Attractors with vanishing rotation number

Abstract

Abstract. Given an orientation-preserving homeomorphism of the plane, a rotation number can be associated with each locally attracting fixed point. Assuming that the homeomorphism is dissipative and the rotation number vanishes we prove the existence of a second fixed point. The main tools in the proof are Carathéodory prime ends and fixed point index. The result is applicable to some concrete problems in the theory of periodic differential equations.

Attractors with irrational rotation number

“…The boundary in R 2 of the region of attraction has infinitely many connected components. This is consistent with Theorem 10 in [14]. There it is proved that if ∞ is accessible from U and ∂U has a finite number of components then there is a periodic orbit in…”

“…In this paper we continue the work initiated in [14] on the connections between Rotation Numbers and Stability Theory. As in the previous paper we assume that h is a dissipative homeomorphism of the plane having an assymptotically stable fixed point, say x * = h(x * ).…”

“…We claim that Γ is not contained in U , for otherwise we could apply Proposition 6 in [14] to deduce that the rotation number vanishes. Indeed the result in [14] dealt with positively invariant rays but the same proof works for negative invariance.…”

“…For d = 1 it is sufficient to exclude 2-cycles and the global attraction is equivalent to h is dissipative and F ix(h 2 ) = {x * }. It was proved in [14] that this is also a characterization of global attraction for d = 2 if h is orientation-reversing. In this setting Bonino proved in [6] that the existence of a k-cycle with k ≥ 2 implies the existence of a 2-cycle.…”

“…We refer to the papers [4,7,2] for more information. In the previous paper [14] the case ρ = 0 (or more generally ρ ∈ Q) was treated, now we plan to study the case ρ ∈ R \ Q.…”