Bifurcation phenomena near homoclinic systems: A two-parameters analysis
The bifurcations of periodic orbits in a class of autonomous three-variable, nonlinear-differential-equation systems possessing a homoclinic orbit associated with a saddle focus with eigenvalues (rho +- i..omega.., lambda), where Vertical Barrho/lambdaVertical Bar<1 (Sil'nikov's condition), are studied in a two-parameters space. The perturbed homoclinic systems undergo a countable set of tangent bifurcation followed by period-doubling bifurcations leading to a periodic orbits which may be attractors if Vertical Barlambda/lVertical Bar<1/2. The accumulation rate of the critical parameter values at the homoclinic system is exp(-2..pi..Vertical Barrho/..omega..Vertical Bar). A global mechanism for the onset of homoclinicity in strongly contractive flows is analyzed. Cusp bifurcations with bistability and hysteresis phenomena exist locally near the onset of homoclinicity. A countable set of these cusp bifurcations with scaling properties related to the eigenvalues rho +- i..omega.. of the stationary state are shown to occur in infinitely contractive flows. In the two-parameter space, the periodic orbit attractor domain exhibits a spiral structure globally, around the set of homoclinic systems, in which all the different periodic orbits are continuously connected.
- Research Organization:
- Faculte des Sciences, Universite Libre de Bruxelles, Campus Plaine, C.P. 231, 1050 Bruxelles, Belgium
- DOE Contract Number:
- AS05-81ER10947
- OSTI ID:
- 6791115
- Journal Information:
- J. Stat. Phys.; (United States), Journal Name: J. Stat. Phys.; (United States) Vol. 35:3; ISSN JSTPB
- Country of Publication:
- United States
- Language:
- English
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