Past exam of the differential equations and applications course of the University of Oxford / 2008 / A / 1 / a

Set $y=\dot{x}$. The phase-plane system is $\dot{x}=y$, $\dot{y}=-9x+ϵ(1-4x^2)y$. The origin is its only equilibrium, and its Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ -9& ϵ\end{array}\right)$. ThusFor the stated small positive $ϵ$, the eigenvalues have positive real part and nonzero imaginary part: the origin is an unstable focus.