Quasi-stability method in study of asymptotic behavior of dynamical systems

In this survey, we have made an attempt to present the contemporary ideas and methods of investigation of qualitative dynamics of infinite dimensional dissipative systems. Essential concepts such as dissipativity and asymptotic smoothness of dynamical systems, global and fractal attractors, determining functionals, regularity of asymptotic dynamics are presented. We place the emphasis on the quasi-stability method developed by I. Chueshov and I. Lasiecka. The method is based on an appropriate decomposition of the difference of the trajectories into a stable and a compact parts. The existence of this decomposition has a lot of important consequences: asymptotic smoothness, existence and finite dimensionality of attractors, existence of a finite set of determining functionals, and (under some additional conditions) existence of a fractal exponential attractor. The rest of the paper shows the application of the abstract theory to specific problems. The main attention is paid to the demonstration of the scope of the quasi-stability method.