Theorems in dynamical systems

In dynamical systems, "theorems" refer to established results that describe the behavior of systems over time under certain conditions. Dynamical systems are mathematical models used to describe the evolution of points in a given space according to specific rules, often represented by differential equations or discrete mappings.

The Denjoy-Wolff theorem is a result in complex analysis, particularly in the field of iterated function systems and the study of holomorphic functions. It characterizes the dynamics of holomorphic self-maps of the unit disk, specifically focusing on the behavior of iterates of such functions.

The Hartman–Grobman theorem is a result in the field of differential equations and dynamical systems, named after mathematicians Philip Hartman and Robert Grobman. The theorem provides a powerful tool for analyzing the local behavior of nonlinear dynamical systems near equilibrium points.

The Poincaré–Bendixson theorem is a fundamental result in the field of dynamical systems, particularly concerning the behavior of continuous dynamical systems in two dimensions. It addresses the long-term behavior of trajectories in a planar (2-dimensional) system described by a set of ordinary differential equations.

Sharkovskii's theorem is a result in the field of dynamical systems, particularly concerning the behavior of continuous functions on the unit interval \([0, 1]\) and the periodic points of these functions. The theorem provides a remarkable ordering of natural numbers that relates to the existence and types of periodic points in continuous functions.