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?
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$\begingroup$ This question is much better than your previous. First, which are your definitions of stability, asymptotic stability? $\endgroup$– BrethloszeNov 26, 2017 at 4:26
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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.
answered Nov 26, 2017 at 4:58
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$\begingroup$ 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 :) $\endgroup$– EvgenyNov 26, 2017 at 15:05