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:
[1]

H. L. An and Y. Chen, The function cascade synchronization scheme for discrete-time hyperchaotic systems, Commun Nonlinear Sci Numer Simulat, 14 (2009), 1494-1501. doi: 10.1016/j.cnsns.2008.04.011. Google Scholar

[2]

G. Baier and M. Klein, Maximum hyperchaos in generalized Hénon maps, Phys Lett A, 151 (1990), 281-284. doi: 10.1016/0375-9601(90)90283-T. Google Scholar

[3]

J. Carr, Applications of Centre Manifold Theory Springer-Verlag, New York, 1981. doi: 0-387-90577-4. Google Scholar

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J. H. Curry, On the Hénon transformation, Commun Math Phys, 68 (1979), 129-140. doi: 10.1007/BF01418124. Google Scholar

[5]

H. R. Dullin and J. D. Meiss, Generalized Hénon maps: The cubic diffeomorphisms of the plane, Phys D, 143 (2000), 262-289. doi: 10.1016/S0167-2789(00)00105-6. Google Scholar

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R. L. FilaliS. HammamiM. Benrejeb and P. Borne, On synchronization, anti-synchronization and hybrid synchronization of 3D discrete generalized Hénon map, Nonlinear Dynamics and Systems Theory, 12 (2012), 81-95. Google Scholar

[7]

A. S. Gonchenko and S. V. Gonchenko, Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps, Phys. D, 337 (2016), 43–57, arXiv: 1510. 02252v2 doi: 10.1016/j.physd.2016.07.006. Google Scholar

[8]

S. V. GonchenkoI. I. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Int J Bifurcat Chaos, 15 (2005), 3493-3508. doi: 10.1142/S0218127405014180. Google Scholar

[9]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-1140-2. Google Scholar

[10]

M. Hénon, A two-dimensional mapping with a strange attractor, Commun math Phys, 50 (1976), 69-77. doi: 10.1007/BF01608556. Google Scholar

[11]

D. L. Hitzl and F. Zele, An exploration of the Hénon quadratic map, Phys D, 14 (1985), 305-326. doi: 10.1016/0167-2789(85)90092-2. Google Scholar

[12]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory 2nd edition, Springer-Verlag, New York, 1998. Google Scholar

[13]

H. K. Lam, Synchronization of generalized Hénon map using polynomial controller, Phys Lett A, 374 (2010), 552-556. doi: 10.1016/j.physleta.2009.11.035. Google Scholar

[14]

E. N. Lorenz, Compound windows of the Hénon map, Phys D, 237 (2008), 1689-1704. doi: 10.1016/j.physd.2007.11.014. Google Scholar

[15]

A. C. J. Luo and Y. Guo, Dynamical Systems: Discontinuity, Stochasticity and Time-Delay Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5754-2. Google Scholar

[16]

F. R. Marotto, Chaotic behavior in the Hénon mapping, Commun Math Phys, 68 (1979), 187-194. doi: 10.1007/BF01418128. Google Scholar

[17]

S. Michael, Once more on Hénon map: Analysis of bifurcations, Chaos, Solitons and Fractals, 7 (1996), 2215-2234. doi: 10.1016/S0960-0779(96)00081-1. Google Scholar

[18]

C. Mira, Chaotic Dynamics World Scientific, Singapore, 1987. doi: 10.1142/0413. Google Scholar

[19]

E. Ott, Chaos in Dynamical Systems 2nd edition, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511803260. Google Scholar

[20]

H. Richter, The generalized Hénon maps: Examples for higher dimensional chaos, Int J Bifurcat Chaos, 12 (2002), 1371-1384. doi: 10.1142/S0218127402005121. Google Scholar

[21]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos CRC Press, New York, 1999. Google Scholar

[22]

S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

[23]

S. Winggins, Introduction to Applied Nonlinear Dynamical System and Chaos Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7. Google Scholar

[24]

Y. J. Xue and S. Y. Yang, Synchronization of generalized Hénon map by using adaptive fuzzy controller, Chaos, Solitons and Fractals, 17 (2003), 717-722. doi: 10.1016/S0960-0779(02)00490-3. Google Scholar

[25]

Z. Y. Yan, Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller, Phys Lett A, 342 (2005), 309-317. doi: 10.1016/j.physleta.2005.04.049. Google Scholar

show all references

References:
[1]

H. L. An and Y. Chen, The function cascade synchronization scheme for discrete-time hyperchaotic systems, Commun Nonlinear Sci Numer Simulat, 14 (2009), 1494-1501. doi: 10.1016/j.cnsns.2008.04.011. Google Scholar

[2]

G. Baier and M. Klein, Maximum hyperchaos in generalized Hénon maps, Phys Lett A, 151 (1990), 281-284. doi: 10.1016/0375-9601(90)90283-T. Google Scholar

[3]

J. Carr, Applications of Centre Manifold Theory Springer-Verlag, New York, 1981. doi: 0-387-90577-4. Google Scholar

[4]

J. H. Curry, On the Hénon transformation, Commun Math Phys, 68 (1979), 129-140. doi: 10.1007/BF01418124. Google Scholar

[5]

H. R. Dullin and J. D. Meiss, Generalized Hénon maps: The cubic diffeomorphisms of the plane, Phys D, 143 (2000), 262-289. doi: 10.1016/S0167-2789(00)00105-6. Google Scholar

[6]

R. L. FilaliS. HammamiM. Benrejeb and P. Borne, On synchronization, anti-synchronization and hybrid synchronization of 3D discrete generalized Hénon map, Nonlinear Dynamics and Systems Theory, 12 (2012), 81-95. Google Scholar

[7]

A. S. Gonchenko and S. V. Gonchenko, Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps, Phys. D, 337 (2016), 43–57, arXiv: 1510. 02252v2 doi: 10.1016/j.physd.2016.07.006. Google Scholar

[8]

S. V. GonchenkoI. I. OvsyannikovC. Simó and D. Turaev, Three-dimensional Hénon-like maps and wild Lorenz-like attractors, Int J Bifurcat Chaos, 15 (2005), 3493-3508. doi: 10.1142/S0218127405014180. Google Scholar

[9]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-1140-2. Google Scholar

[10]

M. Hénon, A two-dimensional mapping with a strange attractor, Commun math Phys, 50 (1976), 69-77. doi: 10.1007/BF01608556. Google Scholar

[11]

D. L. Hitzl and F. Zele, An exploration of the Hénon quadratic map, Phys D, 14 (1985), 305-326. doi: 10.1016/0167-2789(85)90092-2. Google Scholar

[12]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory 2nd edition, Springer-Verlag, New York, 1998. Google Scholar

[13]

H. K. Lam, Synchronization of generalized Hénon map using polynomial controller, Phys Lett A, 374 (2010), 552-556. doi: 10.1016/j.physleta.2009.11.035. Google Scholar

[14]

E. N. Lorenz, Compound windows of the Hénon map, Phys D, 237 (2008), 1689-1704. doi: 10.1016/j.physd.2007.11.014. Google Scholar

[15]

A. C. J. Luo and Y. Guo, Dynamical Systems: Discontinuity, Stochasticity and Time-Delay Springer-Verlag, New York, 2010. doi: 10.1007/978-1-4419-5754-2. Google Scholar

[16]

F. R. Marotto, Chaotic behavior in the Hénon mapping, Commun Math Phys, 68 (1979), 187-194. doi: 10.1007/BF01418128. Google Scholar

[17]

S. Michael, Once more on Hénon map: Analysis of bifurcations, Chaos, Solitons and Fractals, 7 (1996), 2215-2234. doi: 10.1016/S0960-0779(96)00081-1. Google Scholar

[18]

C. Mira, Chaotic Dynamics World Scientific, Singapore, 1987. doi: 10.1142/0413. Google Scholar

[19]

E. Ott, Chaos in Dynamical Systems 2nd edition, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511803260. Google Scholar

[20]

H. Richter, The generalized Hénon maps: Examples for higher dimensional chaos, Int J Bifurcat Chaos, 12 (2002), 1371-1384. doi: 10.1142/S0218127402005121. Google Scholar

[21]

C. Robinson, Dynamical Systems: Stability, Symbolic Dynamics, and Chaos CRC Press, New York, 1999. Google Scholar

[22]

S. Smale, Differentiable dynamical systems, Bulletin of the American Mathematical Society, 73 (1967), 747-817. doi: 10.1090/S0002-9904-1967-11798-1. Google Scholar

[23]

S. Winggins, Introduction to Applied Nonlinear Dynamical System and Chaos Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-4067-7. Google Scholar

[24]

Y. J. Xue and S. Y. Yang, Synchronization of generalized Hénon map by using adaptive fuzzy controller, Chaos, Solitons and Fractals, 17 (2003), 717-722. doi: 10.1016/S0960-0779(02)00490-3. Google Scholar

[25]

Z. Y. Yan, Q-S synchronization in 3D Hénon-like map and generalized Hénon map via a scalar controller, Phys Lett A, 342 (2005), 309-317. doi: 10.1016/j.physleta.2005.04.049. Google Scholar

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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