How to find the Lyapunov function for the following system?

Question

How to find the Lyapunov function for the following ODE system \begin{align} \dot{x} &= ax - x^3 + y\\ \dot{y} &= x^3 \end{align} where $a$ is just a constant.

Answer 1

In general, Lyapunov functions will be "bowl-shaped", i.e. even powered in spatial variables. For a Lyapunov function $L(x,y)$, solution trajectories are decreasing, i.e. $$ \frac{d}{dt}L(x(t),y(t))=\nabla L(x(t),y(t))\cdot(\dot{x},\dot{y})=\frac{\partial}{\partial x}L(x(t),y(t))\dot{x} + \frac{\partial}{\partial y}L(x(t),y(t))\dot{y} < 0 $$

One usually tries Lyapunov functions of the form $L(x,y)=\alpha x^m + \beta y^n$ and picks $\alpha$, $\beta$, $m$ and $n$ to make the time derivative negative-definite. Try this Lyapunov function in your case and pick the correct exponents to make everything negative-definite.

The condition that $dL/dt$ be negative definite can be weakened by requireing that $dL/dt$ be negative definite almost everywhere that you care about.

Edit:

This function does not have any corresponding Lyapunov functions. Linearizing around the sole fixed point at $(0,0)$ gives degenerate stability. Graphical analysis (or numerical simulation) shows that the origin is always unstable for any value of $a$. Hence, there is no stability candidate for a Lyapunov function and analysis via a Lyapunov function is not a valid strategy here.

Existence of a Lyapunov function is a method for extending the stability of a fixed point from a neighborhood in which linear stability analysis is valid to a larger region, and potentially globally.