Nonautonomous Dynamics pp 351-413 | Cite as

# Gradientlike Dynamical Systems

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## Abstract

In this chapter we give a complete description of the structure of compact global (forward) attractors for nonautonomous perturbations of autonomous gradientlike dynamical systems under the assumption that the original autonomous system has a finite number of hyperbolic stationary solutions. We prove that the perturbed nonautonomous (in particular \(\tau \)-periodic, quasiperiodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) system has exactly the same number of invariant sections (in particular, perturbed systems have the same number of \(\tau \)-periodic, quasiperiodic, Bohr almost periodic, almost automorphic, recurrent in the sense of Birkhoff) solutions. It is shown that a compact global (forward) attractor of a nonautonomous perturbed system coincides with the union of unstable manifolds of this finite number of invariant sections.