The three-dimensional generalized Hénon map: Bifurcations and attractors.

We study dynamics of a generic quadratic diffeomorphism, a 3D generalization of the planar Hénon map. Focusing on the dissipative, orientation preserving case, we give a comprehensive parameter study of codimension-one and two bifurcations. Periodic orbits, born at resonant, Neimark-Sacker bifurcations, give rise to Arnold tongues in parameter space. Aperiodic attractors include invariant circles and chaotic orbits; these are distinguished by rotation number and Lyapunov exponents. Chaotic orbits include Hénon-like and Lorenz-like attractors, which can arise from period-doubling cascades, and those born from the destruction of invariant circles. The latter lie on paraboloids near the local unstable manifold of a fixed point.

[1]  D. Turaev,et al.  Doubling of invariant curves and chaos in three-dimensional diffeomorphisms. , 2021, Chaos.

[2]  Amanda E Hampton,et al.  Anti-integrability for Three-Dimensional Quadratic Maps , 2021, SIAM J. Appl. Dyn. Syst..

[3]  A S Gonchenko,et al.  On scenarios of the onset of homoclinic attractors in three-dimensional non-orientable maps. , 2021, Chaos.

[4]  A. Kazakov,et al.  On discrete Lorenz-like attractors. , 2021, Chaos.

[5]  Zhaosheng Feng,et al.  Bifurcation analysis of the three-dimensional Hénon map , 2017 .

[6]  S. Gonchenko,et al.  Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps , 2015, 1510.02252.

[7]  S. Gonchenko,et al.  Homoclinic tangencies to resonant saddles and discrete Lorenz attractors , 2015, 1509.00264.

[8]  Xu Zhang,et al.  Chaotic polynomial maps , 2015, Int. J. Bifurc. Chaos.

[9]  Dmitry Turaev,et al.  Simple Scenarios of Onset of Chaos in Three-Dimensional Maps , 2014, Int. J. Bifurc. Chaos.

[10]  Leon Glass,et al.  Bifurcation structures in two-dimensional maps: The endoskeletons of shrimps , 2013 .

[11]  C. Simó,et al.  Richness of dynamics and global bifurcations in systems with a homoclinic figure-eight , 2013 .

[12]  Julien Clinton Sprott,et al.  Classification of three-dimensional quadratic diffeomorphisms with constant Jacobian , 2009 .

[13]  James D. Meiss,et al.  Quadratic Volume-Preserving Maps: Invariant Circles and Bifurcations , 2008, SIAM J. Appl. Dyn. Syst..

[14]  J. Meiss,et al.  Nilpotent normal form for divergence-free vector fields and volume-preserving maps , 2007, 0706.1575.

[15]  Dmitry Turaev,et al.  Three-Dimensional HÉnon-like Maps and Wild Lorenz-like attractors , 2005, Int. J. Bifurc. Chaos.

[16]  I. I. Ovsyannikov,et al.  CHAOTIC DYNAMICS OF THREE-DIMENSIONAL H ENON MAPS THAT ORIGINATE FROM A HOMOCLINIC BIFURCATION , 2005, nlin/0510061.

[17]  Hendrik Richter,et al.  The Generalized HÉnon Maps: Examples for Higher-Dimensional Chaos , 2002, Int. J. Bifurc. Chaos.

[18]  Dmitry Turaev,et al.  An example of a wild strange attractor , 1998 .

[19]  James D. Meiss,et al.  Computing periodic orbits using the anti-integrable limit , 1998, chao-dyn/9802014.

[20]  J. Meiss,et al.  Quadratic volume preserving maps , 1997, chao-dyn/9706001.

[21]  Robert S. MacKay,et al.  Some flesh on the skeleton: The bifurcation structure of bimodal maps , 1987 .

[22]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[23]  Robert L. Devaney,et al.  Shift automorphisms in the Hénon mapping , 1979, Hamiltonian Dynamical Systems.

[24]  M. Hénon,et al.  A two-dimensional mapping with a strange attractor , 1976 .

[25]  J. Meiss,et al.  Quadratic volume preserving maps: an extension of a result of Moser , 1999 .

[26]  Jason A. C. Gallas,et al.  Dissecting shrimps: results for some one-dimensional physical models , 1994 .

[27]  Lennart Carleson,et al.  The Dynamics of the Henon Map , 1991 .