 Softening Rayleigh-Duffing oscillator

The softening restoring force and cubic velocity damping give the energy $E=(z^{\prime }{)}^2/2+z^2/2-z^4/4$ with $\dot{E}=-A(z^{\prime }{)}^2-(z^{\prime }{)}^4$. The origin and the two saddle points $z=\pm 1$ persist for all parameters. The origin undergoes a supercritical Hopf bifurcation at $A=0$. A heteroclinic cycle and a subsequent saddle-node bifurcation of periodic orbits are distinct global changes in its softening Rayleigh-Duffing phase portrait.