# Is there nonlinear system have both stable and asymptotically stable equilibrium points?

A nonlinear dynamical system can have multiple equilibrium points with different characteristics.

I know that a pendulum with friction model

$$\dot x_1 = x_2$$

$$\dot x_2 = -\dfrac{Mgl}{I} \sin(x_1) - \dfrac{k}{I}x_2$$

can have an asymptotically stable equilibrium at $(\theta, \dot \theta) = (0,0)$ and an unstable equilibria at $(\theta, \dot \theta) = (\pi, 0)$

Is there a dynamical system that have both a stable and an asymptotically stable equilibria in its system?

• This question (v1) seems like a list question. – Qmechanic Apr 10 '16 at 10:47

## 1 Answer

Take a potential energy of the form $$U(x)= x(x-1)\:.$$ Next consider a smooth function $g: \mathbb R \to [0,1]$ such that vanishes in an open neighborhood of $0$ and takes the constant value $1$ form $x= 1/4$ on. Finally consider the dynamical system defined by $$\dot{x}=y$$ $$\dot{y}= -U'(x) - cg(x)y$$ for some $c>0$. With these choices, $(x,y)=(0,0)$ is stable but not asymptotically stable, $(x,y)=(1/2,0)$ is unstable, and $(x,y)=(1,0)$ is stable and asymptotically stable. The proof of the first and third fact easily arises from Liapunov's theorem (or Lagrange-Dirichlet's one) the second via linearisation.