Equilibrium-Like Solutions of Asymptotically Autonomous Differential\n Equations

2017/12/21 by Axel Jänig, Jänig, Axel
Biochemistry, Genetics and Molecular Biology · Mathematics · #37B30 #37B55 #Caveolin-1 and cellular processes #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1712.08096

openalex publication_date 2017/12/21 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We analyze the chain recurrent set of skew product semiflows obtained from\nnonautonomous differential equations -- ordinary differential equations or\nsemilinear parabolic differential equations. For many gradient-like dynamical\nsystems, Morse-Smale dynamical systems e.g., the chain recurrent set contains\nonly isolated equilibria. The structure in the asymptotically autonomous\nsetting is richer but still close to to the structure of a Morse-Smale\ndynamical system.\n The main tool used in this paper is a nonautonomous flavour of Conley index\ntheory developed by the author. We will see that for a class of good equations,\nthe Conley index can be understood in terms of equilibria (in a generalized\nmeaning) and their connections. This allows us to find specific solutions of\nasymptotically autonomous equations and generalizes properties of Morse-Smale\ndynamical systems to the asymptotically autonomous setting.\n

Related

• Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations
• The generic gradient-like structure of certain asymptotically autonomous\n semilinear parabolic equations
• The Morse equation in the Conley index theory for discrete multivalued dynamical systems
• Conley Index Theory and the Attractor-Repeller Decomposition for Differential Inclusions
• Dynamical Systems: Discrete, Continuous and Hybrid