Jumps of the fundamental solution matrix in discontinuous systems and applications
type: Item
instance of: scholarly article
title: Jumps of the fundamental solution matrix in discontinuous systems and applications
zbMATH Open document ID: 1121.34010
author: Peter Kukučka
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-04-26T00:00:00Z
review text: 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.
reviewed by: Jan Melin
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 34A09
zbMATH DE Number: 5146766
zbMATH Keywords: transversal intersection
zbMATH Keywords: periodic solution
MaRDI profile type: Publication
full work available at URL: j.na.2006.03.037
OpenAlex ID: W1982205265
cites work: Applications of global analysis to specific nonlinear eigenvalue problems
cites work: Bifurcation of periodic solutions in differential inclusions
cites work: Bifurcations in nonlinear discontinuous systems
cites work: Q4003574
DOI: 10.1016/J.NA.2006.03.037
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description: scientific article; zbMATH DE number 5146766