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

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
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