Bifurcation analysis of the three-dimensional Hénon map

In this paper, we consider the dynamics of a generalized three-dimensional Henon map. Necessary and sufficient conditions on the existence and stability of the fixed points of this system are established. By applying the center manifold theorem and bifurcation theory, we show that the system has the fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation under certain conditions. Numerical simulations are presented to not only show the consistence between examples and our theoretical analysis, but also exhibit complexity and interesting dynamical behaviors, including period-10, -13, -14, -16, -17, -20, and -34 orbits, quasi-periodic orbits, chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors. These results demonstrate relatively rich dynamical behaviors of the three-dimensional Henon map.

1. Introduction. The well known Hénon map with the constant Jacobian determinant was proposed by Hénon [10] as a simplified model to the Poincaré section of the Lorenz system [14]. This map is a two-dimensional discrete-time system with a single quadratic non-linearity and exhibits chaotic behavior for the parametric values a = 1.4 and b = 0.3. Since the late 1970's, the Hénon map has been extensively studied because of its generality and wide applications. For example, Marotto [16] and Michael [17] analyzed the bifurcation and chaos of the Hénon map with respect to the certain parameters. Curry [4] presented some numerical experiments on the existence of two distinct strange attractors for some parametric values of the Hénon map by using the characteristic exponent, frequency spectrum and a theorem of Smale [22]. The detailed bifurcation diagrams of the Hénon map were described by Mira [18]. By introducing the positive and negative iterative mappings, Luo and Guo [15] investigated the bifurcation and stability of periodic solutions and chaotic layers. Subsequently, different types of the generalized Hénon map were proposed. Dullin and Meiss [5] discussed the dynamics of a single, generalized Hénon map. Baier and Klein [2] considered a generalized Hénon map with a single quadratic nonlinearity, where the dimension of this map can be higher than 2.
The n-dimensional generalized Hénon map was presented by Richter [20]. In the past decades, considerable attention has been dedicated to the three-dimensional systems due to its significant role in the study of dynamical properties for the highdimensional systems [1,2,8,11,13,25,24,6]. For instance, an interesting three dimensional generalization of the Hénon map was introduced in [11]. Maximum hyperchaos in the generalized Hénon map containing a single quadratic term was investigated in [2]. It was shown in [8] that the three-dimensional Hénon-like map possesses the wild Lorenz-type strange attractors. The existence of strange pseudohyperbolic attractors for the three-dimensional Hénon map was presented in [7]. As we know, many nonlinear systems have special dynamical behaviors with the values of parameters given in the certain intervals or regions. As the values of parameters vary, qualitative structure of orbits and dynamical properties of systems, such as bifurcations and chaotic phenomena, may change dramatically [9]. One of interesting problems on nonlinear dynamical systems is to find how the dynamical behaviors and properties of orbits change and evolve as the values of parameters vary [19].
The aim of this paper is to investigate a three-dimensional Hénon map. At this stage we restrict our attention to the existence of the fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by making use of the center manifold theorem [3,9,23] and bifurcation theory [9,12,23]. Numerical simulations on bifurcation diagrams, Lyapunov exponents and phase portraits are undertaken which agree well with our analytical results. In addition, after analyzing Lyapunov exponents of the system, we demonstrate the orbits of period-10, -13, -14, -16, -17, -20, and -34, attracting invariant cycles, quasi-periodic orbits, coexisting chaotic attractors, period-doubling bifurcation from period-10 leading to chaos, cascades of perioddoubling bifurcation in orbits of period-2, -4, -8, · · · , and nice chaotic behaviors which appear and disappear suddenly.
The rest of this paper is organized as follows. In Section 2, we introduce the generalized three-dimensional Hénon map and discuss the existence and local stability of the fixed points of the associated system. In Section 3, we explore sufficient conditions of the existence of codimension one bifurcations, including the fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation. In Section 4, numerical simulations are presented to illustrate complexity and dynamical behaviors of our system. Section 5 is a brief conclusion.
(2) The fixed point P 1 (x 1 , 0, −x 1 ) is stable if and only if one of the following conditions holds: 3. Codimension one bifurcations. In this section, we investigate the fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation at the fixed point of system (2). We choose the coefficient a as a bifurcation parameter for analyzing the fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation by means of the center manifold theorem [3,9,23] and bifurcation theory [9,12,23].
3.1. Fold bifurcation. Consider the fold bifurcation at the unique fixed point P 0 of system (2), i.e, a = a 0 = − 1 4 . The eigenvalues of the Jacobian matrix at P 0 are λ 1 = 1 and λ 2, We translate the fixed point P 0 to the origin and takeā as a new dependent variable, then system (2) becomes where f 1 (x,ȳ,z,ā) = −x 2 . For b > 0, we find an invertible matrix by making use of corresponding eigenvectors of the 4 × 4 matrix in the map (5): Use the transformation of (x,ȳ,z,ā) The map (5) can be rewritten as where By the center manifold theorem [9], we know that there exists a center manifold W c (0) around the origin, which can be expressed as follows and Since the center manifold must satisfy By comparing the corresponding coefficients of the above two equations, we have Consider the map on the center manifold given by It is easy to see that Hence, the fixed point (u, v) = (0, 0) is a fold bifurcation point for the map (7). For b = 0, let By using the transformation of (x,ȳ,z,ā) T = T 1 (u, v, w, µ) T , the map (5) becomes To simplify the map (8) on the center manifold, we assume that the center manifold is of the form whereh Thus, the map restricted to the center manifold is given bỹ From the map (9), we find This implies that the fixed point (u, v) = (0, 0) is a fold bifurcation point for the map (9). For b = −s 2 < 0 (s is a nonzero real number), let By using the transformation of (x,ȳ,z,ā) Applying the center manifold theorem again, we know that there exists a center manifold for (10) which can be written as After substitution, by a straightforward computation on equating the corresponding coefficients, it gives Thus, the map restricted to the center manifold is given bỹ Sincẽ Based on the above discussions, we obtain the following results immediately.
Theorem 3.1. System (2) undergoes a fold bifurcation at the fixed point P 0 if the conditions b = ±1 and a = a 0 hold. Moreover, two fixed points bifurcate from P 0 for a > a 0 , coalesce as the fixed point P 0 at a = a 0 , and disappear for a < a 0 .

Flip bifurcation.
In this subsection, we consider the flip bifurcation occurring at the fixed point P 1 of system (2). When a = a * , the fixed point is The eigenvalues of the associated Jacobian matrix at the fixed point P 1 are λ 1 = −1 and λ 2, We translate the fixed point P 1 into the origin and takeā as a variable, then system (2) can be rewritten as where In the case of b = 0, we get an invertible matrix by making use of corresponding eigenvectors of the 4 × 4 matrix in the map (12): Use the transformation (x,ȳ,z,ā) where Similarly, we reduce the map (13) on the center manifold, which can be represented as for u and v being sufficiently small. Assume that the center manifold is of the form: So the center manifold must satisfy By comparing the coefficients of u 2 , uv, and v 2 in the above two equations, respectively, we get Then, the map (13) restricted to the center manifold can be expressed as where In order for the map (14) to undergo a flip bifurcation, we require that two discriminatory quantities k 1 and k 2 are not zero, where In the case of b = 0, we take the invertible matrix Use the transformation (x,ȳ,z,ā) T = T 2 (u, v, w, µ) T . Then the map (12) can be rewritten as  We reduce the map (15) on a center manifold which can be expressed as Assume thath 3 (u, v) andh 4 (u, v) are of the following forms, respectively, 3 . By a direct computation on the center manifold, we find Thus, the map (15) restricted to the center manifold is given bỹ Note that Thus, the fixed point (u, v) = (0, 0) is a flip bifurcation point for the map (16). In addition, if − 1 3 < b < 1, then |λ 2,3 | < 1 and k 2 > 0. We summarize our discussions into the following results.
Theorem 3.2. The system (2) undergoes a flip bifurcation at the fixed point P 1 when b = − 1 3 , 1 and a = a * . Moreover, if − 1 3 < b < 1 (b < − 1 3 or b > 1) holds, then the period-2 orbits that bifurcate from the fixed point P 1 are stable (unstable). Letx = x −x,ȳ = y −ỹ, andz = z −z. Then the fixed point (x,ỹ,z) can be translated to the origin, and system (2) becomes where f (x,ȳ,z) = −x 2 . Using the corresponding eigenvectors of the 3 × 3 matrix in the map (17), we get an invertible matrix: By making use of the transformation (x,ȳ,z) T = T 3 (u, v, w) T , the map (17) becomes We simplify the map (18) on the center manifold by assuming that the manifold is of the form

Substituting it into (18) yields
Equating the corresponding coefficients to zero gives .
Thus, the map (18) restricted to the center manifold is given as follows where In order for the map (19) to undergo a Neimark-Sacker bifurcation, we require that the following discriminatory quantity l 1 is not equal to zero [9,23,21]: where the sign of l 1 determines the stability of the invariant circle, and By a direct calculation, we get . Note that if −1 < b < 1, then |λ 3 | < 1. By the fact that b < − 1 3 ( b = −1, − 1 2 ), it follows that l 1 < 0 for −1 < b < − 1 3 and b = − 1 2 . Consequently, we summarize our result as the following theorem.
Then the system (2) undergoes a Neimark-Sacker bifurcation at the fixed point P 1 for a = a h . Moreover, if −1 < b < − 1 3 and b = − 1 2 , then the bifurcation is supercritical and an attracting invariant closed curve bifurcates from the fixed point for a > a h .
In Figure 1, it shows the stability region and bifurcation region of system (2) in the parametric space.   Figure 2 (A), we observe that if a < a 0 , there has no fixed point; if a = a 0 , there is only one fixed point P 0 ; if a > a 0 , there are two fixed points P 1 (x 1 , 0, −x 1 ) and P 2 (x 2 , 0, −x 2 ) bifurcating from P 0 . It shows that the number of the fixed points varies as the value of the parameter a changes. We also find that x 2 is unstable and x 1 is stable for a ∈ (a 0 , a h ) in Figure 2 (A). This agrees well with Lemma 2.1 and Theorem 3.1. Furthermore, we see that the fixed point P 1 loses its stability at a h ∼ 0.0344 and an attracting invariant closed curve bifurcates from the fixed point P 1 for a > a h . This is in good agreement with 2.1 and Theorem 3.2, we know that the fixed point P 1 is stable for a ∈ (a 0 , a * ) and loses its stability when a > a * . The flip bifurcation occurs at a = a * and the period-2 orbits that bifurcate from P 1 are stable. All these phenomena are clearly presented in Figure 2 (B). Case 3. Let a = 0.2, and b ∈ (−0.8, 0.8) be the bifurcation parameter. According to Lemma 2.1, the fixed point P 1 is stable for b ∈ (−0.5333, 0.3292). From Theorem 3.2, the flip bifurcation occurs at P 1 for b ∼ 0.3292. From Theorem 3.3, the supercritical Neimark-Sacker bifurcation occurs at P 1 for b ∼ −0.5333. All these phenomena are clearly presented in Figure 2 (C) which agree well with Lemma 2.1 and Theorem 3.1-3.3.

4.2.
More numerical simulations of system (2). In this subsection, we show that new complex dynamical behaviors change as the parameters of system (2) vary by using the bifurcation diagrams, maximum Lyapunov exponents and phase portraits.
The bifurcation diagrams in the three-dimensional (a, b, x) space are shown in Figure 3. For the bifurcation diagrams in the two-dimensional space, we consider the bifurcation parameters in the following three cases: (i) changing a in the range 0 ≤ a ≤ 0.4, and fixing b = −0.6;    , we observe that there is a stable fixed point for a ∈ (0, 0.0344) and the fixed point loses its stability at a ∼ 0.0344. The Neimark-Sacker bifurcation occurs at a ∼ 0.0344 and an attracting invariant cycle appears for a > 0.0344. Then, we can see the process of the period-doubling bifurcation which is from the period-10 orbits to chaos and these chaotic regions are interspersed with several periodic windows. For example, we can observe orbits of period-17 for a = 0.294, period-34 for a = 0.31, and period-14 for a = 0.361. Phase portraits for various values of a are shown in Figure 5, which clearly depicts how a smooth invariant curve bifurcates from the stable fixed point and an invariant curve to chaotic attractors. From Figure 5, we observe that there are an attracting invariant circle, quasi-periodic orbits which are routines to chaos, period-20 orbits, ten-coexisting chaotic attractors, and chaotic sets. In particular, of zero, which correspond to quasi-period orbits or coexistence of chaos and quasiperiod orbits. For a ∈ (0.23, 0.4), the majority of maximum Lyapunov exponents are positive with a few being negative, which shows that there exist stable fixed points or stable period windows in the chaotic region.
For case (ii). The bifurcation diagram of system (2)  a < 0.11, loses its stability at a = 0.11, and then a cascade of period-2, -4, -8, · · · orbits emerge. From Figures 7 (B) and (D), we observe that there exist positive Lyapunov exponents when a exceeds 0.11, which indicates that the chaotic sets arise. This means that the system tends to a stable state at the very beginning, then circulates along period-doubling orbits, and finally follows irregular chaotic sets with some period-2 orbits. For example, the chaotic behavior suddenly disappears for a ∼ 0.8446, and then the chaotic behavior suddenly appears for a ∼ 0.8447 and converges to period-2 orbits for a ∼ 0.845. Phase portraits of four-coexisting chaotic attractors for a = 0.835 and a chaotic attractor for a = 0.8445 are illustrated in Figures 7 (E) and (F), respectively.
Case (iii). The bifurcation diagram of system (2) in the (b, x) plane for a = 0.23 is displayed in Figure 8 Figure 9 (E), and a chaotic attractor at b = 0.691 is shown in Figure 9 (F).

5.
Conclusions. In this study, we investigated the complex dynamical behaviors of a generalized three-dimensional Hénon map. It is shown that system (2) can undergo the fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation by applying the center manifold theorem and bifurcation theory. Moreover, system (2) displays some interesting dynamical behaviors, including invariant circle, orbits of period-10, -13, -14, -16, -17, -20, and -34, quasi-periodic orbits, the new nice types of four-, eight-, and ten-coexisting chaotic attractors, period-doubling bifurcation from period-10 leading to chaos, cascades of period-doubling bifurcation in orbits of period-2, -4, -8, · · · , and chaotic sets. These results show far richer dynamics of the generalized three-dimensional Hénon map, and dynamical properties are different from that of the standard Hénon map. These results can provide us fundamental and useful information to further better understand the dynamic complexity of the higher-dimensional Hénon map arising in different scientific fields.