Localization/nonuniqueness problems for periodic orbits of polynomial autonomous systems

In this paper we present sufficient conditions guaranteeing that a polynomial autonomous system has a periodic orbit Γ₁ passing through some point x₁ provided there is another point x₂ which is contained in some periodic orbit Γ₂ . We consider two cases: the case of equal minimal periods and the case of nonequal periods. Our approach described for periodic orbits is applied to polynomial autonomous systems with homoclinic orbits and heteroclinic orbits as well. One example is examined