Characteristics of Poincaré Recurrences

Abstract

The analysis of Poincaré recurrences is one of the fundamental problems in the theory of dynamical systems. Poincaré recurrence means that practically any phase trajectory starting from some point of the system phase space passes arbitrarily close to the initial state an infinite number of times. H. Poincaré called these phase trajectories stable according to Poisson. Since Poincaré’s day, the analysis of the dynamics of Poisson stable systems has been an active topic of research in both mathematics and physics. The fundamental importance of this problem is evidenced by the fact that the very idea that a system should return over time to a neighborhood of its initial state is used much more widely than in mathematical theory alone. Thus, in a certain sense, it has become one of the philosophical concepts of modern science.