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i was given the following assignment: FitzHugh proposed the dynamical system

[tex] \dot{x}=-x(x-a)(x-1)-y+I [/tex]

[tex]\dot{y}=b(x-\gamma y)[/tex]

to model neurons. I is the input signal, x is the activity, y is the recovery and 0<a<1, [itex]\gamma[/itex]>0, b>0 are constants. Prove that there is a critical value [itex]I_0[/itex] 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 [Broken]). I find the Jacobian to be

[tex] J=\begin{pmatrix} -a&-1\\b&-b\gamma \end{pmatrix}[/tex]

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.

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# Homework Help: Finding the Hopf Bifurcation

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