Lyapunov Functions in the Global Analysis of Chaotic Systems

We present an overview of the development of the direct Lyapunov method in the global analysis of chaotic systems and describe three directions of application of the Lyapunov functions: in the methods of localization of global attractors, where the estimates of dissipativity in Levinson’s sense are obtained, in the problems of existence of homoclinic trajectories, and in the estimation of the dimensions of attractors. The efficiency of construction of Lyapunov-type functions is demonstrated. In particular, the Lyapunov dimension formula is proved for the attractors of the Lorentz system.