Systems with a~homoclinic curve of multidimensional saddle-focus, and spiral chaos

I. M. Ovsyannikov ; L. P. Shilnikov

Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 415-443

Résumé

Consider the space $\mathscr B^1$ of dynamical systems having an isolated equilibrium point $O$ of saddle-focus type with a one- or two-dimensional unstable manifold and a trajectory $\Gamma$ homoclinic at $O$. The following results are proved: 1. Systems with structurally unstable periodic motions are dense in $\mathscr B^1$. 2. Systems with a countable set of stable periodic motions are dense in the open subset $\mathscr B^1_s$ of $\mathscr B^1$ comprised of systems whose second saddle parameter $\sigma_2$ is negative. 3. Neither the subset $\mathscr B^1_u$ of $\mathscr B^1$ consisting of systems satisfying $\sigma_2>0$ nor any sufficiently small neighborhood of $\mathscr B^1_u$ in the space of all dynamical systems contains a system with stable periodic motions in a sufficiently small neighborhood of the contour $O\cup\Gamma$.