A path crossing lemma and applications to nonlinear second order equations under slowly varyng perturbations

  • Anna Pascoletti
  • Fabio Zanolin
Keywords: Paths, Connected sets, Planar maps, Periodic points, Chaotic dynamics, Nonlinear equations, Slowly varying coefficients.


We present some recent results on the existence of periodic solutions and chaotic like dynamics for second order scalar nonlinear ODEs. The equations under consideration belong to a simple class of perturbed planar Hamiltonian systems with slowly varying periodic coefficients, a typical example being given by the pendulum equation with moving support. Although there is already a broad literature on this subject, our approach, based on the concept of stretching along the paths, appears new in this context. In particular, our method is global in nature and stable with respect to small perturbations of the coefficients. Thus it applies even when some small friction terms are inserted into the equations. The main tool on which all our results are based is a topological lemma (that we call path crossing lemma) which was already implicitly used by Poincaré (1883-1884) [51], as well as by Butler (1976) [8] and Conley (1975) [12] and subsequently “rediscovered” and applied in many different contexts. For this reason, the first part of this paper is devoted to a detailed exposition of the Crossing Lemma and its connections with other topological results.