Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations
The paper deals with the second-order differential equation \[ x''=f(\theta_th,x,x'). \tag{1} \] Here, \((H,d)\) is a compact metric space and \(\theta=(\theta_t)_{t\in R}\) is a dynamical system on \(H\). \(H\) is supposed to be minimal with respect to \(\theta\), that is \(\theta\) has no proper closed invariant subset in \(H\). Further \(f(h,x,p)\) is continuous on \(H\times (a,b)\times\mathbb R)\), satisfies the Lipschitz condition in \((x,p)\) and is strictly increasing in \(x\). Moreover \(f\) fulfils a quadratic growth condition in \(p\). The authors provide descriptions of the structure and behavior of bounded solutions of (1). They also prove the stability of the structure of bounded solutions with respect to parameter perturbations in \(f\). The proofs use the theory of pullback attractors for cocycle dynamical systems.
scientific article; zbMATH DE number 5530619