This post is a tribute to Alexnadr Lyapunov whose work on dynamical systems provided novel insights in understanding nonlinear dynnamics.

Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE

in the equation \ref{test}

Consider an autonomous nonlinear system of the form: \begin{equation} \dot{x}=f(x), \quad x(0)=x_0 \label{eq:auton_sys}, $$ where $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ is called vector field.

The vector $\bar{x} \in \mathbb{R}^n$ is an equilibrium point of~\eqref{eq:auton_sys} if $f(\bar{x})=0$. Let $\phi(t,x_0)$ denote the associated flow, i.e. the solution of~\eqref{eq:auton_sys} at time $t$ with initial condition $x_0$. The region of attraction (ROA) associated with $\bar{x}$ is defined as: $$ \mathcal{R}_{\bar{x}}:=\big\{x_0 \in \mathbb{R}^n: \lim_{t\to\infty} \phi(t,x_0)= \bar{x} \big\}. $$ That is, $\mathcal{R}_{\bar{x}}$ is the set of all initial states that eventually converge to $\bar{x}$.