Periodic Solutions and Chaotic Dynamics in Forced Impact Oscillators

Abstract

It is shown that a periodically forced impact oscillator may exhibit chaotic dynamics on two symbols, as well as an infinity of periodic solutions. Two cases are considered, depending on if the impact velocity is finite or infinite. In the second case, the Poincaré map is well defined by continuation of the energy. The proof combines the study of phase-plane curves together with the "stretchingalong-paths" notion.

Periodic bouncing solutions of the Lazer-Solimini equation with weak repulsive singularity

“…For the analogous equation with attractive nonlinearity (that is, changing the sign of the second term of the left-hand side of the equation), the notion of bouncing solution has been adequately defined and studied in a number of papers [4,5,7,9,13,12,15]. In contrast, it remains unexplored for the repulsive case.…”

Chaotic Dynamics in an Impact Problem

“…Actually, in [11] we proved that if f is regular then for every real number ω sufficiently large, there exists a solution with rotation number ω. Among the huge amount of results concerning such model we cite [4,7,10,13,17]. It is worth mentioning also the paper by Dolgopyat [5] dealing with non-gravitational potentials and the paper by Kunze and Ortega [9] dealing with non-periodic functions f .…”

“…See [8,14] for more details. On this line, in a recent paper, Ruiz-Herrera and Torres [17] considered the application of a general topological tool based on stretching techniques. In this way, they constructed some particular periodic functions f for which the corresponding model of bouncing ball shows chaotic dynamics.…”

Combinatorial approach to detection of fixed points, periodic orbits, and symbolic dynamics

“…The method in this paper has potential applications to computer assisted proofs in dynamical systems. In this area, the method of correctly aligned windows has been used extensively to prove the existence of fixed points, periodic orbits, symbolic dynamics, and much more, when combined with other tools; some references include [24,11,20,22,2,7,6,17,3]. As mentioned earlier, the correct alinement of windows requires that the crossing of the windows is topologically non-trivial.…”