Periodic orbit

Definition

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be locally Lipschitz, and consider the autonomous system $x' = f(x)$. A solution $\gamma : \mathbb{R} \to U$ is periodic if it is not constant and there is a real number $T > 0$ with $\gamma(t + T) = \gamma(t)$ for each $t \in \mathbb{R}$. The smallest such $T$ is the period of $\gamma$. A periodic orbit, or closed orbit, is the trajectory $\gamma(\mathbb{R})$ of a periodic solution $\gamma$.

Remarks

A non-constant periodic solution has a smallest period $T > 0$, because $\gamma$ is continuous and not constant. A periodic orbit is a simple closed curve in $U$. Let a maximal solution $\gamma$ that is not constant satisfy $\gamma(t_1 + T) = \gamma(t_1)$ for one $t_1$ and one $T > 0$. Then, by uniqueness, $\gamma$ is defined on $\mathbb{R}$ and is periodic. For the harmonic oscillator $x' = y$, $y' = -x$, each circle $x^2 + y^2 = r^2$ with $r > 0$ is a periodic orbit of period $2\pi$.

Explanations

L1 human

A periodic orbit is a motion that repeats itself exactly after a fixed time. In the phase space it is a closed loop that the state goes around again and again.