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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The transverse homoclinic dynamics and their bifurcations at nonhyperbolic fixed points


Author: Bo Deng
Journal: Trans. Amer. Math. Soc. 331 (1992), 15-53
MSC: Primary 58F14; Secondary 34C23, 34C37, 58F15
MathSciNet review: 1024768
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Abstract: The complete description of the dynamics of diffeomorphisms in a neighborhood of a transverse homoclinic orbit to a hyperbolic fixed point is obtained. It is topologically conjugate to a non-Bernoulli shift called $ \{ {\sum,\sigma } \}$. We also obtain a more or less complete picture, referred to as the net weaving bifurcation, when the fixed point of such a system is undergoing the generic saddle-node bifurcation. The idea of homotopy conjugacy is naturally introduced to show that systems whose fixed points undergo the pitchfork, transcritical, periodic doubling, and Hopf bifurcations are all homotopically conjugate to our shift dynamics $ \{ {\sum,\sigma } \}$ in a neighborhood of a transverse homoclinic orbit. These bifurcations are also examined in the context of the spectral decomposition with respect to the maximal indecomposable nonwandering sets.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1024768-9
PII: S 0002-9947(1992)1024768-9
Keywords: Transverse homoclinic point, saddle-node bifurcation, symbolic system, topological conjugacy
Article copyright: © Copyright 1992 American Mathematical Society



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