RENDICONTI DEL SEMINARIO MATEMATICO

Rend. Sem. Mat. Univ. Pol. Torino - Vol. 65, 1 (2007)Subalpine Rhapsody in DynamicsD. Papini - F. Zanolin ∗SOME RESULTS ON PERIODIC POINTS AND CHAOTICDYNAMICS ARISING FROM THE STUDY OF THENONLINEAR HILL EQUATIONSAbstract. We study fixed point theorems for maps which satisfy a property of stretchinga suitably oriented topological space Z along the paths connecting two disjoint subsets Zl−and Zr− of Z. Our results reconsider and extend previous theorems in [56, 59, 60] where thecase of two-dimensional cells (that is topological spaces homeomorphic to a rectangle of theplane) was analyzed. Applications are given to topological horseshoes and to the study of theperiodic points and the symbolic dynamics associated to discrete (semi)dynamical systems.1. Introduction1.1. A motivation from the theory of ODEsIn the study of boundary value problems for nonlinear ODEs, the shooting method,in spite of being sometimes considered as an old fashioned technique, is still a quitepowerful and effective tool in various different situations. For instance, as a samplemodel, let us consider the generalized Sturm−Liouville problem for a second orderequation of the formu ′′ + f (t, u, u ′ ) = 0, (u(t 0 ), u ′ (t 0 )) ∈ Ŵ 0 , (u(t 1 ), u ′ (t 1 )) ∈ Ŵ 1 ,where f = f (t, x, y) : [t 0 , t 1 ] × R 2 → R is a continuous function satisfying a locallyLipschitz condition with respect to (x, y) and Ŵ 0 and Ŵ 1 are two unbounded closedconnected subsets of the plane R 2 . Using the shooting method, one can start from theCauchy problem (for which we have the uniqueness of the solutions and their continuousdependence upon the initial values)u ′′ + f (t, u, u ′ ) = 0, (u(t 0 ), u ′ (t 0 )) = (x 0 , y 0 ) := z 0 ,with z 0 ∈ Ŵ 0 and, having denoted by ζ(·;t 0 , z 0 ) the corresponding solution of theequivalent first order system in the phase-plane(1) x ′ = y, y ′ = − f (t, x, y),with ζ(t 0 ) = z 0 , look for the intersections between Ŵ 1 and the set Ŵ0 ′ := {ζ(t 1; t 0 , z 0 ) :z 0 ∈ Ŵ 0 }. Clearly, any point z 1 ∈ Ŵ0 ′ ∩ Ŵ 1 is the value (u(t 1 ), u ′ (t 1 )) correspondingto a solution u(·) of the original Sturm−Liouville problem and different points in theintersection of Ŵ 1 with Ŵ0 ′ are associated to different solutions as well. In the simpler∗ Supported by the MIUR project ”Equazioni Differenziali Ordinarie e Applicazioni” (PRIN 2005).115