Multiple periodic solutions and complex dynamics for second order ODEs via linked twist maps

Abstract

We consider some nonlinear second order scalar ODEs of the form x + f (t, x) = 0, where f is periodic in the t-variable and show the existence of infinitely many periodic solutions as well as the presence of complex dynamics, even in the case of certain apparently "simple" equations. We employ a topological approach based on the study of linked twist maps (and suitable modifications of their geometry).

Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps

“…3 In our setting, it is also possible to have an invariant set Λ which contains as a dense subset the periodic points of φ and such that the counterimage (by the semiconjugacy g) of any periodic sequence (s i ) i in Σ p contains a periodic point of φ having the same period of (s i ) i (see [20] for the details).…”

“…Now we are ready to present the abstract setting for our method. Borrowing notation and terminology from [20], we start with the following definitions. By path γ in a metric space X we mean a continuous mapping γ : [t 0 , t 1 ] → X and we setγ := γ ([t 0 , t 1 ]).…”

“…In this framework, and when φ splits as the composition of two maps, we have the following theorem which is adapted to our context from [20]. …”

“…In many applications (see for example the one we present in Section 5 or the ones in [20,21]) one deals with two systems of the form (H) i , i = 1, 2, each one possessing a center C i , i = 1, 2, with C 1 = C 2 . Then, condition (LC) is satisfied if a closed orbit O 1 of system (H) 1 intersects transversally a closed orbit O 2 of system (H) 2 at a point z 0 = (x 0 , y 0 ).…”

Chaotic dynamics in periodically forced asymmetric ordinary differential equations

“…See [29] for a proof of Theorem 1.1 and [30] for some remarks and extensions. In the applications of Theorem 1.1 to the ODE models considered in this paper, we will have X = R 2 and the maps ψ r , ψ s will be the Poincaré maps associated to some planar systems.…”