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# Bifurcation analysis of the three-dimensional Hénon map

This work was supported by National Science Foundation of China (No. 61134005, 11272024 and 10971009).
• In this paper, we consider the dynamics of a generalized three-dimensional Hénon 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 Hénon map.

Mathematics Subject Classification: Primary: 37L10, 65P20; Secondary: 65P30.

 Citation: • • Figure 1.  The stability region and bifurcation region of system (2) in the (b, a)-plane.

Figure 2.  (A)-(B) bifurcation diagrams of system (2) in the (a, x) plane: (A) b = −0.6, and (B) b = 0.4; (C) bifurcation diagram of system (2) in the (b, x) plane with b ∈ (−0.8, 0.8) and a = 0.2. Here, the fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation are labeled as "SN", "PD" and "NS", respectively.

Figure 3.  Bifurcation diagrams of system (2) in the threedimensional (a, b, x) space.

Figure 4.  (A) bifurcation diagram of system (2) in the (a, x) plane (a ∈ (0, 0.4)) for b = −0.6; (B) maximum Lyapunov exponent corresponding to (A); (C) the local amplified bifurcation diagram of (A) for a ∈ (0.22, 0.32).

Figure 5.  (A)-(H) phase portraits for various values of a corresponding to Figure 4 (A).

Figure 6.  (A)-(C) phase portraits for a = 0.385 in the (x, y) plane, the (x, z) plane, and the (y, z) plane.

Figure 7.  (A) bifurcation diagram of system (2) in the (a, x) plane (a ∈ (0, 1)) for b = 0.4; (B) maximum Lyapunov exponent corresponding to (A); (C) the local amplified bifurcation diagram of (A) for a ∈ (0.81, 0.85); (D) maximum Lyapunov exponent corresponding to (C); (E)-(F) chaotic attractors for a = 0.835 and a = 0.8445, respectively.

Figure 8.  (A) bifurcation diagram of system (2) in the (b, x) plane with b ∈ (−0.8, 0.8) and a = 0.23; (B) maximum Lyapunov exponent corresponding to (A); (C) the local amplified bifurcation diagram of (A) for b ∈ (−0.75, −0.5); (D) maximum Lyapunov exponent corresponding to (C); (E) the local amplified bifurcation diagram of (A) for b ∈ (0.64, 0.7); (F) maximum Lyapunov exponent corresponding to (E).

Figure 9.  In (A)-(C), phase portraits corresponding to Figure 8 (C): (A) b = −0.72, (B) b = −0.6, and (C) b = −0.53. In (D)-(F), phase portraits corresponding to Figure 8 (E): (D) b = 0.66, (E) b = 0.675, and (F) b = 0.691.

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