1
$\begingroup$

Let $$ x'_{1} = x_{2} + x_{2}^3$$ $$ x'_{2} = -x_{1} - x_{1}^3 $$. show that origin is stable but not asymptotically stable. Can linearization method be use in this case?

$\endgroup$
1
  • $\begingroup$ This question is much better than your previous. First, which are your definitions of stability, asymptotic stability? $\endgroup$
    – Brethlosze
    Commented Nov 26, 2017 at 4:26

1 Answer 1

0
$\begingroup$

The linearized system at the origin is $$x'_1=x_2 ,\qquad x'_2=-x_1 , $$ which is a centre, making linearization useless. (There's no way of saying from this what effect the nonlinear terms might have; even a tiny perturbation make destroy a centre, in either a stable or an unstable way.)

Instead, try to find a constant of motion: $dx_2/dx_1 = x_2'/x_1' = \cdots/\cdots$, etc.

$\endgroup$
1
  • $\begingroup$ The constant of motion (or first integral) exists here, so you could have said that it will only lead to Lyapunov unstable or stable but not asymptotically stable equilibria :) $\endgroup$
    – Evgeny
    Commented Nov 26, 2017 at 15:05

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .