Periodic attractors, strange attractors and hyperbolic dynamics near homoclinic orbits to saddle-focus equilibria

We discuss dynamics near homoclinic orbits to saddle-focus equilibria in three-dimensional vector fields. The existence of periodic and strange attractors is investigated not in unfoldings, but in families for which each member has a homoclinic orbit. We consider how often, in the sense of measure, periodic and strange attractors occur in such families. We also discuss the fate of typical orbits, and establish that despite the possible existence of attractors, a large proportion of points from a small vicinity of the homoclinic orbit, lies outside the basin of an attractor.

[1]  D. Saari,et al.  Stable and Random Motions in Dynamical Systems , 1975 .

[2]  Dmitry Turaev,et al.  On models with non-rough Poincare´ homoclinic curves , 1993 .

[3]  Raymond Kapral,et al.  Bifurcation phenomena near homoclinic systems: A two-parameter analysis , 1984 .

[4]  Alain Arneodo,et al.  Oscillators with chaotic behavior: An illustration of a theorem by Shil'nikov , 1982 .

[5]  L. P. Šil'nikov,et al.  A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE , 1970 .

[6]  Colin Sparrow,et al.  Local and global behavior near homoclinic orbits , 1984 .

[7]  S. Childress,et al.  Topics in geophysical fluid dynamics. Atmospheric dynamics, dynamo theory, and climate dynamics. , 1987 .

[8]  James A. Yorke,et al.  How often do simple dynamical processes have infinitely many coexisting sinks? , 1986 .

[9]  C. Tresser.  About some theorems by L.P. Sil'nikov , 1983 .

[10]  S. Newhouse,et al.  The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms , 1979 .

[11]  M. Hopkins,et al.  Nilpotence and Stable Homotopy Theory II , 1992 .

[12]  Floris Takens,et al.  HYPERBOLICITY AND THE CREATION OF HOMOCLINIC ORBITS , 1987 .

[13]  L. Shilnikov,et al.  Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. , 1996, Chaos.

[14]  D. Turaev,et al.  Homoclinic Tangencies of an Arbitrary Order in Newhouse Domains , 2001 .

[15]  Floris Takens,et al.  Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations : fractal dimensions and infinitely many attractors , 1993 .

[16]  J. A. Rodríguez,et al.  Coexistence and Persistence of Strange Attractors , 2002 .

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

[18]  Generic Diffeomorphisms with Superexponential Growth of Number of Periodic Orbits , 1999, math/9902159.

[19]  Pierre Gaspard,et al.  Complexity in the bifurcation structure of homoclinic loops to a saddle-focus , 1997 .

[20]  S. Newhouse,et al.  Diffeomorphisms with infinitely many sinks , 1974 .

[21]  Marcelo Viana,et al.  Abundance of strange attractors , 1993 .

[22]  Boris Hasselblatt,et al.  Introduction to the Modern Theory of Dynamical Systems: INTRODUCTION: WHAT IS LOW-DIMENSIONAL DYNAMICS? , 1995 .

[23]  B. Fiedler,et al.  Infinite modal maps and homoclinic bifurcations , 2000 .

[24]  Eduardo Colli.  Infinitely many coexisting strange attractors , 1998 .

[25]  Leonid P Shilnikov,et al.  ON SYSTEMS WITH A SADDLE-FOCUS HOMOCLINIC CURVE , 1987 .

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

[27]  Françoise Argoul,et al.  Homoclinic chaos in chemical systems , 1993 .