On systems with a saddle-focus homoclinic curve. (English. Russian original) Zbl 0628.58044

Ovsyannikov, I. M.; Shil’nikov, L. P.

Math. USSR, Sb. 58, 557-574 (1987); translation from Mat. Sb., Nov. Ser. 130(172), No. 4(8), 552-570 (1986).

The paper is devoted to \(C^ r\)-smooth (r\(\geq 3)\) dynamical systems (vector fields) on \(C^ r\)-manifolds of dimension \(\geq 3\) with an isolated equilibrium point O. This point O is required to be a saddlefocus and have a homoclinic curve (some additional technical conditions are also required to hold). Such systems constitute a codimension 1 submanifold \({\mathcal B}^ 1\) of the space of all systems. It is proved that in \({\mathcal B}^ 1\) there is a dense subset of systems with a nonstable periodic orbit and a nonstable homoclinic Poincaré curve. Also, the systems with a countable set of stable periodic orbits are dense in an open subset of \({\mathcal B}^ 1\). And the systems with a countable set of completely unstable periodic orbits are dense in another open subset of \({\mathcal B}^ 1\).

Reviewer: N.Ivanov

MSC:

37G99

Local and nonlocal bifurcation theory for dynamical systems

Keywords:

saddlefocus; homoclinic curve; nonstable periodic orbit