Fundamental matrix solutions of piecewise smooth differential systems
Section snippets
Introduction and background
Our purpose in this paper is to survey definitions and properties of the fundamental matrix solution associated to piecewise smooth differential equations. Many of the results we give are available in the literature, but are not all readily available. Moreover, some of the extensions we consider herein, such as when there is sliding motion on intersection of surfaces, appear new.
We study differential equations with discontinuous right-hand side, and more precisely equations in which the
Fundamental matrix solution: cross and/or slide on one surface
The fundamental matrix solution associated to the linearized system is a very useful tool in performing stability and bifurcation study of a smooth dynamical systems. It is natural to suspect that it should be a useful tool also for nonsmooth dynamical systems. In this and the next sections, we consider the fundamental matrix solutions for piecewise smooth systems. In this section, we look at the case in which we cross and/or slide on one surface. In the next section we also consider the case
Fundamental matrix solution: cross and/or slide on two surfaces
Now, suppose that the state space is split into four regions R1, R2, R3 and R4 by two intersecting hypersurfaces Σ1 and Σ2 which are defined by the scalar functions and , that is:(see Fig. 5). Consider the system with discontinuous right-hand side:with initial point . The functions h1(x) and h2(x) are assumed
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