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Differential Equations

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Locally Linear System with Pure Imaginary Eigenvalues

DIFFEQ-BKRMFN

Consider the critical point $(0,0)$ of the system

$\dot{x}=-y-x^3$,
$\dot{y}=x-y^3$.

This is a locally linear system, but the eigenvalue analysis for stability fails here since the eigenvalues are $\pm i$. Consider the Lyapunov function $V=x^2+y^2$ which is clearly positive definite. Using this Lyapunov function, we can determine that $(0,0)$ is

A

We can't figure out the stability using $V$.

B

Unstable.

C

Stable but not asymptotically stable.

D

Asymptotically stable.