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Matrix semigroups and differential equations: Ergodism and asymptotic behaviour of solutions with or without impulses, a survey. (English) Zbl 0893.34050
The author presents a condensed survey on the theory of matrix semigroups, ergodism and asymptotic behaviour of differential equations using exponential semigroup theory, mainly without stating proofs. This work can be considered as a continuation of the author’s paper [J.-P. Caubet, Proc. 5th Int. Colloq. Differ. Eq., Utrecht, VSP, 61-72 (1995; Zbl 0843.60101)]. Here he presents statements and theorems on nonstationary matrix semigroups and their structure, isomorphism of cones between equivalent semigroups, continuous deformation of the extremal laws, generalized constants of variation formula, Taylor expansion for solution-propagation, ergodic behaviour and exit cones. The matrix semigroup is assumed to have a finite absolute value as its nonnegative majorant.
The paper is supplemented by an illustrative example of a nonlinear differential equation with limit cycles where the author studies the asymptotic structure of corresponding matrix semigroups (i.e. limit circles with different radii) depending on entrance data.
This paper can be recommended presuming the reader is heavily acquainted with semigroup theory.
Reviewer: Henri Schurz
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
47H20 Semigroups of nonlinear operators
Full Text: DOI
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