Narrow Results

In this paper we will find continuous periodic orbits passing near infinity for a class of polynomial vector fields in ℝ³. We consider polynomial vector fields that are invariant under a symmetry with respect to a plane Σ and that possess a "generalized heteroclinic loop" formed by two singular points e⁺ and e^- at infinity and their invariant manifolds Γ and Λ. Γ is an invariant manifold of dimension 1 formed by an orbit going from e^- to e⁺, Γ is contained in ℝ³ and is transversal to Σ. Λ is an invariant manifold of dimension 2 at infinity. In fact, Λ is the two-dimensional sphere at infinity in the Poincaré compactification minus the singular points e⁺ and e^-. The main tool for proving the existence of such periodic orbits is the construction of a Poincaré map along the generalized heteroclinic loop together with the symmetry with respect to Σ.

The quadratic map of the interval displays one attractor for each parameter value. Conservative maps of the plane display infinite coexistence of stability islands around periodic orbits. Between these two extremes, dissipative systems of the plane are known to have infinite coexistence of sinks as a generic property, yet very hard to detect. We investigate how more and more coexistence is gained as the area-contraction rate b → 1. In this paper, we show a sequence of simple sinks gaining coexistence, and investigate the convergence properties of its bifurcation values. The sinks are simple, or primary, due to their geometrical structure.

This paper considers the weakened Hilbert’s 16th problem for symmetric planar perturbed polynomial Hamiltonian system. A $Z_{13}$-equivariant planar vector field of degree 12 is deduced to find as many as possible limit cycles and their configuration patterns. By using bifurcation theory of planar dynamical system and the method of detection function, we have obtained that, under the thirteenth-order perturbation, the above $Z_{13}$-equivariant planar perturbed Hamiltonian vector field of 12-degree has at least 117 limit cycles. Moreover, this paper also shows the configuration of compound eyes of the corresponding perturbed systems.