Fixed points and stability of them

Question

Find the fixed points and classify them for the system of equations:

$$x'=v$$ $$v'=-x+wx^3$$ $$w'=-w$$

$$0=v$$ $$0=-x+wx^3$$ $$1=wx^2$$ $$0=w$$

is the only fixed point (0,0,0)??

jacobian: \begin{array}{lcr} \mbox0 & 1 & 0 \\ \mbox-1+3wx^2 & 0 & x^3 \\ \mbox0 & 0 & -1 \\ \end{array}

evaluated at the fixed point:

\begin{array}{lcr} \mbox0 & 1 & 0 \\ \mbox-1 & 0 & 0 \\ \mbox0 & 0 & -1 \\ \end{array}

eigenvalues: -1, i, -i

Answer 1

(0,0,0) is indeed your only fixed point, but when you evaluated $-1 +3wx^2$, you got 0 instead of -1 like you should have. Switching to -1 in row 2, column 1, you get eigenvalues of $-1,\pm i$. Hence you have a nonhyperbolic fixed point, you have stability but not asymptoptically stable if I recall correctly for this case.