Finding the Hopf Bifurcation in the FitzHugh Model

[standardflop]

Hello all,
i was given the following assignment: FitzHugh proposed the dynamical system

$$\dot{x}=-x(x-a)(x-1)-y+I$$

$$\dot{y}=b(x-\gamma y)$$

to model neurons. I is the input signal, x is the activity, y is the recovery and 0<a<1, $\gamma$>0, b>0 are constants. Prove that there is a critical value $I_0$ for which there exists a Hopf bifurcation and discuss the stability of the periodic orbit.

I think one can find the bifurcation by using the Hopf Bifurcation Theorem (stated eg. at http://planetmath.org/encyclopedia/HopfBifurcationTheorem.html ). I find the Jacobian to be

$$J=\begin{pmatrix} -a&-1\\b&-b\gamma \end{pmatrix}$$

but this matrix has eigenvalues independent of the parameter I. How can i investigate when the real part of the eigenvalues of J have a positive derivative (with respect to I) when they seem not to be a function of I?

Any help will be greatly appreciated.

Regards.

 

[HallsofIvy]

You seem to be misunderstanding the problem. The problem is not to find the Hopf bifurcation point in terms of I but to determine the value of I such that there is a Hopf bifurcation.

 

[standardflop]

You seem to be misunderstanding the problem. The problem is not to find the Hopf bifurcation point in terms of I but to determine the value of I such that there is a Hopf bifurcation.

Yes; But isent this exactly what the Hopf Theorem states? I mean, if you find that it is possible to have purely imaginary eigenvalues to J with a positive derivative of the real part at some point $I_0$, then this $I_0$ is a Hopf bifurcation point, and i have thus proven that such a point exists for the FitzHugh system.