Localization of invariant compact sets of dynamical systems. (English. Russian original) Zbl 1133.34342
Differ. Equ. 41, No. 12, 1669-1676 (2005); translation from Differ. Uravn. 41, No. 12, 1597-1604 (2005).
The paper is a continuation of previous work of the author related to the problem of localization of invariant compact sets of dynamical systems. Important applications are, for instance, localization of sets containing attractors, periodic trajectories, separatrices, etc. The problem reduces to the study of the least upper bound and the greatest lower bound for a function on a certain set. This new problem has a special form and possesses some interesting properties. The theorem justifying the method for localization of invariant compact sets is stated, and a sufficient condition for the absence of invariant compact sets in a subset of the phase space is derived as a corollary of the main result. The technique is illustrated on the example of the Lorenz system. Efficiency of different approaches to localization problem is discussed, and an iterative procedure for localization of invariant compact sets is suggested.
Reviewer: Svitlana P. Rogovchenko (Famagusta)
MSC:
| 34D45 | Attractors of solutions to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
Keywords:
dynamical systems; invariant compact sets; attractors; localization problem; iterative procedureReferences:
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