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Successive homoclinic tangencies to a limit cycle. (English) Zbl 0886.34028
Summary: The dynamics near a perturbed degenerate homoclinic connection to a periodic orbit in three dimensions is modeled by a two-parameter map. One parameter controls the passage of the manifolds of the orbit through one another, and the other breaks the degeneracy and causes the manifolds to intersect transversely. An analysis of the map recovers the results of P. Gaspard and X.-J. Wang [J. Stat. Phys. 48, 151-199 (1987)], relating to the accumulation of saddle-node bifurcations of periodic orbits on a single homoclinic tangency, and in addition shows that the local behavior of these orbits at the two tangencies can be linked together giving closed loops in period versus parameter plots. These analytic results are then compared with numerical results from a three-dimensional system of ordinary differential equations.

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
Full Text: DOI
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