M. McCracken

Other Bifurcation Theorems

Several authors have published generalizations of the Hopf Bifurcation. In particular, Chafee [1] has eliminated the condition that the eigenvalue λ(µ) cross the imaginary axis with nonzero speed. In this case, bifurcation to periodic orbits occurs, but it is not possible to predict from eigenvalue conditions exactly how many families of periodic orbits will bifurcate from the fixed point. Chafee’s result gives a good description of the behavior of the flow of the vector field near the bifurcation point. See also Bautin [1] and Section 3C.

The Hopf Bifurcation Theorem for Diffeomorphisms

Let X be a vector field and let γ be a closed orbit of the flow ⌽t of X. Let P be a Poincare map associated with γ. (See §2B). Suppose there is a circle σ that is invariant under P. Then it is clear that Ut⌽t(σ) is an invariant torus for the flow of X (see Figure 6.1).If we have a one parameter family of vector fields and closed orbits Xµ and γµ , it is quite conceivable that for small µ, γµ might be stable, but for large µ it might become unstable and a stable invariant torus take its place. Recall that γµ is stable (unstable) if the eigenvalues of the derivative of the Poincare map Pµ have absolute value 1). (See §2B). The Hopf Bifurcation Theorem for diffeomorphisms gives conditions under which we may expect bifurcation to stable invariant tori after loss of stability of γµ. The theorem we present is due to Ruelle-Takens [1]; we follow the exposition of Lanford [1] for the proof.

The Canonical Form

We shall give here, following Lanford [1], an elementary and straightforward derivation of the canonical form for the mapping Φµ: ℝ2 → ℝ2. Recall that we have already arranged things so that

How to use the Stability Formula; An Algorithm

{{\phi }_{\mu }}\left( {\begin{array}{*{20}{c}} x \\ y \\ \end{array} } \right) = \left( {1 + \mu } \right)\;\left( {\begin{array}{*{20}{c}} {\cos \theta \left( \mu \right)} \\ {\sin \theta \left( \mu \right)} \\ \end{array} \quad \begin{array}{*{20}{c}} { - \sin \theta \left( \mu \right)} \\ {\cos \theta \left( \mu \right)} \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} x \\ y \\ \end{array} } \right) + 0\left( {{{{\text{r}}}^{2}}} \right)

The Center Manifold Theorem

We want to organize the second, third, and fourth degree terms by making further coordinate changes. It will be convenient to identify R2 with the complex plane by writing

Some Spectral Theory

{\text{z}} = {\text{x}} + {\text{iy}}