# Hopf bifurcation and limit cycles

$$dV/dt=10(V-\frac{V^3}{3}-R+I_{input})$$ $$dR/dt=0.8(-R+1.25V+1.5)$$

Use $I_{input}$ as the relative parameter to prove that there these equations undergo 2 hopf bifurcations and indicate whether each is subcritical or supercritical.

Setting both equal to 0 to find the equilibrium:

$0=V-\frac{V^3}{3}-R+I_{input}$

$0=R+1.25V+1.5$

I guess I don't really know how to continue from here.. find eigenvalues?

jacobian (not sure if I computed this correctly): \begin{array}{lcr} \mbox1-v^2 & -1 \\ \mbox1.25 & -1 \\ \end{array}

So I know that a supercritical hopf bifurcation is a stable LC around an unstable equilibrium and a subcritical hopf bifurcation is a an unstable LC around a stable equilibrium.

• Why would someone downvote this question..? – Math Major Nov 14 '14 at 17:53
• Is $I_{input}$ a constant or time varying? – Amzoti Nov 14 '14 at 21:46

1) You forgot the constants when you calculated the Jacobian matrix.

2) In general, when you want to prove theorically that a certain bifurcation occurs in a system, the way is verify that your system is under the hypothesis of the associate theorem (in this case the Hopf bifurcation theorem). In the case of local bifurcations (the ones associate to equilibria points), the first step is to check what are the eigenvalues of the linear part of the system (evaluated in the equilibrium point).