Singular perturbation techniques for non-smooth dynamical systems

• Grant number: 12/09770-0
• Support Opportunities: Regular Research Grants
• Duration: August 01, 2012 - July 31, 2014
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Paulo Ricardo da Silva
• Grantee: Paulo Ricardo da Silva
• Host Institution: Instituto de Biociências, Letras e Ciências Exatas (IBILCE). Universidade Estadual Paulista (UNESP). Campus de São José do Rio Preto. São José do Rio Preto , SP, Brazil

Abstract

We consider a differential equation $\dot{p}=X(p),\quad p\in\R^3$ with discontinuous right-hand side and discontinuities occurring on a variety $\Sigma.$ We discuss several aspects of the dynamics of the sliding mode which occurs when for any initial condition near $p\in \Sigma$ the corresponding solution trajectories are attracted to $\Sigma$. First we suppose that $\Sigma=H^{-1}(0)$ where $H$ is a polynomial function and $0\in\R$ is a regular value. In this case $\Sigma$ is locally diffeomorphicto the set $\mathcal{F}=\{(x,y,z)\in\R^3; z=0\}$ (Filippov). Secondwe suppose that $\Sigma$ is the inverse image of a non-regularvalue. We focus our attention when $\Sigma$ is locally diffeomorphic to one of the following sets $\mathcal{D}=\{(x,y,z)\in\R^3; xy=0\}$ (double crossing); $\mathcal{T}=\{(x,y,z)\in\R^3; xyz=0\}$ (triple crossing); $\mathcal{C}=\{(x,y,z)\in\R^3; z^2-x^2-y^2=0\}$ (cone) and$\mathcal{W}=\{(x,y,z)\in\R^3; zx^2-y^2=0\}$ (Whitney's umbrella). (AU)