Requirement of Lyapunov Stability in Asymptotic Stability

Question

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:

1. Lyapunov Stable
2. 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.

Answer 1

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.)

Answer 2

One classic example of a point that satisfies property (2) but is not Lyapunov stable is the saddle point.

A saddle point is an equilibrium point of a dynamical system where some trajectories converge to the point, while others diverge away from it. More precisely, a saddle point has at least one eigenvalue with positive real part and at least one eigenvalue with negative real part.

Consider the following two-dimensional dynamical system:

$$dx/dt = x$$ $$dy/dt = -y$$

The origin $(0, 0)$ is an equilibrium point of this system. The linearization of the system at the origin yields the matrix:

$A = [1, 0; 0, -1]$

The eigenvalues of $A$ are 1 and -1, which means that the origin is a saddle point.

Now, let us consider an epsilon ball around the origin. Any trajectory that starts within this ball will either converge to the origin along the x-axis or diverge away from the origin along the y-axis. Thus, property (2) holds for the origin.

However, the origin is not Lyapunov stable since the trajectories diverge away from the origin along the y-axis. Any small perturbation in the y-direction will cause the trajectory to move further away from the origin, and the distance between the trajectory and the origin will grow exponentially with time.

Therefore, the origin satisfies property (2) but is not Lyapunov stable.