The existence proof for Hopf bifurcations

In a Hopf bifurcation a pair of complex conjugate eigenvalues of the linearization of a dynamical system $\dot x=f(x,\alpha)$ at a stationary point pass through the imaginary axis. This has been discussed in a previous post. Often textbook results (e.g. Theorem 3.3 in Kuznetsov’s book) concentrate on the generic case where two additional conditions are satisfied. One of these is that the first Lyapunov coefficient is non-vanishing. The other is that the eigenvalues pass through the imaginary axis with non-zero velocity. The existence of periodic solutions can be obtained if only the second of these conditions are satisfied. This was already included in the original paper of Hopf in 1942. Hopf states his results only in the case of analytic systems but this should perhaps be seen as a historical accident. A similar result holds with mucher weaker regularity assumptions. It is proved under the assumption of $C^2$ dependence on $x$ and $C^1$ dependence on $\alpha$ in Hale’s book on ordinary differential equations. This has consequences for the case where the second genericity assumption is not satisfied. Let $\lambda$ be an eigenvalue which passes through the imaginary axis for $\alpha=0$ and suppose that the derivatives of ${{\rm Re}\lambda}$ with respect to $\alpha$ vanish up to order $2k$ for an integer $k$ but that the derivative of order $2k+1$ does not vanish. Then it is possible to replace $\alpha$ by $\alpha^{2k+1}$ as parameter and after this change the second genericity assumption is satisfied. Even if the original right hand side was analytic in $\alpha$ the transformed right hand side is in general not $C^2$. It is, however, $C^1$ and so the version of the theorem in Hale’s book applies to give the existence of periodic solutions. This theorem applies to a two-dimensional system but it then also evidently applies in general by a centre manifold reduction.

The theorem is proved as follows. The problem is transformed to polar coordinates $(\rho,\theta)$ and then $\rho$ is written as a function of $\theta$. In this way a non-autonomous scalar equation with $2\pi$-periodic coefficients is obtained and the aim is to find a $2\pi$-periodic solution. The first step is to reformulate the task as a fixed-point problem with the property that if a fixed point is periodic it will be a solution of the original problem. Then it is shown using the Banach fixed point theorem(in a minor variant of the local existence theorem for ODE using Picard iteration) that there always exists a fixed point depending on a certain new parameter. This fixed point is only periodic if the result of substituting it into the right hand side of the original equation has mean value zero. This condition can be written as $G(\alpha,a)=0$. Applying the implicit function theorem to $G$ shows the existence of a solution of $G(\alpha(a),a)=0$ for $a$ small. This completes the proof.

Summing up, there are two types of theorem about Hopf bifurcation, a ‘coarse’ theorem of the type just sketched with weak hypotheses and a weak but still very interesting conclusion and a ‘fine’ theorem which gives stronger conclusions but needs a stronger hypothesis (non-vanishing of the Lyapunov coefficient and its sign). In his original paper Hopf proved both types. Are there also ‘rough’ versions of theorems about other bifurcations?

One Response to “The existence proof for Hopf bifurcations”

1. hydrobates Says:

I noticed that one important part of what I wrote here is not correct. The transformed system need not be continuously differerentiable
in general. Thus the argument I sketched above with the transformation is not valid. The ‘coarse’ version of the theorem on Hopf bifurcations can nevertheless be useful in other ways.

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