Jumps of the fundamental solution matrix in discontinuous systems and applications (Q878528)

The author studies discontinuous systems in case of trajectories crossing the discontinuity surfaces transversally. Three types of systems are considered: 1) \(\dot{x}=f(t,x)\), where \(f\) is discontinuous; 2) \(F(t,x,\dot{x})=0\), where the implicit function theorem cannot be used; 3) \(F(t,x,\dot{x})=0\), where the equation has more than one solution with respect to \(\dot{x}\). Theorems are proved concerning systems with respect to unique solutions, and the behaviour of the solutions are studied. Theorems concerning the fundamental solution matrix are addressed. It is also considered how the discontinuities causes jumps of this matrix of the corresponding variational equation. Non-smooth Poincaré mappings are studied regarding autonomous systems \(\dot{x}=f(x)\). Eigenvalues of the derivative of such mappings of periodic solutions are considered. Finally a couple of examples are choosen to demonstrate the uselfulness of the theory.