# Rank One Strange Attractors in Periodically Kicked Lorenz System with Time-Delay

Authors: Wenjie Yang; Yiping Lin; Yunxian Dai
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

Rank one strange attractor in periodically kicked Lorenz system with time-delay is investigated. Our discussion is based on the theory of rank one maps formulated by Wang and Young. First, we develop the rank one chaotic
theory to delayed systems. It is shown that strange attractors occur when periodically kicked delayed system undergoes a
generic Hopf bifurcation. Then we use the theory to the periodically kicked Lorenz system with delay, and derivation of conditions
for Hopf bifurcation and rank one chaos along with the results of numerical simulations are presented.


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## Body

## 1. Introduction

In the 1970s, chaos characterized by SRB measure was first introduced by Sinai, Ruelle, and Bowen [1–3]. Since then, many results on SRB measure, strange attractor, Lyapunov exponent, and their applications were obtained [4–6]. In 2001, Wang and Young gave simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors [7]. In 2008, Wang and Young accomplished a more comprehensive understanding of the complicated geometric and dynamical structures of a specific class of nonuniformly hyperbolic homoclinic tangles. For certain differential equations, through their well-defined computational process, the existence of the indicated phenomenon of rank one chaos was verified [8]. In 2009, Chen and Han studied the existence of rank one chaos in the neighborhood of a periodically kicked stable limit cycle close to a heteroclinic cycle of a planar equation [9]. In 2012, Fang studied the synchronization between rank one chaotic systems without and with delay using linear delayed feedback control method [10]. In 2013, Wang and Young gave a dynamical study of certain classes of rank one attractors [11].The system we consider in this paper is the well-known Lorenz system with time-delay. Since Lorenz presented the Lorenz system as (1) in [12], people had considerable interest in the three-dimensional autonomous chaotic system (see, e.g., [13, 14]):(1)x˙=ay-x,y˙=cx-xz-y,z˙=xy-bz.The following Lorenz system with time-delay has been researched [15]:(2)x˙t=ayt-τ2-xt-τ1,y˙t=cxt-τ1-xt-τ1zt-τ3-yt-τ2,z˙t=xt-τ1yt-τ2-bzt-τ3.The authors found that a simple time-delay transforms the Lorenz system to the generalized Chen system or the generalized Lu¨ system without any coefficient changes.In this paper, we consider the derivative ofx(t) with respect to a(y(t)-x(t-τ)) and the derivative of y(t) with respect to a time-periodic forcing term εPT. Then we consider the following system:(3)x˙t=ayt-xt-τ,y˙t=cxt-xtzt-yt+εPTyt,z˙t=xtyt-bzt,where a, b, and c are real parameters. The stability of equilibrium, the bifurcating of periodic solutions, and the rank one chaos of the periodic kicked delayed system are investigated.This paper is organized as follows. In Section2, we give preliminaries about the rank one theory. In Section 3, we derived the rank one theory for delayed systems. In Section 4, we consider rank one chaos for the periodically kicked Lorenz system, and at last, numerical simulations are presented in Section 5.


## 2. Preliminaries

To properly motivate the studies presented in this paper, we first give a brief overview on the studies of rank one strange attractors, which can be constructed in the following way according to [16].Letu∈Rm, let m≥2 be the phase variable, and let t∈R be the time. Consider the following system of equations:2.1εdudt=Aμu+fμu+εPTΦu,where Aμ is a real m by m matrix and fμ(u) is a vector valued real analytic function in u defined on a given neighborhood of u=0 such that f(0)=0, Duf(0)=0. Both Aμ and fμ(u) are smooth dependents of parameter μ around μ=0. ε is a parameter that controls the magnitude of the forcing. Φ(u) is a real vector valued function of u that represents the shape of the forcing and PT=∑n=-∞∞δ(t-nT). When ε=0, the undisturbed system is2.10dudt=Aμu+fμu.Firstly, we give several definitions.Definition 1 (see [17]).
Letf:M→M be a diffeomorphism of a compact Riemannian manifold onto itself. One says f is anAnosov diffeomorphism if the tangent space at every x∈M is split into Eu(x)⊕Es(x) where Eu and Es are Df-invariant subspaces, Df|Eu is uniformly expanding, and Df|Es is uniformly contracting. A compact f-invariant set Λ⊂M is called anattractor if there is a neighborhood U of Λ called itsbasin such that fnx→Λ for every x∈U. Λ is called anAxiom A attractor if the tangent bundle over Λ is split into Eu⊕Es as above.Definition 2 (see [17]).
Letf be a C2 diffeomorphism with an Axiom A attractor Λ. Then there is a unique f-invariant Borel probability measure μ on Λ that is characterized by each of the following (equivalent) conditions:(i)
μ has absolutely continuous conditional measures on unstable manifolds;(ii)
consider(4)hμf=∫det⁡DfEudμ,wherehμ(f) is the metric entropy of f;(iii)
there is a setV⊂U having full Lebesgue measure such that, for every continuous observable φ:U→R, we have, for every x∈V,(5)1n∑i=0n-1φfix⟶∫φdμ;(iv)
μ is the zero-noise limit of small random perturbations of f.Then the invariant measureμ is called theSinai-Ruelle-Bowen measure orSRB measure of f.Definition 3.
Given a mapf, theLyapunov exponent is defined at x0 as follows: (6)λx0=limN→+∞⁡1Nln⁡∏i=0N-1f′xi=limN→+∞⁡1N∑i=0N-1ln⁡f′xi,where xi≜f(xi-1)(i=1,2,…,N-1).Forf such that f(U-)⊂U where U⊂M is open, let Ω=⋂n≥0‍fn(U-) be the attractor. Bystrange attractor, here we refer to attractors with the following three properties, see [18].(SA1)
Positive Lyapunov exponents: For Lebesgue, almost everyx∈U; the orbit of x has a positive Lyapunov exponent.(SA2)
Existence of SRB measure and basin property:
(a)
f admits at least one and at most finitely many ergodic SRB measures all of which have no zero Lyapunov exponents, denoted by μ1,μ2,…,μk.(b)
For Lebesgue, almost everyx∈U; ∃j=j(x) such that, for every continuous function ϕ:U→R,(7)1n∑i=0n-1ϕfix⟶∫ϕdμj.(SA3)
Statistical properties of dynamical observations:
(a)
For every ergodic SRB measureμ and every Ho¨lder continuous function ϕ:Ω→R, the sequence {ϕ∘fi}i=0,1,… obeys a central limit theorem; that is, if ∫ϕdμ=0, then 1/n∑i=0n-1‍ϕ∘fi converges in distribution to the normal distribution, and the variance is strictly positive unless ϕ∘f=ψ∘f-ψ for some ψ.(b)
Suppose that, for someN≥1, fN has a SRB measure μ that is mixing. Then given Ho¨lder exponent η, ∃τ=τ(η)<1 such that, for all φ,ψ:Ω→RHo¨lder with exponent η, ∃K=K(φ,ψ) such that, for all n≥1,(8)∫φ∘fnψdμ-∫φdμ∫ψdμ≤Kφ,ψτn.Now, we consider2.1ε. Assume the following.(A1) Let{λi}i=1m be the eigenvalues of Aμ. There is a conjugated pair(9)λ1,2=αμ±iωμ,such that α(0)=0, ω(0)≠0, (d/dμ)α(0)≠0, and there exists c>0 such that Re(λi)<-c, i≥3.Then the flow on the central manifoldWc of system 2.10 can be explicitly written by using a complex variable z in the following normal form:(10)dzdt=αμ+iωμz+g20μz22+g11μzz¯+g02μz¯22+g21μz2z¯2+⋯.According to the standard normal form theory, there exists a change of variables near identity, which we write as(11)z=Z+ΨZ,Z¯=Z+∑2≤j+k≤LψjkZjZ¯kj!k!,which transfers (10) to(12)dZdt=αμ+iωμZ+k1μZ2Z¯+⋯.Using the formula given in [19], we have(13)k10=i2ωg20g11-2g112-13g022+g212.Assume the following.(A2)Re(k1(0))<0. Suppose that a real linear coordinate change u→Lμu∶=(ξ,η,w)T transfers 2.1ε into(14)dξdt=αξ+ωη+fξ+εΦξPTt,dηdt=-ωξ+αη+fη+εΦηPTt,dwdt=Asw+ωη+fw+εΦwPTt,where ξ, η are scalars. w is an m-2-vector, A(s) is an (m-2) by (m-2) matrix of stable eigenvalues, and (15)fξ,fη,fwT=LμfμLμ-1ξ,η,wT,Φξ,Φη,ΦwT=LμΦLμ-1ξ,η,wT.Let(r^,s^)∈R+×Sm-1=r^s^ be such that (ξ,η,w)T=r^s^, and define(16)Ψξr^,s^,Ψηr^,s^,Ψwr^,s^T=Φξ,Φη,ΦwT.Further let{s^0=(cosθ,sinθ,0)∈Sm-1,θ∈[0,2π)} be the unit circle in (ξ,η)-plane in (ξ,η,w)-space, and define(17)φθ=cosθΨξ0,s^0+sinθΨη0,s^0.The time-T map of 2.1ε is denoted by Fμ,ε,T, where μ is the bifurcation parameter of the unperturbed 2.10 and ε, T are the parameters of forcing. Assume that(a)
(A1)-(A2) hold for2.1ε;(b)
φ(θ) in (13) is a Morse function;(c)
μ, ε are such that 0<μ≪1 and 0<ε≪1.The following Theorem is obtained by Wang and Oksasoglu [16].Theorem 4.
Let the values ofμ and ε be fixed and assume that (a)–(c) hold. Regard period T of the forcing as a parameter and denote FT=Fμ,ε,T. Then there exists a constant K1, determined exclusively by φ(θ), such that if(18)εIm⁡k10Re⁡k10>K1,then there exists a positive measure set Δ⊂(μ-1,∞) for T, so that, for T∈Δ, FT has a strange attractor Λ admitting no periodic sinks. That is to say, there exists an open neighborhood U of Λ in Rm such that FT has a positive Lyapunov exponent for Lebesgue almost every point in U. Furthermore, FT admits an ergodic SRB measure, with respect to which almost every point of U is generic.


## 3. Rank One Strange Attractors of Delayed System

In this section we consider a nonlinear delayed differential equation:3.1εdudt=Lμut+fμut+εPTΦut,where ut(θ)=u(t+θ), θ∈[-τ,0] for τ>0, Lμ:C[-τ,0]→Rn is a linear operator, and fμ:C[-τ,0]→Rn is a nonlinear term satisfying fμ(0)=0, Dufμ(0)=0. fμ and Lμ depend on μ analytically for |μ| is sufficiently small, and PT=∑n=-∞∞δ(t-nT). When ε=0, the undisturbed system is3.10dudt=Lμut+fμut.For linear systemu˙=Lμut, there is an n×n matrix η(·,μ):[-τ,0]→Rn2, such that for any ϕ∈C[-τ,0](19)Lμϕ=∫-τ0dηθ,μϕθ.Let the spectral set ofLμ(20)σμ=λ∣det⁡λI-LμeλθI=0satisfy the following.(B1) There is a simple conjugated pair(21)λ1,2=aμ±iωμ,such that a(0)=0, ω(0)=ω0>0, (d/dμ)a(0)≠0, and there exists c>0 such that, for any λ∈σ(μ), λ≠λ1,2; Re(λi)<-c, i≥3.Then the flow on the central manifoldWc of system 3.10 can be written by using Hassard’s method [20] in the following form:(22)z˙=iω0z+gz,z¯=iω0z+g202z2+g11zz¯+g022z¯2+g212z2z¯+⋯,at μ=0. Let(23)k10=i2ω0g20g11-2g112-13g022+g212.Suppose the following.(B2)Re(k1(0))<0. Then we know that system 3.10 has a supercritical Hopf bifurcation near the equilibrium.We define forϕ∈C[-τ,0](24)Aμϕ=dϕdθ,θ∈-τ,0,∫-τ0dηs,μϕs=Lμϕ,θ=0,Rϕ=0,θ∈-τ,0,fμϕ+εPTΦϕ,θ=0.Sincedut/dθ=dut/dt, 3.1ε becomes(25)u˙t=Aμut+Rut.For θ=0 (25) is 3.1ε.Following Hassard’s method, we know thatz satisfies equation(26)zt=q∗θ,utθ,where q∗(θ) is a eigenvector corresponding to eigenvalue -iω0 of A∗ which is the adjoint operator of A(0).Whenμ=0, let A=A(0) and we define(27)Wt,θ=utθ-ztqθ-z¯tq¯θ=utθ-2Re⁡ztqθ=Wzt,z¯t,θ.Then from3.1ε and the definitions of A, A∗, we obtain(28)z˙=q∗θ,u˙tθ=q∗θ,Autθ+Rutθ=q∗θ,Autθ+q∗θ,Rutθ=A∗q∗θ,utθ+q∗θ,Rutθ=iω0q∗θ,utθ+q∗θ,Rutθ.At θ=0,(29)z˙=iω0z+q∗¯0f0Wz,z¯,0+2Re⁡zq0+εPTq∗¯0ΦWz,z¯,0+2Re⁡zq0=iω0z+gz,z¯+εPTq∗¯0ΦWz,z¯,0+2Re⁡zq0.We can calculate(30)W˙=u˙t-z˙q-z¯˙q¯=AW+Hz,z¯,0=H20θ2z2+H11θzz¯+H02θ2z¯2+H30θ6z3+⋯.Then (25) can be written as(31)z˙=iω0z+g202z2+g11zz¯+g022z¯2+g212z2z¯+⋯+εPTq∗¯0ΦWz,z¯,0+2Re⁡zq0,W˙=AW+Hz,z¯,0.LetW∈B, where B is a Banach space, and let z=x+iy in (31). Define(32)Ψxx,y=Re⁡q∗¯0ΦWx,y,0+2Re⁡x+iyq0,Ψyx,y=Im⁡q∗¯0ΦWx,y,0+2Re⁡x+iyq0.We further letx=cosθ, y=sinθ, and W=0 in (32); then {s^0=(cosθ,sinθ,0)∈S×B,θ∈[0,2π)} is the unit circle in (x,y)-plane in (x,y,W)-space. Define(33)φθ=cos⁡θΨxs^0+sin⁡θΨys^0.The time-T map of 3.1ε is denoted by Fμ,ε,T where μ is the bifurcation parameter of the unperturbed 3.10 and ε, T are the parameters of forcing. Assume that(a)
(B1)-(B2) hold for3.1ε;(b)
ϕ(θ) in (33) is a Morse function;(c)
μ,ε are such that 0<μ≪1, 0<ε≪1.Then we obtain the following.Theorem 5.
Let the values ofμ and ε be fixed and assume that (a)–(c) hold. Regard period T of the forcing as a parameter and denote FT=Fμ,ε,T. Then there exists a constant K2, determined exclusively by φ(θ), such that if(34)εIm⁡k10Re⁡k10>K2,then there exists a positive measure set Δ⊂(μ-1,∞) for T, so that, for T∈Δ, FT has a strange attractor Λ admitting no periodic sinks. That is to say, there exists an open neighborhood U of Λ such that FT has a positive Lyapunov exponent for Lebesgue almost every point in U. Furthermore, FT admits an ergodic SRB measure, with respect to which almost every point of U is generic.Proof.
We can easy see that (B1) and (B2) in Theorem5 are corresponding to (A1) and (A2) in Theorem 4. After transformations, (33) in (x,y)-plane is corresponding to (17) in (ξ,η)-plane, and they are on the central manifolds. Obviously the conditions in Theorem 4 are satisfied.


## 4. Analysis of Rank One Strange Attractors in Delayed Lorenz System

We consider the corresponding undisturbed system of (3) as follows:(35)x˙t=ayt-xt-τ,y˙t=cxt-xtzt-yt,z˙t=xtyt-bzt.Obviously, system (35) always has equilibria Ei∗(xi∗,yi∗,zi∗), i=1,2,3, where(36)x1∗=0,y1∗=0,z1∗=0,x2∗=y2∗,y2∗=bz2∗,z2∗=c-1,x3∗=y3∗,y3∗=-bz3∗,z3∗=c-1.LetE∗=(x∗,y∗,z∗) be the arbitrary equilibrium, and denote u=(x1(t),y1(t),z1(t))T, where x1(t)=x(t)-x∗, y1(t)=y(t)-y∗, and z1(t)=z(t)-z∗ still denote x1(t), y1(t), and z1(t) by x(t), y(t), and z(t), respectively; then the linearized system of the corresponding equation at E∗ is as follows:(37)u˙=Aut+But-τ,where A=(aij)3×3, B=(bij)3×3, a12=a, a21=c-z∗, a22=-1, a23=-x∗, a31=y∗, a32=x∗, a33=-b, b11=-a, and all the others of aij and bij are 0.The characteristic equation of system (37) is(38)λ3+α1λ2+α2λ+α3+ae-λτλ2+β1λ+β2=0,where α1=1+b, α2=b+x∗2-a(c-z∗), α3=ax∗2-ab(c-z∗), β1=1+b, and β2=b+x∗2. We consider the following cases.(1) One hasE∗=E1∗. The coefficients of (38) are α1=1+b, α2=b-ac, α3=-abc, β1=α1, and β2=b.Whenτ=0, (38) becomes(39)Γλ=λ3+a+α1λ2+α2+aβ1λ+α3+aβ2=0.Routh-Hurwitz criterion implies that ifH1a+b+1>0,baa+b+3+c1+b-a>0,2ab2c-1aa+b+3+c1+b-a>0,then (38) has at least one positive real root, so when H1 holds, E1∗ is unstable equilibrium.(2) One hasE∗=E2∗, or E∗=E3∗. Due to the symmetry of E2∗ and E3∗, here we only consider E∗=E2∗. When τ=0, the coefficients of system (38) are α1=1+b, α2=bc-a, α3=ab(c-2), β1=α1, and β2=bc. According to Routh-Hurwitz criterion, if H1 holds, then all roots of (38) have negative real parts.Suppose thatiω(ω>0) is a root of (38) in the imaginary axis. Substituting it in (38) and separating the real and imaginary parts, we have(40)aβ1ωcos⁡ωτ-aβ2-ω2sin⁡ωτ=ω3-α2ω,aβ1ωsin⁡ωτ+aβ2-ω2cos⁡ωτ=α1ω2-α3,which is equivalent to(41)ω6+e1ω4+e2ω2+e3=0.Letz=ω2; then (41) becomes(42)Gz≜z3+e1z2+e2z+e3=0,where e1=α12-2α2-a2, e2=2a2β2+α22-a2β12-2α1α3, and e3=α32-a2β22.Lemma 6.
For (42), one has the following.(i)
Ife3<0, then (42) has at least one positive root.(ii)
Ife3⩾0, then (42) has at least one positive root if and only if there exists z∗>0 such that G′(z∗)=0 and G(z∗)⩽0.We assume that (42) has three positive roots z1, z2, and z3 and ω1=z1, ω2=z2, and ω3=z3.According to (40), we have(43)τkj=1ωkarcsin⁡ωk5+α1β1-α2-β2ωk3+α2β2-α3β1ωkaβ12ωk2+aβ2-ωk22+2jπ,where k=1,2,3 and j=0,1,…; then ±iωk is a pair of imaginary roots of (38) with τk(j).Define(44)τ0=τk00=mink=1,2,3⁡τk0,ω0=ωk0.In order to investigate the distribution of the roots of (38), we need to introduce the following lemmas [21].Lemma 7.
Consider the exponential polynomial(45)Pλ,e-λτ1,…,e-λτm=λn+p10λn-1+⋯+pn-10λ+pn0=p11λn-1+⋯+pn-11λ+pn1e-λτ1+⋯+p1mλn-1+⋯+pn-1mλ+pnme-λτm,where τi≥0, i=1,2,…,m, and pj(i) (i=0,1,…,m; j=1,2,…,n) are constants. As (τ1,τ2,…,τm) vary, the sum of the order of the zeros of P(λ,e-λτ1,…,e-λτm) on the open right half plane can change if and only if there are zeros crossing the imaginary axis.Letλ(τ)=α(τ)+iω(τ) be the root of (38) near τ=τkj satisfying (46)ατkj=0,ωτkj=ωk.Then, the following transversal condition holds.Lemma 8.
Suppose thatzk=ωk2 and G′(zk)≠0; then Reλ(τk(j))/dτ has the same sign as G′(zk).Proof.
Substitutingλ(τ) into (38) and taking the derivative with respect to τ, we obtain(47)3λ2+2α1λ+α2+ae-λτ2λ+β1-τλ2+β1λ+β2dλdτ=aλe-λτλ2+β1λ+β2.Therefore,(48)dλdτ-1=3λ2+2α1λ+α2eλτaλλ2+β1λ+β2+2aλ+aβ1aλλ2+β1λ+β2-τλ.When τ=τk(j), λ=iωk, k=1,2, we have(49)3λ2+2α1λ+α2eλττ=τkj=α2-3ωk2cos⁡ωkτkj-2α1ωksin⁡ωkτkj+2α1ωkcos⁡ωkτkj+α2-3ωk2sin⁡ωkτkj·i,λλ2+β1λ+β2τ=τkj=-β1ωk2+ωkβ2-ωk2i,2aλ+aβ1τ=τkj=aβ1+2aωki.
According to (48), (49), and (42), we have(50)dReλdττ=τkj-1=Re⁡3λ2+2α1λ+α2eλτaλλ2+β1λ+β2τ=τkj+Re⁡2λ+β1λλ2+β1λ+β2τ=τkj=1Λ-2aωk4+ωk22aβ2-aβ1+ωk43β1-2α1+ωk22α1β2-α2β1·cos⁡ωkτkj+3ωk5+ωk32α1β1-α2-3β2+α2β2ωk·sin⁡ωkτkj=1Λ3ωk6+2α12-2α2-a2ωk4+2a2β2+α22-a2β12ωk2=zk3zk2+2e1zk+e2Λτ=τkj-1=zkG′zkΛ,where Λ=a2β12ωk4+a2ωk2(β2-ωk2)2>0.
Therefore,Reλ(τk(j))/dτ has the same sign as G′(zk).
Now we apply the Hopf bifurcation theorem for functional differential equations [22] and get the following results.Theorem 9.
Suppose thatH1 and Lemma 6 are satisfied. Then(i)
whenτ∈[0,τ0), equilibrium E2∗ of system (35) is locally asymptotically stable;(ii)
whenτ>τ0, equilibrium E2∗ of system (35) is unstable;(iii)
whenτ=τ0, then system (35) undergoes Hopf bifurcation at equilibrium E2∗.Now, we will derive the explicit formulae determining the direction and stability of these periodic solutions bifurcating from equilibriumE2∗. Let τ=τ0+μ, t=τt-, and omit “-” above t; we rewrite (3) as(51)u˙t=Lμut+fμ,ut+εPTΦϕ,Lμϕ=τ0+μAϕ10ϕ20ϕ30+τ0+μBϕ1-1ϕ2-1ϕ3-1,fμ,ϕ=τ0+μ0-ϕ10ϕ30ϕ10ϕ20,Φϕ=τ0+μ0ϕ200,where A, B are as in (37). We write Lμϕ as the form Lμϕ=∫-10‍dηθ,μϕ(μ), where (52)dηθ,μ=τ0+μAδθ+τ0+μBδθ+1,where δ(θ) is the Dirac delta function. Then (3) can be written as(53)u˙t=Aμut+Rut,where(54)Aμϕ=dϕθdθ,θ∈-1,0,∫-10dηs,μϕs,θ=0,Rϕ=0,θ∈-1,0,fμϕ+εPTΦϕ,θ=0.Assume thatq(θ) is the eigenvector of A(0) corresponding to iω0τ0; then A(0)q(θ)=iω0τ0q(θ). It follows from the definition of A(0) that(55)iω0+ae-iω0τ0-a0-1iω0+1x∗-x∗-x∗iω0+bqθ=000.Thus, we can easily computeq(θ)=(1,α,β)Teiω0τ0θ, where(56)α=ae-iω0τ0+iω0a,β=x∗1+αb+iω0.Similarly, it can be verified thatq∗(s)=D(1,α∗,β∗)eiω0τ0 is the eigenvector of A∗ corresponding to -iω0τ0, where(57)α∗=aeiω0τ0-iω0iω0-bx∗2-b+iω0,β∗=x∗α∗iω0-b.In order to assure the bilinear inner product q∗(s),q(θ)=1, we have(58)q∗s,qθ=D¯1,α¯∗,β¯∗1,α,βT-∫-10∫ξ=0θD¯1,α¯∗,β¯∗e-iξ-θω0τ0dηθ1,α,βT·eiξω0τ0dξ=D¯1+αα¯∗+ββ¯∗-∫-101,α¯∗,β¯∗θeiθω0τ0dηθ1,α,βT=D¯1+αα¯∗+ββ¯∗-aτ0e-iω0τ0.Therefore, we can chooseD¯ as(59)D¯=1+αα¯∗+ββ¯∗-aτ0e-iω0τ0-1=1Re⁡D¯-1+iIm⁡D¯-1,where Re(D¯-1)=1-aτ0cos(ω0τ0)+Re(α)Re(α∗)+Im(α)Im(α∗)+Re(β)Re(β∗)+Im(β)Im(β∗), Im(D¯-1)=aτ0sin(ω0τ0)+Re(α∗)Im(α)-Re(α)Im(α∗)+Re(β∗)Im(β)-Re(β)Im(β∗).We can getq∗(s),q¯θ=0.In what follows, we will obtain the coordinates to describe the center manifoldC0 at τ=τ0.Notice thatut(θ)=(xt(θ),yt(θ),zt(θ))T=zq(θ)+z¯q¯(θ)+W(t,θ); then we have(60)xt0=z+z¯+W2010z22+W1110zz¯+W0210z¯22+W3010z36+W2110z2z¯2+⋯,yt0=zα+z¯α¯+W2020z22+W1120zz¯+W0220z¯22+W3020z36+W2120z2z¯2+⋯,zt0=zβ+z¯β¯+W2030z22+W1130zz¯+W0230z¯22+W3030z36+W2130z2z¯2+⋯.Thus, from (29) we have(61)gz,z¯=q∗¯0f0z,z¯=D¯τ01,α¯∗,β¯∗0-xt0zt0xt0yt0=D¯τ0β¯∗xt0yt0-α¯∗xt0zt0.Comparing the coefficients with (31), we can obtain the following formulae:(62)g20=2D¯τ0αβ∗¯-βα∗¯,g11=2D¯τ0Reαβ∗¯-Reβα∗¯,g02=2D¯τ0α¯β∗¯-β¯α∗¯,g21=2D¯τ0β∗¯12W2010α¯+12W2020+W1110α+W1120-α∗¯12W2010β¯+12W2030+W1110β+W1130.In order to obtaing21, we need to compute W20(θ) and W11(θ). Substituting the corresponding series into (31) and comparing the coefficients, we obtain(63)A-2iτ0ω0W20θ=-H20θ,AW11θ=-H11θ.For θ∈[-1,0), from (31), we know that(64)Hz,z¯,θ=-q¯∗0f0qθ-q∗0f¯0q¯θ=-gqθ-g¯q¯θ.Comparing the coefficients with (31), we have(65)H20θ=-g20qθ-g¯02q¯θ,H11θ=-g11qθ-g¯11q¯θ.From (63), (64), and the definition of A, we can easily get(66)W˙20θ=2iτ0ω0W20θ+g20qθ+g¯02q¯θ.Notice that q(θ)=(1,α,β)Teiθω0τ0; hence(67)W20θ=ig20τ0ω0q0eiθω0τ0+ig¯023τ0ω0q¯0e-iθω0τ0+E1e2iθω0τ0,where E1=(E1(1),E1(2),E1(3))∈R3 is a constant vector. Similarly, we can obtain(68)W11θ=-ig11τ0ω0q0eiθω0τ0+ig¯11τ0ω0q¯0e-iθω0τ0+E2,where E2=(E2(1),E2(2),E2(3))∈R3 is a constant vector.In what follows, we will seek appropriateE1 and E2. According to the definition of A and (63), we have(69)∫-10dηθ,0W20θ=2iτ0ω0W200-H200,(70)∫-10dηθ,0W11θ=-H110.From (31), we have(71)H200=-g20q0-g¯02q¯0+2τ00-βα,(72)H110=-g11q0-g¯11q¯0+2τ00-Re⁡βRe⁡α.Substituting (67) and (71) into (69) and noticing that(73)iω0τ0I-∫-10eiω0τ0θdηθq0=0,-iω0τ0I-∫-10e-iω0τ0θdηθq0=0,we have(74)2iω0τ0I-∫-10e2iω0τ0θdηθE1=2τ00-βα,that is,(75)2iω0+ae-2iω0τ0-a0-12iω0+1x∗-x∗-x∗2iω0+bE1=20-βα.According to Cramer’s criteria, the solution of (75) is described by (76)E1=22iω0+ae-2iω0τ0-a0-12iω0+1x∗-x∗-x∗2iω0+b-1·0-βα=2Aa1a2a3,where, a1=-aβ(2iω0+b)-ax∗α,a2=-β(2iω0+ae-2iω0τ0)(2iω0+b)-x∗α(2iω0+ae-2iω0τ0), a3= -βx∗(2iω0+ae-2iω0τ0+a)+α(2iω0+ae-2iω0τ0)(2iω0+1)-aα, and(77)A=2iω0+ae-2iω0τ0-a0-12iω0+1x∗-x∗-x∗2iω0+b.Similarly, substituting (68) and (72) into (70), we have(78)E2=2a-a0-11x∗-x∗-x∗b-10Re⁡βRe⁡α=2BabRe⁡β-ax∗Re⁡αabRe⁡β-ax∗Re⁡α2ax∗Re⁡β,where (79)B=a-a0-11x∗-x∗-x∗b.Thus, we can compute the following values:(80)μ2=-Re⁡k10Re⁡λ′τk,β2=2Re⁡k10,where k1(0)=i/2ω0τ0(g20g11-2|g11|2-1/3|g02|2)+g21/2, determining the quantities of bifurcating periodic solutions in the center manifold at the critical value τ0; that is, μ2 determines the directions of the Hopf bifurcation: if Re{λ′(τ0)}>0, μ2>0 (resp., μ2<0), then the Hopf bifurcation is supercritical (resp., subcritical) and the periodic solutions exist for τ>τ0(τ<τ0). β2 determines the stability of the bifurcation periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(β2>0).Furthermore, we can get the definite expressions ofRe(k1(0)) and Im(k1(0)). From (32), we have(81)Ψxx,y=Re⁡q∗¯0Φ2Re⁡x+iyq0=Re⁡D¯τ01,α¯∗,β¯∗·0x+iyα+x-iyα¯0=Re⁡D¯τ0α¯∗x+iyα+x-iyα¯=2τ0x·Re⁡α-yIm⁡αRe⁡α∗Re⁡D¯-1+Im⁡α∗·Im⁡D¯-1,Ψyx,y=Im⁡q∗¯0Φ2Re⁡zq0=2τ0xRe⁡α-yIm⁡αRe⁡α∗Im⁡D¯-1-Im⁡α∗·Re⁡D¯-1.Let x=cosθ and y=sinθ, then(82)φθ=cos⁡θΨxs^0+sin⁡θΨys^0=2τ0δ1Re⁡αcos2⁡θ-δ2Im⁡αsin2⁡θ+δ2Re⁡α-δ1Im⁡αcos⁡θsin⁡θ,where δ1=(Re(α∗)Re(D¯-1)+Im(α∗)Im(D¯-1))and δ2=(Re(α∗)Im(D¯-1)-Im(α∗)Re(D¯-1)). Then φ(θ) is easy to verify being a Morse function from (82). According to Theorem 5, if Re(k1(0))<0, then there exists K2 such that when εIm(k1(0))/Re(k1(0))>K2, we can get an observable rank one chaos.


## 5. Numerical Simulations

Now we move to the parameters for which rank one attractors are observable, so we consider the following system:(83)x˙t=1.9yt-xt-τ,y˙t=8.009xt-xtzt-yt+εPTyt,z˙t=xtyt-2.25zt.System (83) has a positive equilibrium E2∗(3.9712,3.9712,7.0090). It is not difficult to get the following linear system of (83) at E2∗ without the disturbed term εPTy(t):(84)u˙1t=-1.9xt-τ+1.9yt,u˙2t=xt-yt-3.9712zt,u˙3t=3.9712xt-3.9712yt-2.25zt.It follows that the characteristic equation of system (84) is (85)λ3+3.2500λ2+16.1203λ+25.6885+1.9e-λτλ2+3.2500λ+18.0203=0.From the discussion of Section 4, we have ω0≐3.9355, τ0≐0.375, Re{λ′(0)}≐0.1975, Re{k1(0)}≐-0.0027, μ2≐0.0139, and β2≐-0.0055, so we know that the equilibrium E2∗ is asymptotically stable when ε=0 and τ∈[0,0.375) (see Figure 1). Further, we know that the bifurcation is supercritical and the stable periodic solution emerges from the equilibrium (see Figure 2).Figure 1
The trajectories and the phase graph of system (83) with T=98, ε=0, and τ=0.2; E2∗ is asymptotically stable.
(a)
(b)
(c)
(d)Figure 2
The trajectories and the phase graph of system (83) with T=98, ε=0, and τ=0.4; E2∗ becomes unstable, and a stable periodic solution bifurcates from equilibrium E2∗.
(a)
(b)
(c)
(d)We also get|Im(k1(0))/Re(k1(0))|≐70.6999. Choose T in interval (98,100), which is quite “large” to afford a long relaxation period between consecutive kicks of the external force, and choose ε in (0,0.4). In Figure 3, we present a rank one strange attractor at (τ,ε,T)=(0.4,0.326,98). In Figure 4, a rank one strange attractor at (τ,ε,T)=(0.4,0.326,100) is shown. In Figure 5, largest Lyapunov exponent λ versus ε for system (83) with T=98 and T=100 is given. We can see that there exists a positive measure set Δ⊂(0.4-1,∞) for T, such that, for T∈Δ, the time-T map FT of system (83) has a strange attractor. Largest Lyapunov exponent λ versus T for system (83) with τ=0.4, ε=0.326, and T varying from 0 to 120 is given in Figure 6. The results of our numerical simulations are in perfect agreement with the predictions of the rank one theory in Section 3.Figure 3
The trajectories and the phase graph of system (83) with T=98, ε=0.326, and τ=0.4; E2∗ is unstable, and a rank one strange attractor occurs.
(a)
(b)
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(d)Figure 4
The trajectories and the phase graph of system (83) with T=100, ε=0.326, and τ=0.4; E2∗ is unstable, and a rank one strange attractor occurs.
(a)
(b)
(c)
(d)Figure 5
Largest Lyapunov exponentλ versus ε for system (83) with T=98 (a), T=100 (b), τ=0.4, and ε varying from 0 to 0.4.
(a)
(b)Figure 6
Largest Lyapunov exponentλ versus T for system (83) with ε=0.326, τ=0.4, and T varying from 0 to 120.



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*Source: 915614-2015-05-12.xml*