Classification and unfoldings of \(1:2\) resonant Hopf bifurcation
The paper concerns the interaction of two Hopf bifurcations in the case of 1:2 resonance, i.e. the bifurcation of periodic solutions from a stationary solution of an autonomous differential equation whose linearization has eigenvalues \(\pm i\) and \(\pm 2i\) and no other eigenvalues of the form \(ki\), where \(k\) is an integer. No other restrictions on the equation (such as requiring it to be Hamiltonian or time-reversible or equivariant) are assumed. Working in spaces of periodic functions and using the Lyapunov-Schmidt reduction, only periodic solutions (near the stationary one) with period near \(2\pi\) are investigated (and not such with larger minimal period or homoclinic or heteroclinic solutions), but for those solutions the results are rigorous and complete. Under a specific non-degeneracy condition, the generic cases of equations with no distinguished control parameter and such with one distinguished bifurcation parameter are completely classified and unfolded.
bifurcation of periodic solutions from a stationary solution
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