# Enhancing Ikeda Time Delay System by Breaking the Symmetry of Sine Nonlinearity

Authors: Xiaojing Gao
Journal: Complexity
 (2019)

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## Abstract

In the present contribution, an asymmetric central contraction mutation (ACCM) model is proposed to enhance the Ikeda time delay system. The modified Ikeda system model is designed by introducing a superimposed tanh function term into the sine nonlinearity term. Stability and Hopf bifurcation characteristics of the system are analyzed theoretically. Numerical simulations, carried out in terms of bifurcation diagrams, Lyapunov exponents spectrum, phase portraits, and two-parameter (2D) largest Lyapunov exponent diagrams are employed to highlight the complex dynamical behaviors exhibited by the enhanced system. The results indicate that the modified system has rich dynamical behaviors including limit cycle, multiscroll hyperchaos, chaos, and hyperchaos. Moreover, as a major outcome of this paper, considering the fragile chaos phenomenon, the ACCM-Ikeda time delay system has better dynamical complexity and larger connected chaotic parameter spaces (connectedness means that there is no stripe corresponding to nonchaotic dynamics embedded in the chaos regions).


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## Body

## 1. Introduction

Chaotic system has many specific properties, such as initial state and parameters sensitivity, unpredictability, and topological mixing [1–3]. Although the equation model of a chaotic system is deterministic, it is impossible to predict its long-term behavior. These meaningful properties make chaotic systems widely studied and applied in many disciplines [4–10]. Specially, with the rapid development of information society chaos, secure communication has already been established to be a good candidate for transmission of confidential message [11–13]. In addition, chaotic systems have important applications in chaos-based random number generation [14] and sensors [15].In recent years, time delay (TD) systems have become the subject of active research, which stems from the following reasons: (i) the existence of TD in nonlinear systems makes systems more complex. It has been found that communication systems based on low-dimensional chaotic systems (having a single positive LE) are insecure because their dynamics can be easily reconstructed by Takens’ embedding theorem [16]. TD chaotic systems modelled by nonlinear delay differential equations (DDEs) exhibit infinite dimensional phase space that cannot be anticipated by a low-dimensional system. As such, time delay systems can provide a higher level of computational security against embedding reconstruction. (ii) TD systems provide hyperchaos with multiple positive Lyapunov exponents (LEs) [17]. Due to these reasons, a number of simple and well characterized TD systems have been designed to produce chaos and hyperchaos [18–20]. Dynamics of nonlinear time-delay systems have been studied [19, 21].In this paper, we focus our attention towards the simple and well characterized first-order Ikeda TD system:(1)x˙t=−axt+bsinxτ,where a∈R and b∈R are positive parameters, τ∈R+ is the time delay constant, and xτ≡xt−τ. From the theoretical point of view, some works have been carried out with the aim of inherent dynamics of the Ikeda TD system. In [22], the chaotic dynamics of first-order scalar delay differential equations, including Ikeda equations, have been investigated. An analytical framework for study of dynamics of breathers in slow-fast Ikeda optoelectronic systems was proposed in [23]. A study on the biorhythmic behavior and its control in an optoelectronic oscillator was presented in [24]. From the viewpoint of engineering, some research studies have focused on the performance enhancement of the system for random number generation, optical chaotic secure communication, chaotic image encryption, optical secret communication, and so on [25–31].It is noted that chaotic dynamical properties such as high complexity and wide chaotic parameter range are strongly required in some scenarios, including secure communication and random number generation. But researchers found that existing chaotic systems have certain limitations in different aspects. One concern is that many systems suffered from the fragile chaos phenomenon [32]. Fragile chaos means that a system has small chaotic parameter regions or has some nonchaotic structures embedded in the chaotic parameter zone. Small perturbation modulation of a parameter of the system is possible to destroy the chaotic attractors and transform the chaotic oscillator to period oscillations, as the parameter will fall into a nonchaotic region easily. Moreover, due to the limitation of physical devices, the range of physical parameters is always restricted, which will cause performance degradation in many cases. A typical scenario is chaotic secure communication. The security level of a chaos-based scheme is significantly dependent on the chaotic parameter range [33].Considering the fragile chaos phenomenon, this paper proposes an asymmetric central contraction mutation (ACCM) model to enhance the Ikeda TD system. The modified TD system with ACCM is studied. We carry out stability analysis to identify the parameter zone for which the system shows a stable equilibrium response. And we simulate the system model numerically to show that with the variation of delay and other system parameters, the system exhibits stable limit cycle, chaos, hyperchaos, and multiscroll hyperchaos over the whole three-dimensional (3D) parameter space (given bya, b, and τ). Single parameter bifurcation diagrams, phase plots, and two-parameter (2D) Lyapunov exponent diagrams [34] are employed to explore the dynamics of the system. The ACCM model not only can enhance the dynamical complexity of the original Ikeda TD system in the chaotic range, but also can produce chaos in the nonchaotic range. It is worth mentioning that the improved system has larger chaotic parameter zones with good connectivity.The rest of the paper is arranged in the following order. Section2 presents the related mathematical model of the proposed ACCM-Ikeda time delay system. Analysis of stability and Hopf bifurcation are shown in Section 3. Simulation results of the ACCM-Ikeda time delay system and discussions are presented in Section 4, followed by performance comparison in Section 5. Section 6 draws conclusion.


## 2. System Description

We propose the following Ikeda TD system with Asymmetric Central Contraction Mutation (ACCM):(2)x˙t=−axt+bsinxτ+tanhxτ+tanhxτ−2,where a>0 and b>0 are real positive system parameters. xτ≡xt−τ, and τ∈R+ is the intrinsic time delay of the system.Sketches of the nonlinearitiessinxτ, tanhxτ, and gxτ=sinxτ+tanhxτ+tanhxτ−2 are depicted in Figures 1(a)–1(c), respectively. It can be found from Figure 1(c) that the sketch of nonlinearity gxτ showing a variation law similar to that of the sinxτ function, but with asymmetric central contraction, emerging more humps and valleys in the curve within the same parameter range. Therefore, we named the modified model as the ACCM-Ikeda TD system, which will have expected dynamic characteristics, such as better complexity, more larger and connected (meaning that there is no transient nonchaotic window interspersed in the chaotic parameter zone) chaotic parameter zone than the seed Ikeda time delay system (will be discussed in Section 5).Figure 1
Sketches of (a)sinxτ, (b) tanhxτ and tanhxτ−2, and (c) gxτ (dotted line).
(a)
(b)
(c)


## 3. Stability and Hopf Bifurcation Analysis

Consider the TD system (2) expressed as(3)x˙t=fx,xτ=−axt+bgxτ.Equilibrium pointsx∗ of (3) are obtained by solving(4)fx∗,x∗=0.


### 3.1. Linearization near Equilibrium

Defining a small perturbationδ=x−x∗ in equilibrium solution and using first-order Taylor’s approximation, we get a linearized equation of (3) as(5)dδdt=dxdt=fx,xτ=fx+δ,xτ+δτ=fx∗,x∗+∂1fx∗,x∗δ+∂2fx∗,x∗δτ=pδ+qδτ,where xτ≡xt−τ, δτ≡δt−τ, p=∂1fx∗,x∗=−a, and q=∂2fx∗,x∗=bg′x∗ are partial derivatives of fx,xτ with respect to variables x and xτ evaluated at x∗,x∗, respectively. We can get the characteristic equation as(6)λ=−a+qe−λτ.


### 3.2. Stability of Equilibrium

For the stability analysis, we use the methodology discussed in [22]. An equilibrium point x∗ is asymptotically stable if all the roots λi of characteristic equation (6) satisfy(7)Reλi<0,for∀i.Ifτ=0, then condition (7) takes the form(8)−a+q<0.Writeλ=u+iω, u,ω∈R. Stability of equilibrium will change if λ crosses imaginary axis at λ=iω. The characteristic equation in this case becomes(9)iω=−a+qe−iωτ.Usinge−iωτ=cosωτ−isinωτ and separating real and imaginary parts in (9), we get(10)qcosωτ=a,qsinωτ=−ω.This gives(11)q2=a2+ω2.Then, we can get(12)ω=q2−a2.This is possible if and only if(13)q>a.


### 3.3. Critical Surfaces

Forb>a, from (10) we can obtain the critical surfaces expressed as(14)τk=cos−1a/bg′x∗+2kπq2−a2,forg′x∗<0,2π−cos−1a/bg′x∗+2kπq2−a2,forg′x∗>0.Differentiating characteristic equation (6) with respect to τ, we get(15)dλdτ=qe−λτ−λ−τdλdτ.Noting thatqe−λτ=λ+a, we can have(16)dλdτ=−λλ+a1+τλ+a,and hence(17)dλdτu=0=ω2−iaω1+aτ+iωτ.On critical surfaces (14),(18)dλdτ=Redλdτu=0=ω21+aτ2+ωτ2,du/dτ>0 on each of the critical surfaces τk. This implies that there does not exist any eigenvalue with negative real part across the critical surfaces (14). Thus, there is only one possible stability region (under condition (8)) enclosed by τ=0 and the critical surface τ0 closest to it.Fora>0, we have following main results:Theorem 1.
Supposex∗ is an equilibrium solution of the ACCM-Ikeda TD system (3) and q=∂2fx∗,x∗=bg′x∗, then we can get the following:(1)
Ifq∈−∞,−a, then the stability region of x∗ in τ,a,q parameter space is located between the plane τ=0 and τ0. The equation undergoes Hopf bifurcation at this value.(2)
Ifq∈a,∞, then x∗ is unstable for any τ≥0.(3)
Ifq∈−a,a, then x∗ is stable for any τ≥0.


## 3.1. Linearization near Equilibrium

Defining a small perturbationδ=x−x∗ in equilibrium solution and using first-order Taylor’s approximation, we get a linearized equation of (3) as(5)dδdt=dxdt=fx,xτ=fx+δ,xτ+δτ=fx∗,x∗+∂1fx∗,x∗δ+∂2fx∗,x∗δτ=pδ+qδτ,where xτ≡xt−τ, δτ≡δt−τ, p=∂1fx∗,x∗=−a, and q=∂2fx∗,x∗=bg′x∗ are partial derivatives of fx,xτ with respect to variables x and xτ evaluated at x∗,x∗, respectively. We can get the characteristic equation as(6)λ=−a+qe−λτ.


## 3.2. Stability of Equilibrium

For the stability analysis, we use the methodology discussed in [22]. An equilibrium point x∗ is asymptotically stable if all the roots λi of characteristic equation (6) satisfy(7)Reλi<0,for∀i.Ifτ=0, then condition (7) takes the form(8)−a+q<0.Writeλ=u+iω, u,ω∈R. Stability of equilibrium will change if λ crosses imaginary axis at λ=iω. The characteristic equation in this case becomes(9)iω=−a+qe−iωτ.Usinge−iωτ=cosωτ−isinωτ and separating real and imaginary parts in (9), we get(10)qcosωτ=a,qsinωτ=−ω.This gives(11)q2=a2+ω2.Then, we can get(12)ω=q2−a2.This is possible if and only if(13)q>a.


## 3.3. Critical Surfaces

Forb>a, from (10) we can obtain the critical surfaces expressed as(14)τk=cos−1a/bg′x∗+2kπq2−a2,forg′x∗<0,2π−cos−1a/bg′x∗+2kπq2−a2,forg′x∗>0.Differentiating characteristic equation (6) with respect to τ, we get(15)dλdτ=qe−λτ−λ−τdλdτ.Noting thatqe−λτ=λ+a, we can have(16)dλdτ=−λλ+a1+τλ+a,and hence(17)dλdτu=0=ω2−iaω1+aτ+iωτ.On critical surfaces (14),(18)dλdτ=Redλdτu=0=ω21+aτ2+ωτ2,du/dτ>0 on each of the critical surfaces τk. This implies that there does not exist any eigenvalue with negative real part across the critical surfaces (14). Thus, there is only one possible stability region (under condition (8)) enclosed by τ=0 and the critical surface τ0 closest to it.Fora>0, we have following main results:Theorem 1.
Supposex∗ is an equilibrium solution of the ACCM-Ikeda TD system (3) and q=∂2fx∗,x∗=bg′x∗, then we can get the following:(1)
Ifq∈−∞,−a, then the stability region of x∗ in τ,a,q parameter space is located between the plane τ=0 and τ0. The equation undergoes Hopf bifurcation at this value.(2)
Ifq∈a,∞, then x∗ is unstable for any τ≥0.(3)
Ifq∈−a,a, then x∗ is stable for any τ≥0.


## 4. Numerical Studies

In this section, the system equation (2) is solved numerically using the fourth-order Runge–Kutta algorithm with a step size h=0.01. The initial condition is ϕt=1 for t∈−τ,0 throughout the investigation. Then, the dynamics of the ACCM-Ikeda TD system is investigated through bifurcation diagram, phase plots, and Lyapunov exponent spectrum (LES) considered from different perspectives. The LES is calculated using the method proposed in [35].


### 4.1. Varyinga for b=4 and Time Delay τ=2

System (2) with parameter b=4 and the time delay τ=2 is numerically examined for the single bifurcation parameter a. The bifurcation parameter a varies in the range 0.1≤a≤3. Five largest Lyapunov exponents λ1,2,3,4,5 of all the spectrum versus the bifurcation parameter a is shown in Figure 2(a), which reveals that,(1)
in the regiona∈2.18,3, the equilibrium point x∗ is stable(2)
in the regiona∈1.9,2.17, there is one positive LE (λ1>0) meaning that chaos appears(3)
in the regiona∈1.44,1.89, chaotic and hyperchaotic regime appears alternately(4)
in the region a ∈ [0.1, 1.43], there are two positive LE (λ1 > 0 and λ2 > 0) meaning that hyperchaos appearsFigure 2
(a) The five largest Lyapunov exponentsλ1,2,3,4,5 plotted versus the bifurcation parameter −3≤−a≤−0.1 for parameter b=4 and time delay τ=2. (b) The bifurcation diagram for the same parameter as in case (a).
(a)
(b)These regions of nonchaos and chaos observed in Figure2(a) are best visualized in the bifurcation diagram, where we plot the maxima of xt versus the bifurcation parameter a, as shown in Figure 2(b). A good consistency can be noted between the bifurcation diagram and the LES. The outcomes of this variation are shown in Figure 3.(i)
Fora=2.67, we get equilibrium point x∗=1.44555 and q=bg′x∗=−2.0435. By Case 3 of Theorem 1, the time series xt converges to x∗, as shown in Figure 3(a).(ii)
Fora=2.35, we get equilibrium point x∗=1.53665 and q=bg′x∗=−3.4094. Therefore, in view of Case 1 of Theorem 1, the stability region is located between τ=0 and τ0=0.7564. By setting τ=2, the equilibrium is found unstable leading to a limit cycle, as depicted in Figure 3(b).(iii)
Fora=2.02, we get q=bg′x∗=−4.6290. Therefore, in view of Case 1 of Theorem 1, the stability region is located between τ=0 and τ0=0.4294. By setting τ=2, the equilibrium is found unstable leading to a chaotic attractor, as depicted in Figure 3(c).(iv)
Fora=1.57, a=0.61, and a=0.12, we get q=7.5144, q=7.8277, and q=7.9023. In view of Case 2 of Theorem 1, x∗ is unstable for any τ≥0. We can obtain hyperchaotic attractors (shown in Figures 3(d) and 3(e)) and a multiscroll hyperchaotic attractor (Figure 3(f)), respectively.Figure 3
The initial condition isϕt=1 for t∈−τ,0 throughout the investigation. Parameter b = 4, and time delay τ=2. (a) Time series for a=2.67, the fixed point is stable. Phase plane plots in xt−τ−xt plane for variable a, (b) period limit cycle for a=2.35, (c) chaotic attractor for a=2.02, (d and e) hyperchaotic attractor for a=1.57 and a=0.61, (f) multiscroll hyperchaotic attractor for a=0.12.
(a)
(b)
(c)
(d)
(e)
(f)


### 4.2. Varyingb for a=1 and Time Delay τ=2

System (2) with the parameter a=1 and the time delay τ=2 is numerically examined for the single bifurcation parameter b. The bifurcation parameter b varies in the range 1≤b≤5. Five largest Lyapunov exponents λ1,2,3,4,5 of all the spectrum versus the bifurcation parameter b is shown in Figure 4(a), which reveals that,(1)
in the region1≤b≤2.1, the equilibrium x∗ is stable(2)
in the region2.11≤b≤2.23, chaotic and hyperchaotic regime appears alternately(3)
in the region2.24≤b≤3, there is one positive LE (λ1>0) meaning that the chaos has appeared(4)
in the region 3.1 ≤ b ≤ 5, there are two positive LE (λ1>0 and λ2>0) meaning that hyperchaos appearsFigure 4
(a) The five largest Lyapunov exponentsλ1,2,3,4,5 plotted versus the bifurcation parameter 1≤b≤5 for parameter a=1 and time delay τ=2. (b) The bifurcation diagram for the same parameter as in case (a).
(a)
(b)These regions of nonchaos and chaos observed in Figure4(a) are best visualized in the bifurcation diagram, where we plot the maxima of xt versus the bifurcation parameter b, as shown in Figure 4(b). The outcomes of this variation are shown in Figure 5.(i)
Forb=1.24, we get q=0.3554. x∗ is stable in view of Case 3 of Theorem 1. And the solution converges to x∗=1.22535, as shown in Figure 5(a).(ii)
Forb=1.92, we get q=−2.1159. In view of Case 1 of Theorem 1, the stability region is located between τ=0 and τ0=1.1064. By setting τ=2, the equilibrium is found unstable leading to a limit cycle, as depicted in Figure 5(b).(iii)
Forb=2.52 and b=4.57, we get q=4.7286 and q=8.8743. By Case 2 of Theorem 1, we observe a chaotic attractor in Figure 5(c) and a hyperchaotic attractor in Figure 5(d), respectively.Figure 5
The initial condition isϕt=1 for t∈−τ,0 throughout the investigation. Parameter a = 1, and time delay τ=2. (a) Time series for b=1.24, the fixed point is stable, (b) period limit cycles for b=1.92, (c) chaotic attractors for b=2.52, and (d) hyperchaotic attractors for b=4.57.
(a)
(b)
(c)
(d)


### 4.3. Varying Time Delayτ for a=1 and b=4

System (2) with the parameter a=1 and b=4 is numerically examined for the time delay τ. The time delay τ varies in the range 0.1≤τ≤20. Five largest Lyapunov exponents λ1,2,3,4,5 of all the spectrum versus the time delay τ is shown in Figure 6(a), which reveals that there is a narrow region of chaos (λ1>0) between 0.48≤τ≤0.54 and 0.7≤τ≤1.33. When τ is starting at about 1.45, there is already a permanent hyperchaos (λ1,λ2>0) range.Figure 6
(a) The five largest Lyapunov exponentsλ1,2,3,4,5 plotted versus the time delay 0.1≤τ≤20 for parameters a=1 and b=4. (b) The bifurcation diagram for the same parameter as in case (a).
(a)
(b)These regions of nonchaos and chaos observed in Figure6(a) are best visualized in the bifurcation diagram, where we plot the maxima of xt versus the time delay τ, as shown in Figure 6(b).Fora=1 and b=4, we get x∗=1.853 and q=bg′x∗=−7.3430, then x∗ is stable in view of Case 1 of Theorem 1 and the stability region is located between 0≤τ≤0.2347. The outcomes of this variation are shown in Figure 7.(i)
Forτ=0.2<τ0, x∗ is stable and the solution converges to x∗=1.853, as shown in Figure 7(a)(ii)
Chaos attractors forτ=0.5 and τ=0.97 are shown in Figures 7(b) and 7(c)(iii)
Atτ=2.3 a hyperchaotic attractor is exhibited in Figure 7(d)Figure 7
The initial condition isϕt=1 for t∈−τ,0 throughout the investigation. Parameters a = 1 and b=4. (a) Period limit cycle for τ=0.2, (b and c) chaos attractors for τ=0.5 and τ=0.97, and (d) multiscroll hyperchaotic attractors for τ=2.3.
(a)
(b)
(c)
(d)


## 4.1. Varyinga for b=4 and Time Delay τ=2

System (2) with parameter b=4 and the time delay τ=2 is numerically examined for the single bifurcation parameter a. The bifurcation parameter a varies in the range 0.1≤a≤3. Five largest Lyapunov exponents λ1,2,3,4,5 of all the spectrum versus the bifurcation parameter a is shown in Figure 2(a), which reveals that,(1)
in the regiona∈2.18,3, the equilibrium point x∗ is stable(2)
in the regiona∈1.9,2.17, there is one positive LE (λ1>0) meaning that chaos appears(3)
in the regiona∈1.44,1.89, chaotic and hyperchaotic regime appears alternately(4)
in the region a ∈ [0.1, 1.43], there are two positive LE (λ1 > 0 and λ2 > 0) meaning that hyperchaos appearsFigure 2
(a) The five largest Lyapunov exponentsλ1,2,3,4,5 plotted versus the bifurcation parameter −3≤−a≤−0.1 for parameter b=4 and time delay τ=2. (b) The bifurcation diagram for the same parameter as in case (a).
(a)
(b)These regions of nonchaos and chaos observed in Figure2(a) are best visualized in the bifurcation diagram, where we plot the maxima of xt versus the bifurcation parameter a, as shown in Figure 2(b). A good consistency can be noted between the bifurcation diagram and the LES. The outcomes of this variation are shown in Figure 3.(i)
Fora=2.67, we get equilibrium point x∗=1.44555 and q=bg′x∗=−2.0435. By Case 3 of Theorem 1, the time series xt converges to x∗, as shown in Figure 3(a).(ii)
Fora=2.35, we get equilibrium point x∗=1.53665 and q=bg′x∗=−3.4094. Therefore, in view of Case 1 of Theorem 1, the stability region is located between τ=0 and τ0=0.7564. By setting τ=2, the equilibrium is found unstable leading to a limit cycle, as depicted in Figure 3(b).(iii)
Fora=2.02, we get q=bg′x∗=−4.6290. Therefore, in view of Case 1 of Theorem 1, the stability region is located between τ=0 and τ0=0.4294. By setting τ=2, the equilibrium is found unstable leading to a chaotic attractor, as depicted in Figure 3(c).(iv)
Fora=1.57, a=0.61, and a=0.12, we get q=7.5144, q=7.8277, and q=7.9023. In view of Case 2 of Theorem 1, x∗ is unstable for any τ≥0. We can obtain hyperchaotic attractors (shown in Figures 3(d) and 3(e)) and a multiscroll hyperchaotic attractor (Figure 3(f)), respectively.Figure 3
The initial condition isϕt=1 for t∈−τ,0 throughout the investigation. Parameter b = 4, and time delay τ=2. (a) Time series for a=2.67, the fixed point is stable. Phase plane plots in xt−τ−xt plane for variable a, (b) period limit cycle for a=2.35, (c) chaotic attractor for a=2.02, (d and e) hyperchaotic attractor for a=1.57 and a=0.61, (f) multiscroll hyperchaotic attractor for a=0.12.
(a)
(b)
(c)
(d)
(e)
(f)


## 4.2. Varyingb for a=1 and Time Delay τ=2

System (2) with the parameter a=1 and the time delay τ=2 is numerically examined for the single bifurcation parameter b. The bifurcation parameter b varies in the range 1≤b≤5. Five largest Lyapunov exponents λ1,2,3,4,5 of all the spectrum versus the bifurcation parameter b is shown in Figure 4(a), which reveals that,(1)
in the region1≤b≤2.1, the equilibrium x∗ is stable(2)
in the region2.11≤b≤2.23, chaotic and hyperchaotic regime appears alternately(3)
in the region2.24≤b≤3, there is one positive LE (λ1>0) meaning that the chaos has appeared(4)
in the region 3.1 ≤ b ≤ 5, there are two positive LE (λ1>0 and λ2>0) meaning that hyperchaos appearsFigure 4
(a) The five largest Lyapunov exponentsλ1,2,3,4,5 plotted versus the bifurcation parameter 1≤b≤5 for parameter a=1 and time delay τ=2. (b) The bifurcation diagram for the same parameter as in case (a).
(a)
(b)These regions of nonchaos and chaos observed in Figure4(a) are best visualized in the bifurcation diagram, where we plot the maxima of xt versus the bifurcation parameter b, as shown in Figure 4(b). The outcomes of this variation are shown in Figure 5.(i)
Forb=1.24, we get q=0.3554. x∗ is stable in view of Case 3 of Theorem 1. And the solution converges to x∗=1.22535, as shown in Figure 5(a).(ii)
Forb=1.92, we get q=−2.1159. In view of Case 1 of Theorem 1, the stability region is located between τ=0 and τ0=1.1064. By setting τ=2, the equilibrium is found unstable leading to a limit cycle, as depicted in Figure 5(b).(iii)
Forb=2.52 and b=4.57, we get q=4.7286 and q=8.8743. By Case 2 of Theorem 1, we observe a chaotic attractor in Figure 5(c) and a hyperchaotic attractor in Figure 5(d), respectively.Figure 5
The initial condition isϕt=1 for t∈−τ,0 throughout the investigation. Parameter a = 1, and time delay τ=2. (a) Time series for b=1.24, the fixed point is stable, (b) period limit cycles for b=1.92, (c) chaotic attractors for b=2.52, and (d) hyperchaotic attractors for b=4.57.
(a)
(b)
(c)
(d)


## 4.3. Varying Time Delayτ for a=1 and b=4

System (2) with the parameter a=1 and b=4 is numerically examined for the time delay τ. The time delay τ varies in the range 0.1≤τ≤20. Five largest Lyapunov exponents λ1,2,3,4,5 of all the spectrum versus the time delay τ is shown in Figure 6(a), which reveals that there is a narrow region of chaos (λ1>0) between 0.48≤τ≤0.54 and 0.7≤τ≤1.33. When τ is starting at about 1.45, there is already a permanent hyperchaos (λ1,λ2>0) range.Figure 6
(a) The five largest Lyapunov exponentsλ1,2,3,4,5 plotted versus the time delay 0.1≤τ≤20 for parameters a=1 and b=4. (b) The bifurcation diagram for the same parameter as in case (a).
(a)
(b)These regions of nonchaos and chaos observed in Figure6(a) are best visualized in the bifurcation diagram, where we plot the maxima of xt versus the time delay τ, as shown in Figure 6(b).Fora=1 and b=4, we get x∗=1.853 and q=bg′x∗=−7.3430, then x∗ is stable in view of Case 1 of Theorem 1 and the stability region is located between 0≤τ≤0.2347. The outcomes of this variation are shown in Figure 7.(i)
Forτ=0.2<τ0, x∗ is stable and the solution converges to x∗=1.853, as shown in Figure 7(a)(ii)
Chaos attractors forτ=0.5 and τ=0.97 are shown in Figures 7(b) and 7(c)(iii)
Atτ=2.3 a hyperchaotic attractor is exhibited in Figure 7(d)Figure 7
The initial condition isϕt=1 for t∈−τ,0 throughout the investigation. Parameters a = 1 and b=4. (a) Period limit cycle for τ=0.2, (b and c) chaos attractors for τ=0.5 and τ=0.97, and (d) multiscroll hyperchaotic attractors for τ=2.3.
(a)
(b)
(c)
(d)


## 5. Performance Evaluations

To exhibit the effect of the ACCM in enhancing the chaotic dynamic characteristics of the Ikeda TD system, we compare the chaos and hyperchaos characteristics of the improved system with that of the seed Ikeda TD system. The systems are integrated with a fourth-order Runge–Kutta algorithm, with a fixed time step size equal toh=0.01. Two-parameter (2D) Lyapunov exponent diagrams are employed to explore the dynamics of the systems over the whole three-dimensional (3D) parameter space (given by a, b, and τ).


### 5.1. 2D Largest Lyapunov Exponent Diagrams

The chaotic dynamical behaviors of the ACCM-Ikeda and seed Ikeda TD systems over the wholea,b, a,τ, and b,τ parameter space are presented in this section. All of these 2D parameter space diagrams are obtained by considering the LLE value. The results for a cut of this 3D space at τ=2, b=4, and a=1 are shown in Figures 8–10. The white and faint yellow zones (noted by I) indicate the chaotic dynamics of the system λ1>0. In Figure 8(a), some stripes corresponding to nonchaotic regions embedded in the chaos region can be found. In Figure 8(b), we can find a larger connected chaos region. The ACCM-Ikeda TD system has larger connected chaos region in the a,b space. And from Figures 9 and 10, we can get the same results. The ACCM-Ikeda TD system has larger connected chaos region over the whole a,b, a,τ, and b,τ parameter space. Moreover, it is noteworthy that with increasing b smaller time delay of the ACCM-Ikeda TD system is required to produce chaos (Figure 10(b)).Figure 8
(a) The 2D LE diagram showing the largest Lyapunov exponentλ1 of the Ikeda TD system in the a−b space. (b) The 2D LE diagram showing the largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−b space.
(a)
(b)Figure 9
(a) The 2D LE diagram showing the largest Lyapunov exponentλ1 of the Ikeda TD system in the a−τ space. (b) The 2D LE diagram showing the largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−τ space.
(a)
(b)Figure 10
(a) The 2D LE diagram showing the largest Lyapunov exponentλ1 of the Ikeda TD system in the b−τ space. (b) The 2D LE diagram showing the largest Lyapunov exponent of the ACCM-Ikeda TD system in the b−τ space.
(a)
(b)


### 5.2. 2D Second Largest Lyapunov Exponent

The hyperchaotic dynamical behaviors of the ACCM-Ikeda and seed Ikeda TD systems over the wholea,b, a,τ, and b,τ parameter space are presented in this section. Hyperchaotic dynamical behavior is studied by using the second largest Lyapunov exponent (LE2) as indicators. As shown in Figures 11–13, the white and faint yellow zones (noted by I) indicate the hyperchaotic states of the system (λ2>0 and λ1>0, in Figures 8–10). Clearly in Figures 11–13, the ACCM-Ikeda TD system has larger connected hyperchaotic regions in 2D parameter spaces a,b, a,τ, and b,τ. With parameters −a,b, and time delay τ increase, λ2 of the enhanced ACCM-Ikeda TD system become larger and positive when λ2 of the original Ikeda system is still negative. As such, the ACCM model can transform the chaotic oscillator to hyperchaotic oscillations.Figure 11
(a) The 2D LE diagram showing the second largest Lyapunov exponentλ2 of the Ikeda TD system in the a−b space. (b) The 2D LE diagram showing the second largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−b space.
(a)
(b)Figure 12
(a) The 2D LE diagram showing the second largest Lyapunov exponentλ2 of the Ikeda TD system in the a−τ space. (b) The 2D LE diagram showing the second largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−τ space.
(a)
(b)Figure 13
(a) The 2D LE diagram showing the second largest Lyapunov exponentλ2 of the Ikeda TD system in the b−τ space. (b) The 2D LE diagram showing the second largest Lyapunov exponent of the ACCM-Ikeda TD system in the b−τ space.
(a)
(b)


### 5.3. Largest Lyapunov Exponent

Moreover, the complexity of the two systems is compared by calculating the single parameter Largest Lyapunov Exponent. Lyapunov Exponent can describe the average separation rate of trajectories starting from two extremely close initial states. A positive LE means that the two adjacent trajectories of a dynamical system exponentially separate in each iteration. A dynamical system with a positive LE is regarded as chaotic, and larger LE represents higher dynamical complexity. Figures14(a)–14(c) compare the largest Lyapunov exponents (LLEs) of the ACCM-Ikeda TD system and its associated seed Ikeda system. The largest Lyapunov exponents (LLEs) curve with the change of parameters a, b, and time delay τ, as shown in Figures 11(a) and 11(b), respectively. As can be observed from these figures that the ACCM-Ikeda TD system has larger LLEs, it proves that ACCM can enhance the chaos complexity of the original Ikeda system.Figure 14
The first largest LEs comparisons of the ACCM-Ikeda TD system (2) and the seed Ikeda TD system (1): (a) LLE λ1 comparison for bifurcation parameter a; (b) LLE λ1 comparison for bifurcation parameter b; (c) LLE λ1 comparison for bifurcation parameter τ.
(a)
(b)
(c)


## 5.1. 2D Largest Lyapunov Exponent Diagrams

The chaotic dynamical behaviors of the ACCM-Ikeda and seed Ikeda TD systems over the wholea,b, a,τ, and b,τ parameter space are presented in this section. All of these 2D parameter space diagrams are obtained by considering the LLE value. The results for a cut of this 3D space at τ=2, b=4, and a=1 are shown in Figures 8–10. The white and faint yellow zones (noted by I) indicate the chaotic dynamics of the system λ1>0. In Figure 8(a), some stripes corresponding to nonchaotic regions embedded in the chaos region can be found. In Figure 8(b), we can find a larger connected chaos region. The ACCM-Ikeda TD system has larger connected chaos region in the a,b space. And from Figures 9 and 10, we can get the same results. The ACCM-Ikeda TD system has larger connected chaos region over the whole a,b, a,τ, and b,τ parameter space. Moreover, it is noteworthy that with increasing b smaller time delay of the ACCM-Ikeda TD system is required to produce chaos (Figure 10(b)).Figure 8
(a) The 2D LE diagram showing the largest Lyapunov exponentλ1 of the Ikeda TD system in the a−b space. (b) The 2D LE diagram showing the largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−b space.
(a)
(b)Figure 9
(a) The 2D LE diagram showing the largest Lyapunov exponentλ1 of the Ikeda TD system in the a−τ space. (b) The 2D LE diagram showing the largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−τ space.
(a)
(b)Figure 10
(a) The 2D LE diagram showing the largest Lyapunov exponentλ1 of the Ikeda TD system in the b−τ space. (b) The 2D LE diagram showing the largest Lyapunov exponent of the ACCM-Ikeda TD system in the b−τ space.
(a)
(b)


## 5.2. 2D Second Largest Lyapunov Exponent

The hyperchaotic dynamical behaviors of the ACCM-Ikeda and seed Ikeda TD systems over the wholea,b, a,τ, and b,τ parameter space are presented in this section. Hyperchaotic dynamical behavior is studied by using the second largest Lyapunov exponent (LE2) as indicators. As shown in Figures 11–13, the white and faint yellow zones (noted by I) indicate the hyperchaotic states of the system (λ2>0 and λ1>0, in Figures 8–10). Clearly in Figures 11–13, the ACCM-Ikeda TD system has larger connected hyperchaotic regions in 2D parameter spaces a,b, a,τ, and b,τ. With parameters −a,b, and time delay τ increase, λ2 of the enhanced ACCM-Ikeda TD system become larger and positive when λ2 of the original Ikeda system is still negative. As such, the ACCM model can transform the chaotic oscillator to hyperchaotic oscillations.Figure 11
(a) The 2D LE diagram showing the second largest Lyapunov exponentλ2 of the Ikeda TD system in the a−b space. (b) The 2D LE diagram showing the second largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−b space.
(a)
(b)Figure 12
(a) The 2D LE diagram showing the second largest Lyapunov exponentλ2 of the Ikeda TD system in the a−τ space. (b) The 2D LE diagram showing the second largest Lyapunov exponent of the ACCM-Ikeda TD system in the a−τ space.
(a)
(b)Figure 13
(a) The 2D LE diagram showing the second largest Lyapunov exponentλ2 of the Ikeda TD system in the b−τ space. (b) The 2D LE diagram showing the second largest Lyapunov exponent of the ACCM-Ikeda TD system in the b−τ space.
(a)
(b)


## 5.3. Largest Lyapunov Exponent

Moreover, the complexity of the two systems is compared by calculating the single parameter Largest Lyapunov Exponent. Lyapunov Exponent can describe the average separation rate of trajectories starting from two extremely close initial states. A positive LE means that the two adjacent trajectories of a dynamical system exponentially separate in each iteration. A dynamical system with a positive LE is regarded as chaotic, and larger LE represents higher dynamical complexity. Figures14(a)–14(c) compare the largest Lyapunov exponents (LLEs) of the ACCM-Ikeda TD system and its associated seed Ikeda system. The largest Lyapunov exponents (LLEs) curve with the change of parameters a, b, and time delay τ, as shown in Figures 11(a) and 11(b), respectively. As can be observed from these figures that the ACCM-Ikeda TD system has larger LLEs, it proves that ACCM can enhance the chaos complexity of the original Ikeda system.Figure 14
The first largest LEs comparisons of the ACCM-Ikeda TD system (2) and the seed Ikeda TD system (1): (a) LLE λ1 comparison for bifurcation parameter a; (b) LLE λ1 comparison for bifurcation parameter b; (c) LLE λ1 comparison for bifurcation parameter τ.
(a)
(b)
(c)


## 6. Conclusion

In this paper, a new Ikeda time delay system with asymmetric central contraction mutation (ACCM) is proposed and examined. First, we have analyzed the stability and Hopf bifurcation of the system. The results for different types of stability regions are derived. The conditions on the partial derivatives of functionf at equilibrium points are provided theoretically. The Hopf bifurcation value of delay τ is also mentioned. Next, the dynamical analysis of the system is carried out by visualizing the bifurcation diagram and spectrum of the first five LEs as a function of system parameters a, b, and the time delay τ. In a large range of parameters, rich dynamical behaviors including asymptotic stability, limit cycles, chaos, hyperchaos, and multiscroll chaos are observed. Finally, we have compared the chaotic dynamics of the ACCM-Ikeda TD system and those of the seed Ikeda TD system. Two-parameter (2D) LE diagrams are employed to explore the chaotic dynamics of the system over the whole parameter space. Complexity of the systems is characterized by LLE as a function of single parameter. The simulation results show that the ACCM-Ikeda TD system has higher complexity and larger chaotic and hyperchaotic parameter zones. There is no transient nonchaos window, and the fragile chaos phenomenon is successfully suppressed. Moreover, chaotic regime can be transformed to hyperchaotic regime by introducing the ACCM strategy. As a theoretical extension of the Ikeda TD system, the proposed scheme could be realized by the combination of electrical and optical devices according to some existing research studies [28, 36], and the ACCM-Ikeda TD system has the potential to be used in real world applications such as secure communications and random number generation.



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*Source: 2941835-2019-12-16.xml*