In my Differential Equations course, we defined the equilibrium point $x_0$ of a dynamical system $\frac{dx}{dt} = f(x(t))$ (for $f$ defined on an open subset of $\mathbb R^n$, say $\mathbb R^n$ itself) to be stable if it is:
- Lyapunov Stable
- There is an $\epsilon$ ball around $x_0$ such that the solutions $\varphi$ of this differential equation with initial conditions in this ball satisfy $\lim_{t \to \infty} \varphi(t) = x_0$.
I am trying to find an example of the case where the property (2) holds while the point $x_0$ is not Lyapunov stable.
After some searching, I ran across Homoclinic Bifurcation, which is intuitively how I would expect Lyapunov Stability to fail, but have been unable to find examples of Homoclinic Bifurcation where property (2) holds as well.
Any help would be appreciated.
Best Answer
Hint: Consider the system $$\begin{cases} \dot x = x-y-x(x^2+y^2)+\frac{xy}{\sqrt{x^2+y^2}}\\ \dot y = x+y-y(x^2+y^2)-\frac{x^2}{\sqrt{x^2+y^2}} \end{cases}$$ and its fixed point $(1,0)$. (Converting into polar coordinates might help.)