Discrete Dynamics in Nature and Society

Volume 2015, Article ID 270604, 18 pages

http://dx.doi.org/10.1155/2015/270604

## Complex Dynamics in Generalized Hénon Map

Institute of Mathematics and Physics, Central South University of Forestry and Technology, Changsha, Hunan 410004, China

Received 19 November 2014; Accepted 10 February 2015

Academic Editor: Viktor Avrutin

Copyright © 2015 Meixiang Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The complex dynamics of generalized Hénon map with nonconstant Jacobian determinant are investigated. The conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory and checked up by numerical simulations. Chaos in the sense of Marotto's definition is proved by analytical and numerical methods. The numerical simulations show the consistence with the theoretical analysis and reveal some new complex phenomena which can not be given by theoretical analysis, such as the invariant cycles which are irregular closed graphics, the six and forty-one coexisting invariant cycles, and the two, six, seven, nine, ten, and thirteen coexisting chaotic attractors, and some kinds of strange chaotic attractors.

#### 1. Introduction

The planar mapwas first introduced by Hénon [1] as a planar diffeomorphism that imitated essential stretching and folding properties of the Poincaré map of the Lorenz system. This original Hénon map (1) had a strange attractor with fractal structure and had constant Jacobian determinant . Since the late 1970s, Hénon map (1) served as an important but artificial example to illustrate many analytical results and numerical techniques of dynamical theory. For example, Feit [2] introduced the characteristic exponents in order to estimate strange attractors numerically. Marotto [3] proved analytically that the map had a transversal homoclinic orbit, which implied the existence of the chaotic behavior for some parameter values. Curry [4] presented a lot of numerical experiments on Hénon map by using the characteristic exponent, frequency spectrum, and a theorem of Smale [5]. Mora and Viana [6] proposed the more general theory of strange attractors. Sonis [7] presented a detailed description of bifurcation phenomena by using the analysis approach. Cao [8] proved that there exists a set with positive Lebesgue measure, which corresponded to a map possessing a strange attractor. Luo and Guo [9] investigated the complete bifurcation and stability of the stable and unstable periodic solutions and the chaotic layers by introducing the positive and negative iterative mappings. Zhang [10] verified the existence of Wada basin boundaries in a switched Hénon map.

For the measure preserving Hénon map (), there has been some attention. Such as, Brown [11] proved that the measure preserving Hénon map contained an embedded horseshoe for by using geometric methods and a contradiction argument. Kirchgraber and Stoffer [12] proved this Hénon map existing a transversal homoclinic point for a set of parameters which was not small by using shadowing techniques. Jensen [13] proved that the unstable manifold of a hyperbolic fixed point was the iterated limit of a very simple set.

In this paper, we study the following extension of map (1):where , , and are constants. Following [14], we call this map the generalized Hénon map (GHM), which has nonconstant Jacobian determinant .

Our motivation to study this particular generalized Hénon map is that it appears in the bifurcation analysis of nontransversal homoclinic orbits and heteroclinic cycles and plays an important role in other homoclinic studies. In [15], if a diffeomorphism in has two saddle fixed points connected by two heteroclinic orbits and one of which is nontransversal, then the GHM appears as a rescaled first return map. In [16], If a diffeomorphism in has a codimension 1 homoclinic tangency to a saddle-focus fixed point, the GHM appears when the eigenvalues satisfy some conditions. Moreover, when the so-called effective dimension of the problem in [17] can change, the GHM also can be expected as a rescaled first-return map in other cases of homoclinic and heteroclinic tangencies.

In this paper, the bifurcations and chaos phenomenons in map (2) are investigated. The conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory [18]; chaotic behavior in the sense of Marotto’s definition [19] is proved. And numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting dynamics behaviors.

This paper is organized as follows. In Section 2, the existence and stability of the fixed points are given. In Section 3, the sufficient conditions of existence for fold bifurcation, flip bifurcation, and Hopf bifurcation are given. In Section 4, we first rigorously prove the existence of chaos in the sense of Marotto’s definition and then give an example to check up the analytic results. The numerical simulations are presented in Section 5 to verify the theoretical analysis and display the new and interesting dynamics. Finally, we give a conclusion in Section 6.

#### 2. Existence and Stability of Fixed Points

The fixed points of map (2) satisfy the following equations:

By a simple analysis about (3), it is easy to obtain the following proposition.

Proposition 1. *(i) If , map (2) has a unique fixed point at ,(ii)if , map (2) has two fixed points at and ,(iii)if , map (2) has three fixed points at , , and , where + +.*

*The Jacobian matrix of map (2) evaluated at fixed point is given by The characteristic equation of the Jacobian matrix can be written as where and .*

*A simple calculation shows the stability of the fixed points as the following proposition.*

*Proposition 2. The fixed point of map (2) is stable if one of the following conditions holds:(i);(ii) and and .*

*3. Bifurcations*

*3. Bifurcations**In analysis of the fold bifurcation, flip bifurcation, and Hopf bifurcation of map (2), is used as the bifurcation parameter. By using center manifold theorems and bifurcation theory, the conditions of existences for the bifurcations are given as follows.*

*Theorem 3. Map (2) undergoes a fold bifurcation at if the following conditions are satisfied at the same time:Moreover, if (resp., ), the two fixed points bifurcate from for (resp., ) and coalesce as the fixed point at and disappear for (resp., ) (see Appendix A for the proof).*

*Theorem 4. Map (2) undergoes a flip bifurcation at if the following conditions are satisfied at the same time:Moreover, if (resp., ), the period-2 points that bifurcate from this fixed point are stable (resp., unstable) (see Appendix B for the proof and expressions of and ).*

*Theorem 5. Map (2) undergoes a Hopf bifurcation at if the following conditions are satisfied at the same time:Moreover, if (resp., ) and ; then, an attracting (resp., repelling) invariant closed curve bifurcates from the fixed point for (resp., ) (see Appendix C for the proof and expression of ).*

*4. Existence of Marotto’s Chaos*

*4. Existence of Marotto’s Chaos*

*In this section, we rigorously prove that map (2) possesses chaotic behavior in the sense of Marotto’s definition in [19].*

*Suppose that is the fixed point of map (2). We first give the conditions such that the fixed point is a snap-back repeller. The eigenvalues associated with the fixed point are given by where and .*

*According to Definition 1 in [19], we begin to find a set in which the norm of conjugate complex eigenvalues exceeds 1 for all . Let , which represents the characteristic equation for any point in . If and , then the characteristic equation have a pair of conjugate complex eigenvalues satisfying .*

*Letthen,*

*If , the roots of are , and if , the roots of are .*

*From the above discussion, we obtain the following results.*

*Lemma 6. Supposing that , and , we have the following. (i)If , , , , and or or or , then .(ii)If , , , , and or , then .(iii)If , , , , then .(iv)If , , , , then .(v)If , , , , then .(vi)If , , , , then .(vii)If , , , , then is arbitrary.*

*Lemma 7. If one of the conditions in Lemma 6 is satisfied and the y-coordinate of fixed point satisfies , then is an expanding fixed point of map (2) in .*

*Due to Definition 2 of snap-back repeller in [19], we need to find one point such that , , and for some positive integer .*

*In fact, we haveNow, a map has been constructed to map the point to the fixed point after two iteration if there are solutions different from for (12) and (13). By calculation, the solutions different from for (13) are*

*Substituting (14) into (12), we havewhere .*

*Next, we expect to find a real root of (15) satisfying . Let , then (15) becomesLet ; then, (16) becomeswhere , , and .*

*If the real root of (17) , then (15) has at least a real root .*

*Obviously, if one of the conditions in Lemma 6 is satisfied and and the solution of (15) satisfies , then is a snap-back repeller in . Thus, we have the following theorem.*

*Theorem 8. If one of the conditions in Lemma 6 holds and and the solutions and of (14) and (15) satisfy , then the point is a snap-back repeller of map (2), and hence, map (2) is chaotic in the sense of Marotto’s definition, where is given in Lemma 7.*

*Next, we give specific values of the parameters for illustrating and verifying the conditions in Theorem 8.*

*Example 9. *For , , , , , map (2) has a fixed point , and the eigenvalues of the fixed point are . From Lemma 6, we have , , and , which shows that . We choose ; then, the solution of (15) is and . There exists a point , which is different from and satisfies and . So; is a snap-back repeller.

The bifurcation diagram and the corresponding maximum Lyapunov exponents of map (2) in plane for , , , and are shown in Figures 1(a) and 1(b), respectively. The attractor of map (2) for , , , , and is given in Figure 1(c) (where the maximum Lyapunov exponent ), which is a Maratto’s chaotic attractor.