A theory for non-smooth dynamic systems on the connectable domains. (English) Zbl 1065.34007

This paper studies a certain class of nonsmooth dynamical systems in the plane. Let \(\Omega\) be a region of \(\mathbb{R}^2\), decomposed as the disjoint union \(\Omega = \bigcup_{i=1}^n \Omega_i \). Consider a dynamical system \(\dot{x} = F (x,t;\mu)\) on \(\Omega\), depending on the parameters \(\mu \in \mathbb{R}^n\). It is assumed that \(F\) is smooth and Lipschitz in each of the \(\Omega_i\) (call \(F^{(i)}\) the restriction of \(F\) to \(\Omega_i\)), and \(\Xi \subset \Omega\) is the set on which \(F\) cannot be expressed as a piecewise smooth and continuous vector function. The subdomains \(\Omega_i\) are termed accessible, while \(\Xi\) is termed inaccessible; \(\Omega\) is said to be connectable if the union of the \(\Omega_i\) is connected, and separable otherways (these terms are not standard).
With this notation, the paper studies nonsmooth dynamical systems on connectable subdomains. First of all, properties of separating boundaries based on the characteristics of the flows \(F^{(i)}\) are determined. Then, the local singularity and transversality properties – and correspondingly, the bouncing and tangency of solutions – of a flow at the boundaries between subdomains \(\Omega_i\) are studied. In particular, sufficient and necessary conditions for a local singularity, transversality and bouncing of flows are investigated. The sliding dynamics at boundaries is also considered.


34A36 Discontinuous ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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