A Topological Approach to Bend‐Twist Maps with Applications

Abstract

In this paper we reconsider, in a purely topological framework, the concept of bend-twist map previously studied in the analytic setting by Tongren Ding in (2007). We obtain some results about the existence and multiplicity of fixed points which are related to the classical Poincaré-Birkhoff twist theorem for area-preserving maps of the annulus; however, in our approach, like in Ding (2007), we do not require measure-preserving conditions. This makes our theorems in principle applicable to nonconservative planar systems. Some of our results are also stable for small perturbations. Possible applications of the fixed point theorems for topological bend-twist maps are outlined in the last section.

A Crossing Lemma for Annular Regions and Invariant Sets with an Application to Planar Dynamical Systems

“…We observe that the existence of two solutions is not guaranteed if the minimality of the set is not assumed (see [42,Example 2.8]).…”

“…For a different application of these results to the existence of fixed points and periodic points to planar maps arising from ordinary differential equations, we refer also to [42].…”

A topological degree theory for rotating solutions of planar systems

“…Yet, the area-preserving assumption is quite restrictive, albeit crucial for the existence of fixed-points. An attempt to circumvent this situation is given by the theory of bend-twist maps [10,29], where area-preservation is replaced by some structural assumption on the map along some p. gidoni: A topological degree theory for rotating solutions of planar systems curves in the interior of the annulus, producing a behaviour analogous to the one in the classical case.…”

Multiple periodic solutions for one-sided sublinear systems: A refinement of the Poincaré-Birkhoff approach

“…The original treatment was given in [20] for analytic maps. There are extensions to continuous maps as well [59], [60]. Clearly, the bend-twist map condition is difficult to check in practice, due to the lack of information about the curve Γ (which, in the non-analytic case, may not even be a curve).…”

“…Clearly, the bend-twist map condition is difficult to check in practice, due to the lack of information about the curve Γ (which, in the non-analytic case, may not even be a curve). For this reason, one can rely on the following corollary [20,Corollary 7.3] which also follows from the Poincaré-Miranda theorem (as observed in [59]).…”