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Finding the Hopf Bifurcation

  1. Nov 2, 2007 #1
    Hello all,
    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). 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.

  2. jcsd
  3. Nov 2, 2007 #2


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    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.
  4. Nov 2, 2007 #3
    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 [itex]I_0[/itex], then this [itex]I_0[/itex] is a Hopf bifurcation point, and i have thus proven that such a point exists for the FitzHugh system.
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