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Constructing dynamical systems having homoclinic bifurcation points of codimension two. (English) Zbl 0879.34051
Summary: A procedure is derived which allows for a systematic construction of three-dimensional ordinary differential equations having homoclinic solutions. The equations are proved to exhibit codimension-two homoclinic bifurcation points. Examples include the non-orientable resonant bifurcation, the inclination-flip, and the orbit-flip. In addition, an equation is constructed which has a homoclinic orbit converging to a saddle-focus satisfying Shilnikov’s condition. The vector fields are polynomial and non-stiff in that the eigenvalues are of moderate size.
MSC:
34C37Homoclinic and heteroclinic solutions of ODE
37G99Local and nonlocal bifurcation theory
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