Title of article :
On dynamical properties of general dynamical systems and differential inclusions ✩
Author/Authors :
Desheng Li، نويسنده ,
Issue Information :
دوهفته نامه با شماره پیاپی سال 2002
Pages :
20
From page :
705
To page :
724
Abstract :
Let F be a general dynamical system defined on a complete locally compact metric space X. We give a slightly improved version of the Lyapunov characterization of asymptotic stability in one of our previous works and provide a short self-contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context. Based on these results, we prove that the uniform attractors of F are connected and are stable with respect to perturbations under appropriate conditions. We are also interested in the dynamical properties of differential inclusion: x (t ) ∈ f (x(t)) on Rm. First, we show that if no solutions of the system blow up in finite future time, then its reachable mapping F is a general dynamical system. Then we discuss some asymptotic stability properties of the system. In particular, we prove that if there exists a nonempty compact connected subset M of Rm such that M attracts a neighborhood of itself, then the system has a connected uniform attractor A. We also prove that A is stable with respect to both internal and external perturbations. More precisely, we prove that when λ > 0 is sufficiently small, the perturbed system: x (t ) ∈ conf (x(t)+λ B1)+λ B1 has a connected uniform attractorAλ; moreover, δH(Aλ,A)→0 as λ→0, where δH(·, ·) is the Hausdorff distance in Rm.  2002 Elsevier Science (USA). All rights reserved.
Journal title :
Journal of Mathematical Analysis and Applications
Serial Year :
2002
Journal title :
Journal of Mathematical Analysis and Applications
Record number :
930217
Link To Document :
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