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A new chaotic behavior of a general model of the Henon map
Advances in Difference Equations volume 2014, Article number: 107 (2014)
Abstract
In this paper we are concerned with a general form of the Henon map as a retarded functional equation. The existence of a unique solution is proved. The continuous dependence of the solution and the local stability of fixed points are investigated. Chaos, bifurcation and chaotic attractor of the resulting system are discussed. In addition, we compare our results with the discrete dynamical system of the Henon map.
1 Introduction
Discontinuous (sectionally continuous) dynamical systems have been defined as a problem of retarded functional equation and studied in [1–10]. The generalized timedelayed Henon map was introduced in [11, 12]. In this work we study the discontinuous (sectionally continuous) dynamical system of the Henon map as a problem of retarded functional equation with two different delays
with
where \alpha >0 and \beta <1.
The existence of a unique continuous dependence solution is proved. The local stability of fixed points is studied. The chaos, bifurcation and chaotic attractor are discussed. Comparison with the corresponding discrete dynamical system of the Henon map
is given.
Let f:[0,T]\times {\mathbb{R}}^{n}\to \mathbb{R} and {r}_{1},{r}_{2},\dots ,{r}_{k}\in {\mathbb{R}}^{+}.
Consider the problem of retarded functional equation
with the initial condition
If T is a positive integer, {r}_{k}=k, \varphi (0)={x}_{0}, and t=n=1,2,3,\dots , then problem (1.4)(1.5) will be the discrete dynamical system
This shows that discrete dynamical system (1.6)(1.7) is a special case of the problem of retarded functional equation (1.4)(1.5).
Consider also the singularly perturbed differential difference equation [13]
and the singularly perturbed delay differential equation [14]
{m}_{j}\ge 0, {m}_{j}\in \mathbb{Z}, j=1,\dots ,k.
The limiting cases as \u03f5\to 0 of (1.8) and (1.7) are special cases of retarded functional equation (1.4)(1.5).
Let t\in (0,r], then tr\in (r,0], the solution of (1.4)(1.5) (by the method of steps as in [13–16]) is given by
For t\in (r,2r], then tr\in (0,r], the solution of (1.4)(1.5) is given by
Repeating the process we can easily deduce the solution of (1.4)(1.5) which is given by
which is continuous on each subinterval ((k1)r,kr), k=1,2,3,\dots ,n, but
which implies that the solution of problem (1.4)(1.5) is discontinuous (sectionally continuous) on (0,T].
Now we have the following definitions.
Definition 1 The discontinuous (sectionally continuous) dynamical system is the problem of retarded functional equation (1.4)(1.5).
Definition 2 The fixed points of discontinuous (sectionally continuous) dynamical system (1.4)(1.5) are the solution of the equation
Remark 1 We should notice that the difference equations representing the Henon map in its different cases,
are just special cases of our problem (1.1)(1.2).
2 Existence and uniqueness
Now consider the discontinuous (sectionally continuous) dynamical system of the Henon map (1.1)(1.2). The existence of a unique solution as well as the continuous dependence of the solution on the initial data are proved. We study also the continuous dependence of the solution on the parameter α.
Let {L}^{1}={L}^{1}[0,T], T<\mathrm{\infty}, be the class of Lebesgue integrable functions on [0,T] with the norm
Let D=\{x\in \mathbb{R}:x<k,k=\frac{1}{1\beta }\}\subset {L}^{1}[0,T].
Definition 3 By a solution of problem (1.1)(1.2) we mean that the function x\in {L}^{1} satisfies problem (1.1)(1.2).
Theorem 1 The sufficient condition for the existence of a unique solution of problem (1.1)(1.2) is \beta +2\alpha k<1.
Proof Define the operator F:{L}^{1}\to {L}^{1} on D by
then
This proves that F:D\to D.
Now, for x,y\in D, we have
Thus we can get
If M=\beta +2\alpha k<1, then
So, problem (1.1)(1.2) has, on D, a unique solution x\in {L}^{1}. □
Continuous dependence on the initial conditions
Theorem 2 The solution of discontinuous (sectionally continuous) dynamical system (1.1)(1.2) is continuously dependent on the initial data.
Proof Let x(t) and {x}^{\ast}(t) be the solutions of dynamical system (1.1)(1.2) and the dynamical system of equation (1.1) with the initial data
Then
and we can get
This implies that
That is,
□
Continuous dependence on the parameter α
Theorem 3 The solution of discontinuous (sectionally continuous) dynamical system (1.1)(1.2) is continuously dependent on the parameter α.
Proof Let x(t) and {x}^{\ast}(t) be the solutions of dynamical system (1.1)(1.2) and the dynamical system
with the initial data (1.2), then
which gives
This implies that
That is,
□
3 Fixed points and stability
Exactly like its discrete counter part, dynamical system (1.1)(1.2) has two fixed points which are the solutions of the equation
So, we have
Obviously, they exist only for {(1\beta )}^{2}+4\alpha \ge 0 [15]. To determine the stability of a fixed point, consider a small perturbation from the fixed point by letting
Substituting in (2.2) we get
which implies that the fixed points are asymptotically stable if all roots of the equation
satisfy \lambda <1, where x(t)={x}_{\mathrm{fix}}+\u03f5{\lambda}^{t}. Here we study three cases:

{r}_{1}={r}_{2} then {({x}_{\mathrm{fix}})}_{1} is stable if {(1\beta )}^{2}+4\alpha <4.

{r}_{2}=2{r}_{1} then {({x}_{\mathrm{fix}})}_{1} is stable if 2\beta +4\alpha <3.

{r}_{1}=2{r}_{2} then {({x}_{\mathrm{fix}})}_{1} is stable if \alpha <\frac{3}{4}{(1\beta )}^{2}.
In all the simulations, {r}_{1} and {r}_{2} are rationally dependent.
Figure 1 illustrates the trajectories of (1.3), while Figure 2 illustrates the trajectories of (1.1).
4 Bifurcation and chaos
In this section we show, by numerical experiments illustrated by bifurcation diagrams, that the dynamical behavior of discontinuous (sectionally continuous) dynamical system (1.1)(1.2) is completely affected by the change in both r and T [17]. We consider three cases for different delays {r}_{1} and {r}_{2} as follows.

Case 1: {r}_{1}>{r}_{2}.
Let \beta =0.3 be fixed and vary α from 0 to 1.4 with step size 0.001 and the initial condition ({x}_{0},{y}_{0})=(0.3,0).
Take {r}_{1}=2 and {r}_{2}=1 and t\in [0,150] in (1.1)(1.2) (Figure 3).
Take {r}_{1}=0.50 and {r}_{2}=0.25 and t\in [0,38] in (1.1)(1.2) (Figure 4).
Take {r}_{1}=0.3 and {r}_{2}=0.1 and t\in [0,20] in (1.1)(1.2) (Figure 5).
Take {r}_{1}=0.25 and {r}_{2}=0.15 and t\in [0,15] in (1.1)(1.2) (Figure 6).
We see clearly in Figure 3 the bifurcation from a stable fixed point to a stable orbit of period two at \alpha =0.4, and then the bifurcation from period two to period four at \alpha =0.9. The further period doubling occurs at decreasing increments in α, and the orbit becomes chaotic for \alpha \simeq 1.1.

Case 2: {r}_{1}={r}_{2}.
Take {r}_{1}={r}_{2}=1 and t\in [0,100] in (1.1)(1.2) (Figure 7).
Take {r}_{1}={r}_{2}=2 and t\in [0,200] in (1.1)(1.2) (Figure 8).
Take {r}_{1}={r}_{2}=0.1 and t\in [0,10] in (1.1)(1.2) (Figure 9).
Take {r}_{1}={r}_{2}=0.2 and t\in [0,15] in (1.1)(1.2) (Figure 10).

Case 3: {r}_{1}<{r}_{2}.
Take {r}_{1}=1 and {r}_{2}=2 and t\in [0,200] in (1.1)(1.2) (Figure 11).
Take {r}_{1}=0.1 and {r}_{2}=0.2 and t\in [0,20] in (1.1)(1.2) (Figure 12).
Take {r}_{1}=0.25 and {r}_{2}=0.75 and t\in [0,30] in (1.1)(1.2) (Figure 13).
Take {r}_{1}=0.15 and {r}_{2}=0.25 and t\in [0,50] in (1.1)(1.2) (Figure 14).
5 Chaotic attractor
In this section we are interested in studying the chaotic attractor for three different cases.

Case 1: {r}_{2}>{r}_{1}.
Here we rewrite system (1.1)(1.2) as follows:
It is worth here to mention what we get when we plot the chaotic attractor for system (5.1)(5.2) in this case. Figure 15 shows the chaotic attractor when {r}_{1}=2 and {r}_{2}=1, while Figure 16 shows the chaotic attractor of the same when {r}_{1}=0.25 and {r}_{2}=0.15.

Case 2: {r}_{1}={r}_{2}=r.
Here system (1.1)(1.2) is rewritten as
with
In this case, the chaotic attractor for {r}_{1}={r}_{2}=1 and {r}_{1}={r}_{2}=0.1 looks like in Figures 17 and 18.

Case 3: {r}_{1}>{r}_{2}.
Here we also rewrite system (1.1)(1.2) as follows:
Here we show the chaotic attractor for system (5.5)(5.6). Figure 19 shows the chaotic attractor when {r}_{1}=1 and {r}_{2}=2, while Figure 20 shows the chaotic attractor of the same system but with {r}_{1}=0.15 and {r}_{2}=0.25.
Since the Lyapunov exponent is a good indicator for the existence of chaos [18–21], we compute the Lyapunov characteristic exponents (LCEs) via the Householder QRbased methods described in [22]. LCEs play a key role in the study of nonlinear dynamical systems, and they are a measure of sensitivity of solutions of a given dynamical system to small changes in the initial conditions. One feature of chaos is sensitive dependence on initial conditions; for a chaotic dynamical system, at least one LCE must be positive. Since for nonchaotic systems all LCEs are nonpositive, the presence of a positive LCE has often been used to help determine if a system is chaotic or not. Figure 21 shows the LCEs for system (1.1)(1.2) in the case {r}_{1}>{r}_{2} for \beta =0.3 with the initial conditions ({x}_{0},{y}_{0})=(0,0). With these parameter values, we find that \mathrm{LCE}1=0.3228 and \mathrm{LCE}2=1.2461. While Figure 22 shows the LCEs for the same system in the case {r}_{1}<{r}_{2} for \beta =0.5 with the same initial conditions, we find that \mathrm{LCE}1=0.1318 and \mathrm{LCE}2=0.8153. Finally, Figure 23 shows the LCEs for system (1.1)(1.2) in the case {r}_{1}={r}_{2} for parameter values \beta =0.6 with the same initial conditions. We find that \mathrm{LCE}1=0.3228 and \mathrm{LCE}2=1.2460.
6 Conclusion
The discontinuous (sectionally continuous) dynamical system of the Henon map describes dynamical properties for different values of the parameters {r}_{1},{r}_{2}\in {\mathbb{R}}^{+} when the time t\in [0,T] is continuous. Indeed, the stability of fixed points depends on the values of delay parameters {r}_{1} and {r}_{2} as we have seen. The bifurcation diagrams, as well, depend on the values of delay parameters {r}_{1} and {r}_{2} and the time interval [0,T]. We have also noticed that the chaotic attractor of the discontinuous (sectionally continuous) Henon system in its different versions is also affected by the change in {r}_{1}, {r}_{2} and the time interval [0,T]. On the other hand, from Figures 34, 78, and 1112 it looks like there is a scale that gives identical chaotic behavior. To summarize, our analytical result (3.1) agrees with the numerical simulations.
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ElSayed, A.M.A., ElRaheem, Z.F.E. & Salman, S.M. A new chaotic behavior of a general model of the Henon map. Adv Differ Equ 2014, 107 (2014). https://doi.org/10.1186/168718472014107
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DOI: https://doi.org/10.1186/168718472014107
Keywords
 retarded functional equation
 Henon map
 fixed points
 existence
 uniqueness
 bifurcation
 chaos
 chaotic attractor