Continuity and topological structural stability for nonautonomous random attractors

Author(s)

Caraballo, Tomas ; Langa, Jose A. ; Carvalho, Alexandre N. ; Oliveira-Sousa, Alexandre N.

• Document type: Journal article
• Source: Stochastics and Dynamics; v. 22, n. 07, p. 28-pg., 2022-07-26.

Abstract

In this work, we study the continuity and topological structural stability of attractors for nonautonomous random differential equations obtained by small bounded random perturbations of autonomous semilinear problems. First, we study the existence and permanence of unstable sets of hyperbolic solutions. Then, we use this to establish the lower semicontinuity of nonautonomous random attractors and to show that the gradient structure persists under nonautonomous random perturbations. Finally, we apply the abstract results in a stochastic differential equation and in a damped wave equation with a perturbation on the damping. (AU)

• FAPESP's process: 20/14075-6 - Dynamical systems and their attractors under perturbations
• Grantee: Alexandre Nolasco de Carvalho
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 17/21729-0 - A study about structural stability of atrators for random dynamical systems
• Grantee: Alexandre do Nascimento Oliveira Sousa
• Support Opportunities: Scholarships in Brazil - Doctorate
• FAPESP's process: 22/00176-0 - Assymptotic behavior of stochastic evolutionary equações
• Grantee: Alexandre do Nascimento Oliveira Sousa
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 18/10633-4 - A study of structural stability for random attractors
• Grantee: Alexandre do Nascimento Oliveira Sousa
• Support Opportunities: Scholarships abroad - Research Internship - Doctorate