Lyapunov Function

For the system $\dot{x}=F(x)$ with fixed point $x^{\ast }$. If there is some differentiable function $L:U\to \mathbb{R}$ on the neighbourhood $U\subset {\mathbb{R}}^n$ such that:

1. $L(x)>L(x^{\ast })$ for any $x\in U\setminus \{x^{\ast }\}$
2. $\dot{L}({\varphi }^t(x))<0$ for any ${\varphi }^t(x)\in U\setminus \{x^{\ast }\}$

Then $L$ is a Lyapunov function about $x^{\ast }$

Attracting Fixed Points

If $L$ is a Lyapunov function for $x^{\ast }$ and $U_{c_0}$ is bounded for some $c_0>0$ then $x^{\ast }$ is asymptotically stable.

Exclusion of Periodic Orbits

If we have the Lyapunov function for the fixed point $x^{\ast }$ on $U$ then there is no periodic orbit passing through $U$.