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Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations. (English) Zbl 0990.37039

The paper presents a careful numerical study of two homoclinic codimension-three bifurcations. In these resonant homoclinic flip bifurcations both a resonance between real eigenvalues and some codimension-two flip bifurcation (inclination flip or orbit flip) occur simultaneously.

Using a model equation of Sandstede the authors study with AUTO and HOMCONT transitions between different regions. Although they consider a specific equation, many of the results should hold universally.

In each case, a small sphere around the codimension-three point is considered assuming that the qualitative picture of the bifurcation curves on the sphere does not depend on the radius. Some care has to be paid to a “good” choice of the radius since for large radii new bifurcations appear while it is impossible to distinguish certain bifurcation curves when the radius is too small. A rich variety of bifurcations can be detected including homoclinic-doubling cascades, torus bifurcations and shift dynamics.

The numerical results largely confirm recent theoretical studies and conjectures of A. J. Homburg and B. Krauskopf [J. Dyn. Differ. Equ. 12, 807-850 (2000; Zbl 0990.37041)].

37G20Hyperbolic singular points with homoclinic trajectories
37M20Computational methods for bifurcation problems
34C37Homoclinic and heteroclinic solutions of ODE
37G15Bifurcations of limit cycles and periodic orbits
37C29Homoclinic and heteroclinic orbits
65P30Bifurcation problems (numerical analysis)