<< The main inspiration of (this AA) work is the paper of Llibre and Teixeira (Nonlinear Dyn. 91, No. 1, 249-255, 2018) about Filippov systems formed by two linear centers and having x = 0 as discontinuity line. One of the main conclusions of the paper is that such systems cannot have limit cycles. Actually, either it does not have periodic orbits or every orbit is periodic. Therefore, its dynamics is relatively simple. Inspired by this work and the raising notion of hybrid systems, (AA) wondered what could happen if we allow jumps on the discontinuity line. As a result, (They) discovered not only that limit cycles are allowed with arbitrarily small “perturbations” in the jump, (..), but also that such systems allow chaotic dynamics. Therefore, (AA) conclude that hybrid systems with simple formulation can have rich dynamics. (They) also observe that a complete characterization of the dynamics of X ∈ Xn depends on the characterization of its first return map, which is a piecewise polynomial map on the real line. This, together with the fact that the systems studied here are a generalization of the Filippov systems (..), illustrates that hybrid systems can be seen as a three-fold bridge connecting continuous, piecewise continuous and discrete dynamical systems. >>️
Jaume Llibre, Paulo Santana. Limit cycles and chaos in planar hybrid systems. arXiv: 2407.05151v2 [math.DS]. Oct 1, 2024.
Also: chaos, transition, in https://www.inkgmr.net/kwrds.html
Keywords: gst, limit cycles, chaos, transitions, small perturbations, jumps
