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 ẋ = Ax+f (t, x) x ˙ = A x + f (t, x) and influences the y –variable through ẏ = 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.
Bortolan et al. (Mon,) studied this question.