Multiple Limit Cycle Bifurcation Surfaces and Global Families of Multiple Limit Cycles

Sufficient conditions are given for the local exitence of multiplicity-m limit cycle bifurcation surfaces, Cm, of planar analytic systems depending on n parameters with n ≥ m ≥ 2. In the generic case, the surfaces C2, C3, and C4 are the familiar saddle-node, cusp, and swallow-tail bifurcation surfaces, respectively. The author′s termination principle for global one-parameter families of simple limit cycles of relatively prime, planar, analytic systems is generalized to a termination principle for global one-parameter families of multiple limit cycles which implies that the boundary of a global limit cycle bifurcation surface typically consists of Hopf bifurcation surfaces and/or homoclinic (or heteroclinic) loop bifurcation surfaces.