Lagrange stability

Lagrange stability is a concept in the stability theory of dynamical systems, named after Joseph-Louis Lagrange.

For any point in the state space, $$x \in M $$ in a real continuous dynamical system $$(T,M,\Phi)$$, where $$T$$ is $$\mathbb{R}$$, the motion $$\Phi(t,x)$$ is said to be positively Lagrange stable if the positive semi-orbit $$\gamma_x^+$$ is compact. If the negative semi-orbit $$\gamma_x^-$$ is compact, then the motion is said to be negatively Lagrange stable. The motion through $$x$$ is said to be Lagrange stable if it is both positively and negatively Lagrange stable. If the state space $$M$$ is the Euclidean space $$\mathbb{R}^n$$, then the above definitions are equivalent to $$\gamma_x^+, \gamma_x^-$$ and $$\gamma_x$$ being bounded, respectively.

A dynamical system is said to be positively-/negatively-/Lagrange stable if for each $$x \in M $$, the motion $$\Phi(t,x)$$ is positively-/negatively-/Lagrange stable, respectively.