2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ This question (v1) seems like a list question. $\endgroup$ – Qmechanic Apr 10 '16 at 10:47
1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.