tjji|,j=1,2.If at least one of the λj is positive, then the orbit is classified as
chaotic. Figure 2(b) shows the behavior of the positive Lyapunov exponent
plotted against the number of collisions n with the wall located at y=0 for 10 different initial conditions on the
chaotic sea. The control parameters used in the construction of Figure 2 were ϵ=10−3 and g=5.628×10−5.
The average of the positive Lyapunov exponent for the ensemble of the 10-time
series gives λ¯=1.55±0.01, where the value 0.01 corresponds to the standard deviation of the
ten samples. Finally, in Figure 2(c), we present the behavior of the average
energy of the particle for the simplified model.(a) Phase space generated from iteration of mapping (2.8) for the control
parameters ϵ=10−3 and g=5.628×10−5.
(b) Behavior of the positive Lyapunov exponent of the chaotic sea for the same
control parameters used in (a). The average value obtained was λ¯=1.55±0.01, where the error 0.01 corresponds to the standard deviation of the
10 samples. (c) Behavior of the average energy, E¯,
as function of n.
(a)(b)(c)


## 2.1. A Simplified Hybrid Fermi-Ulam-Bouncer Model

In this section, we discuss a simplification used in
the model. For the complete model, the instant of the impact of the particle
with the moving wall is obtained via a solution of a transcendental equation,
which yields the simulations to be long-time consuming. However, instead of
considering solving transcendental equations, we will use a simplification in
the model which is commonly used in the literature [2]. We will suppose that both
walls are fixed (one of them is fixed at y=0 and the other one is fixed at y=l), but that, when the particle suffers a
collision with one of then (say, with the one at y=0), the particle exchanges momentum and energy
as if the wall was moving. This simplification carries a great advantage of
allowing us to speed up our numerical simulations when compared to the complete
version of the model. Such a simplification also retains the nonlinearity of
the problem. Considering the same dimensionless variables defined for the
complete model, the map for the simplified hybrid Fermi-Ulam-bouncer model can
be written as(2.8)T:{Vn+1=|Vn−2ϵsin(ϕn+1)|,ϕn+1=(ϕn+ΔTn)mod(2π).The modulus function used in the
equation of the velocity on the mapping (2.8) was introduced to preserve the
particle into the region between the walls. The expressions for ΔTn depend on the following conditions:(1)
collision without reflection in the upper
wall. Such a condition is verified ifVn≤2g(1−ϵcos(ϕn)).
In this case, the particle rises and decelerates due to the gravitational field
only until reaches instantaneously the rest. Then the particle is accelerated
downward until it collides with the wall at y=0.
Thus, ΔTn is given by(2.9)ΔTn=2Vng.(2)
Collision with reflection in the upper wall.
For such a collision, the condition that must be observed isVn>2g(1−ϵcos(ϕn)).
In this case, the particle goes in the upward direction and hits the upper
wall. It is then reflected downward and is also accelerated by the gravitational
field. The expression for ΔTn is written as(2.10)ΔTn=Vn−Vn2−2gg.
The mapping (2.8) preserves the phase space measure since that det(J)=±1.The phase space for the mapping (2.8) is shown in Figure 2(a) for the same control parameters used in Figure 1. We have also evaluated numerically the positive Lyapunov exponent for
the chaotic sea of Figure 2(a). It is well known that the Lyapunov exponent has
great applicability as a practical tool that can quantify the average expansion
or contraction rate for a small volume of initial conditions. As discussed in
[41], the Lyapunov
exponents are defined as(2.11)λj=limn→∞1nln|Λj|,j=1,2,where Λj are the eigenvalues of M=∏i=1nJi(Vi,ϕi) and Ji is the Jacobian matrix evaluated over the
orbit (Vi,ϕi).
However, a direct implementation of a computational algorithm to evaluate (2.11)
has a severe limitation to obtain M.
Even in the limit of short n,
the components of M can assume very different orders of magnitude
for chaotic orbits and periodic attractors, yielding impracticable the
implementation of the algorithm. In order to avoid such problem, we note that J can be written as J=ΘT, where Θ is an orthogonal matrix and T is a right triangular matrix. Thus, we rewrite M as M=JnJn−1,…,J2Θ1Θ1−1J1,
where T1=Θ1−1J1.
A product of J2Θ1 defines a new J2′.
In a next step, it is easy to show that M=JnJn−1,…,J3Θ2Θ2−1J2′T1.
The same procedure can be used to obtain T2=Θ2−1J2′, and so on. Using this procedure, the problem
is reduced to evaluate the diagonal elements of Ti: T11i,T22i.
Finally, the Lyapunov exponents are now given by(2.12)λj=limn→∞1n∑i=1nln|Tjji|,j=1,2.If at least one of the λj is positive, then the orbit is classified as
chaotic. Figure 2(b) shows the behavior of the positive Lyapunov exponent
plotted against the number of collisions n with the wall located at y=0 for 10 different initial conditions on the
chaotic sea. The control parameters used in the construction of Figure 2 were ϵ=10−3 and g=5.628×10−5.
The average of the positive Lyapunov exponent for the ensemble of the 10-time
series gives λ¯=1.55±0.01, where the value 0.01 corresponds to the standard deviation of the
ten samples. Finally, in Figure 2(c), we present the behavior of the average
energy of the particle for the simplified model.(a) Phase space generated from iteration of mapping (2.8) for the control
parameters ϵ=10−3 and g=5.628×10−5.
(b) Behavior of the positive Lyapunov exponent of the chaotic sea for the same
control parameters used in (a). The average value obtained was λ¯=1.55±0.01, where the error 0.01 corresponds to the standard deviation of the
10 samples. (c) Behavior of the average energy, E¯,
as function of n.
(a)(b)(c)


## 3. Scaling Analysis

The main goal of this section is to describe a scaling
present in the low-energy regime. We discuss in full detail the investigation
for the simplified and then, at the end of section, we present the
corresponding results for the complete version. Thus, we now discuss the
procedures used to obtain the average velocity on the chaotic low-energy
region. The average velocity (average along an orbit) is defined
as(3.1)V¯(n,ϵ)=1n+1∑i=0nVi.Since we obtain the average
velocity, it is also easy to obtain the deviation of the average velocity,
sometimes also called as the second momenta of V¯.
It is defined as(3.2)ω(n,ϵ)=1M∑i=1MVi2¯(n,ϵ)−Vi¯2(n,ϵ),where the sum on M refers to an average over an ensemble of M different initial conditions. In order to
iterate (3.2) for the simplified version of the model, we evolved our
simulations considering an ensemble of M=5×103 different initial conditions. Figure 3 shows
the behavior of the deviation around the average velocity for the simplified
model as function of n for three different control parameters, as
labelled in the figures. It is easy to see in Figure 3 two different kinds of
behaviors. For short n,
the deviation of the average velocity grows according to a power law and
suddenly it bends toward a regime of saturation for long enough values of n.
The changeover from growth to the saturation is marked by a cross-over
iteration number nx.
It must be emphasized that different values of the parameter ϵ generate different behaviors for short n.
However, applying the transformation n→nϵ2 coalesces all the curves at short n,
as shown in Figure 3(b). We can see that different values of ϵ yield each curve to saturate at distinct
values, thus we can suppose that(a) Behavior of the deviation of the average velocity
for different values of the control parameterϵ for the simplified model. (b) Their initial
collapse after the transformation nϵ2 for the simplified model. The control parameter g was fixed as g=5.628×10−5.
(a)(b)(1)
whenn≪nx, the deviation of the average velocity grows
according to(3.3)ω(nϵ2,ϵ)∝(nϵ2)β,where the exponent β is a critical exponent;(2)
as the iteration number increases,n≫nx,
the deviation of the average velocity approaches a regime of saturation, that
is described as(3.4)ωsat(ϵ)∝ϵα,where the exponent α is also a critical exponent;(3)
the cross-over iteration number that marks the
change from growth to the saturation is written as(3.5)nx∝ϵz,where z is a dynamic exponent.After considering these three initial suppositions, we
are now able to describe the deviation of the average velocity in terms of a
scaling function of the type(3.6)ω(nϵ2,ϵ)=lω(lanϵ2,lbϵ),where l is the scaling factor, a and b are scaling exponents. If we chose properly
the scaling factor l, we can relate the scaling exponents a and b with the critical exponents α, β, and z.We begin considering thatl=(nϵ2)−1/a.
Thus, (3.6) is rewritten as(3.7)ω(nϵ2,ϵ)=(nϵ2)−1/aω1((nϵ2)−b/aϵ),where the function ω1((nϵ2)−b/aϵ)=ω(1,(nϵ2)−b/aϵ) is assumed to be constant for n≪nx.
Comparing (3.3) and (3.7), we obtain β=−1/a.Choosing nowlbϵ=1,
we have l=ϵ−1/b and (3.6) is  given by(3.8)ω(nϵ2,ϵ)=ϵ−1/bω(ϵ−a/bnϵ2),where the function ω2 is defined as ω2(ϵ−a/bnϵ2)=ω(ϵ−a/bnϵ2,1).
It is also assumed as constant for n≫nx.
An immediate comparison of (3.4) and (3.6) gives us α=−1/b.
Given the two different expressions of the scaling factor l,
we obtain a relation for the dynamic exponent z,
that is, given by(3.9)z=αβ−2.Note that the scaling exponents are determined if the
critical exponentsβ and α were numerically obtained. The exponent β is obtained from a power law fitting for the
deviation of the average velocity curves for the parameter ϵ∈[10−4,10−2] for short iteration number. Thus, the average
of these values gives that β=0.502(4)≅1/2.
Figure 4 shows the behaviors of (a) ωsat×ϵ and (b) nx×ϵ.
Applying power law fittings on the figure, we obtain α=0.5003(6)≅0.5 and z=−0.96(1)≅−1.
We can also obtain the exponent z considering (3.9) and the previous values of
both α and β;
we found that z=−1.003(6)≅−1.
Such result indeed agrees with our numerical result. In order to confirm the initial hypotheses,
and since the values of the scaling exponents α, β, and z are now known, we will collapse all the curves
onto a single and universal plots, as demonstrated in Figure 5. With this good collapse of all the curves of
the deviation of the average velocity and considering that the critical
exponents are α≅0.5, β≅0.5 and z≅−1,
we can conclude that the hybrid Fermi-Ulam-bouncer model belongs to the same
class of universality of a periodically corrugated waveguide [42]. It also belongs to the
same class of universality of the Fermi-Ulam [43, 44] itself and that the
presence of a gravitational field does not seem to create a new universality
class, at least for the control parameters considered in the present study.(a) Plot ofωsat as function of the control parameter ϵ for the simplified model. (b) Behavior of the
cross-over number nx against ϵ for the simplified model.
(a)(b)(a) Different curves of theω for four different control parameters for the
simplified model. (b) Their collapse onto a single and universal plots.
(a)(b)Let us now discuss our numerical results for the
complete version of the model. Once the equations of the mapping now be solved
numerically, we have considered an ensemble of less different initial
conditions. Such a consideration is mainly to produce a simulation not so
longer. However, it is still relevant to characterize statistical properties of
the model. For the complete version of the model, we have considered an
ensemble ofM=103 different initial conditions. The behavior of ω for the complete version is rather similar to
that observed for the simplified version of the model. After an extensive
simulation, we obtain that the critical exponents are β=0.50(2)≅0.5, α=0.501(3), and z=−0.95(3).
Evaluating (3.9), we found that z=−1.00(3).
These critical exponents also allow us to produce a good collapse of all the
curves of ω onto a single and universal plots, as shown in
Figure 6. Thus, we can conclude that the scaling
properties are unaffected by considering the simplified or complete versions of
the model.(a) Different
curves of theω for three different control parameters for the
complete model. (b) Their collapse onto a single and universal plots.
(a)(b)


## 4. Final Remarks

As a final
remark of the present paper, we have studied a simplified and the complete
version of the hybrid Fermi-Ulam-bouncer model considering elastic collisions
with the walls. We show that the average energy as well as the deviation around
the average velocity for chaotic orbits for both the complete and simplified
versions of the model exhibit scaling properties with the same critical
exponents. Moreover, we have shown that there is an analytical relation between
the critical exponentsα, β, and z.
Our scaling hypotheses are confirmed by a good collapse of all the curves of ω onto a single and universal plot, therefore,
confirming that this model also belongs to the same class of universality of
the Fermi-Ulam model [43], for the range of control parameters studied, and the
periodically corrugated waveguide [42].



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*Source: 213857-2009-03-15.xml*