Poincaré-Bendixson theorem

The Poincaré-Bendixson theorem is a result in mathematics that describes the long-term behavior of certain two-dimensional systems, such as the motion of a pendulum or predator-prey populations. It states that if a system’s trajectory remains confined within a bounded area and doesn’t settle into a fixed point, then its path must eventually approach a repeating cycle, called a limit cycle. In simple terms, for specific systems, the long-term behavior can only be a stable repeating pattern or a steady state, helping scientists understand potential outcomes of dynamic processes.