An example of a wild strange attractor

It is proved that in the space of -smooth () flows in () there exist regions filled by systems that each have an attractor (here: a completely stable chain-transitive closed invariant set) containing a non-trivial basic hyperbolic set together with its unstable manifold, which has points of non-transversal intersection with the stable manifold. A construction is given for such a wild attractor containing an equilibrium state of saddle-focus type.

[1]  D. Anosov,et al.  Ergodic Properties of Geodesic Flows on Closed Riemannian Manifolds of Negative Curvature , 2020 .

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

[3]  Bjj Orn Sandstede,et al.  Center Manifolds for Homoclinic Solutions , 1995 .

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

[5]  J. Palis,et al.  High dimension diffeomorphisms displaying infinitely many periodic attractors , 1994 .

[6]  W. Kyner.  Invariant Manifolds , 1961 .

[7]  Y. Pesin.  CHARACTERISTIC LYAPUNOV EXPONENTS AND SMOOTH ERGODIC THEORY , 1977 .

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

[9]  Dmitry Turaev,et al.  ON DIMENSION OF NON-LOCAL BIFURCATIONAL PROBLEMS , 1996 .

[10]  J. Feroe.  Homoclinic orbits in a parametrized saddle-focus system , 1993 .

[11]  Vladimir I. Arnold,et al.  Dynamical Systems I , 1988 .

[12]  J. Auslander,et al.  Generalized recurrence in dynamical systems. , 1964 .

[13]  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 .

[14]  L. P. Šil'nikov.  ON A POINCARÉ-BIRKHOFF PROBLEM , 1967 .

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

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

[17]  R. F. Williams,et al.  Structural stability of Lorenz attractors , 1979 .