Limit cycle

Definition

Let $U$ be an open subset of $\mathbb{R}^2$, let $f : U \to \mathbb{R}^2$ be continuously differentiable, and let $\varphi$ be the flow of $x' = f(x)$. Let $y \in U$ be a point whose solution exists for all $t \ge 0$. Its $\omega$-limit set $\omega(y)$ is the set of points $z \in U$ such that $\varphi_{t_k}(y) \to z$ for some sequence $t_k \to \infty$. For a point $y$ whose solution exists for all $t \le 0$, the $\alpha$-limit set $\alpha(y)$ is defined in the same way with $t_k \to -\infty$. A limit cycle is a periodic orbit $\Gamma$ such that $\Gamma = \omega(y)$ or $\Gamma = \alpha(y)$ for some point $y \in U$ that is not on $\Gamma$.

Remarks

A limit cycle is stable if all nearby trajectories approach it as $t \to \infty$, and unstable if they approach it as $t \to -\infty$. It can also be semi-stable. A linear system has no limit cycles: its periodic orbits, if any, fill a region around the origin. The van der Pol oscillator $x'' - \mu (1 - x^2) x' + x = 0$ with $\mu > 0$ has exactly one limit cycle, and it is stable. The Poincaré–Bendixson theorem is a main tool to prove that a limit cycle exists.

Explanations

L1 human

A limit cycle is an isolated loop in the phase plane that nearby motions spiral toward or away from. It models a self-sustained oscillation, such as a heartbeat, whose size does not depend on the initial state.