BIFURCATIONS OF MODE EQUATIONS - SPIN-WAVES

We study the bifurcations of the "spin-wave" equations that describe the parametric pumping of collective modes in magnetic media. We discuss the following dynamical phenomena: (i) sequential excitation of modes via zero-eigenvalue bifurcations; (ii) Hopf bifurcations followed (or not) by Feigenbaum cascades of period doublings; (iii) local and global homoclinic phenomena. Two new organizing centers for "routes to chaos" are identified; in the classification given by Guckenheimer and Holmes, one is the codimension-two local bifurcation, with one pair of imaginary eigenvalues and a zero eigenvalue, to which many dynamical consequences are known; the second, global homoclinic bifurcations associated to splitting of separatrices, in the limit where the system can be considered a Hamiltonian subjected to weak dissipation and forcing.