Interactions of Hopf and Pitchfork Bifurcations
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Non linear interactions between a Hopf bifurcation and a pitchfork-type stationary bifurcation can produce secondary bifurcations of periodic solutions, and tertiary bifurcations of periodic or aperiodic solutions lying on an invariant torus. A complete classification of the resulting bifurcation diagrams is presented, with emphasis on the cases which exhibit tertiary bifurcation. Calculations involving successive transformations to polar normal forms lead to existence theorems for the secondary and tertiary solutions and asymptotic formulae for the invariant torus.
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