Consider the nonlinear dynamical system given by the following differential equations
\begin{cases} \dot{x} = y, \\ \dot{y} = x - x^3 - \gamma y + \delta x^2 y. \end{cases}
I want to demonstrate that, in the particular case where $\gamma = \delta$, the equilibrium point $(1, 0)$ acts as an attractor in a neighborhood of this point. To approach this problem, I have tried to construct a suitable Lyapunov function $V$, wich satisfies
- Be positive definite in a neighborhood of $(1, 0)$, so that $V(x, y) > 0$ for all points $(x, y)$ near $(1, 0)$, except at the equilibrium itself, where $V(1, 0) = 0$.
- Have a negative time derivative in a neighborhood of $(1, 0)$ when evaluated along the trajectories of the system, i.e., $\dot{V} \leq 0$ in that region. This would ensure that $V(x, y)$ decreases over time, indicating that solutions tend to approach the equilibrium $(1, 0)$.
but I have not been able to find such a function. I would appreciate any suggestions or other ways to approach the problem.
