# American Institute of Mathematical Sciences

June  2017, 10(3): 625-645. doi: 10.3934/dcdss.2017031

## Bifurcation analysis of the three-dimensional Hénon map

 1 LMIB-School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China 2 School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA

* Corresponding author: mingzhao@buaa.edu.cn

Received  January 2016 Revised  December 2016 Published  February 2017

Fund Project: 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.

Citation: Ming Zhao, Cuiping Li, Jinliang Wang, Zhaosheng Feng. Bifurcation analysis of the three-dimensional Hénon map. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 625-645. doi: 10.3934/dcdss.2017031
##### References:

show all references

##### References:
The stability region and bifurcation region of system (2) in the (b, a)-plane.
(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.
Bifurcation diagrams of system (2) in the threedimensional (a, b, x) space.
(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).
(A)-(C) phase portraits for a = 0.385 in the (x, y) plane, the (x, z) plane, and the (y, z) plane.
(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.
(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).
: (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.">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.
 [1] Anna Lisa Amadori. Global bifurcation for the Hénon problem. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4797-4816. doi: 10.3934/cpaa.2020212 [2] Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019 [3] Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009 [4] Joel Kübler, Tobias Weth. Spectral asymptotics of radial solutions and nonradial bifurcation for the Hénon equation. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3629-3656. doi: 10.3934/dcds.2020032 [5] Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020 [6] Jean-François Rault. A bifurcation for a generalized Burgers' equation in dimension one. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 683-706. doi: 10.3934/dcdss.2012.5.683 [7] Patrick M. Fitzpatrick, Jacobo Pejsachowicz. Branching and bifurcation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1955-1975. doi: 10.3934/dcdss.2019127 [8] Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237 [9] Gian-Italo Bischi, Laura Gardini, Fabio Tramontana. Bifurcation curves in discontinuous maps. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 249-267. doi: 10.3934/dcdsb.2010.13.249 [10] Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997 [11] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [12] Yangjian Sun, Changjian Liu. The Poincaré bifurcation of a SD oscillator. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1565-1577. doi: 10.3934/dcdsb.2020173 [13] John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805 [14] Carmen Núñez, Rafael Obaya. A non-autonomous bifurcation theory for deterministic scalar differential equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 701-730. doi: 10.3934/dcdsb.2008.9.701 [15] José Caicedo, Alfonso Castro, Arturo Sanjuán. Bifurcation at infinity for a semilinear wave equation with non-monotone nonlinearity. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1857-1865. doi: 10.3934/dcds.2017078 [16] Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $2$D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089 [17] Andrus Giraldo, Neil G. R. Broderick, Bernd Krauskopf. Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021217 [18] Christian Pötzsche. Nonautonomous bifurcation of bounded solutions II: A Shovel-Bifurcation pattern. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 941-973. doi: 10.3934/dcds.2011.31.941 [19] Rafael Labarca, Solange Aranzubia. A formula for the boundary of chaos in the lexicographical scenario and applications to the bifurcation diagram of the standard two parameter family of quadratic increasing-increasing Lorenz maps. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 1745-1776. doi: 10.3934/dcds.2018072 [20] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

2020 Impact Factor: 2.425