| Full text | |
| Author(s): |
da Silva, Guilherme Tavares
;
Martins, Ricardo Miranda
Total Authors: 2
|
| Document type: | Journal article |
| Source: | NONLINEAR DYNAMICS; v. 108, n. 4, p. 28-pg., 2022-04-13. |
| Abstract | |
One of the most common hypotheses on the theory of non-smooth dynamical systems is a regular surface as switching manifold, at which case there is at least well-defined and established Filippov dynamics. However, systems with singular switching manifolds still lack such well-established dynamics, although present in many relevant models of phenomena where multiple switches or multiple abrupt changes occur. In this paper, we leverage a methodology that, through blow-ups and singular perturbation, allows to extend Filippov dynamics to the singular case. Specifically, tridimensional systems whose switching manifold consists of an algebraic manifold with transversal self-intersection are considered. This configuration, known as double discontinuity, represents systems with two switches and whose singular part consists of a straight line, where ordinary Filippov dynamics is not directly applicable. For the general, nonlinear case, beyond defining the so-called fundamental dynamics over the singular part, general theorems on its qualitative behavior are provided. For the affine case, however, theorems fully describing the fundamental dynamics are obtained. Finally, this fine-grained control over the dynamics is leveraged to derive Peixoto-like theorems characterizing semi-local structural stability. (AU) | |
| FAPESP's process: | 18/03338-6 - Global dynamics of piecewise smooth dynamical systems |
| Grantee: | Ricardo Miranda Martins |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 21/08031-9 - Piecewise smooth vector fields on compact manifolds |
| Grantee: | Ricardo Miranda Martins |
| Support Opportunities: | Regular Research Grants |
| FAPESP's process: | 18/13481-0 - Geometry of control, dynamical and stochastic systems |
| Grantee: | Marco Antônio Teixeira |
| Support Opportunities: | Research Projects - Thematic Grants |