Invariant set

Remarks

A set is invariant exactly when it is a union of trajectories. Example: for $x' = -y$, $y' = x$ on $\R^2$, the flow is rotation by the angle $t$, so it preserves $x^2 + y^2$; hence each disc $\set{x^2 + y^2 \le r^2}$ is invariant. Non-example: for $x' = -x$ on $\R$, the interval $[0, 1]$ is positively invariant, since $\varphi_t(x) = e^{-t} x$ stays in $[0, 1]$ for $t \ge 0$. It is not invariant, since $\varphi_{-1}(1) = e > 1$. An equilibrium point and a periodic orbit are invariant sets. For a continuously differentiable function $V \colon U \to \R$ with $\nabla V \cdot f \le 0$ on $U$, each sublevel set $\set{x \in U : V(x) \le c}$ is positively invariant. For a general flow $\varphi$ on a set $X$ the same definition reads $\varphi_t(M) \subseteq M$ for all $t$.