Work

Gradient Structure of Dynamics for Nonautonomous Cascade Systems

Year: 2026

Type: article

Abstract: In this paper we prove the gradient structure of solutions for a nonautonomous cascade system defined on Banach spaces, where the x –variable evolves independently via $$\dot{x} = Ax+f(t,x)$$ x ˙ = A x + f ( t , x ) and influences the y –variable through $$\dot{y} = By+g(x,y)$$ y ˙ = B y + g ( x , y ) . By first analyzing the long–time dynamics of the nonautonomous x –equation and then examining the resulting y –dynamics for each asymptotic state of x , we provide a complete description of the system’s gradient structure in two levels: a more abstract and general, with less hypotheses on f , and a deeper level of description, when the term f ( t , x ) is asymptotically autonomous. Finally, we present a description when the term f ( t , x ) is a small nonautonomous perturbation of an autonomous term.

Source: Journal of Dynamics and Differential Equations

Authors: M. C. Bortolan, M. C. A. Brito, A. N. Carvalho, J. A. Langa

Institutions: Universidade Federal de Santa Catarina, Universidade de São Paulo, Universidad de Sevilla

Language: English

Topic: Nonlinear Differential Equations Analysis

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090