 Trapping region

A trapping region for a continuous-time dynamical system is a compact region whose boundary directs the vector field inward, so every forward trajectory that enters remains confined. For a smooth boundary $h(x)=0$ with interior $h<0$, a sufficient strict condition is $\nabla h\cdot f<0$ on the boundary. The chain rule then prevents a first outward crossing.

For $\dot{x}=-x$, $\dot{y}=-y$, the disk $x^2+y^2\le R^2$ is trapping because the orbital derivative of $x^2+y^2-R^2$ is $-2R^2$ on its boundary. A Lyapunov function often supplies such regions through its sublevel sets.

Forward invariance alone is weaker than strict inward trapping: a boundary can itself contain an invariant trajectory with zero normal velocity. The distinction matters when applying a theorem that requires strict boundary inequalities rather than confinement alone.