Completed and incompleted stochastic webs. (English) Zbl 0761.58049
It is shown that the stochastic web formed by a saddle separatrix net under perturbation has two different types. One is a completed stochastic web. This means that the stable and unstable manifolds of all heteroclinic orbits in a saddle separatrix net of an unperturbed system must be a transversal intersection under perturbation.
The motion on the completed stochastic web is random and spread over the whole web. Another type of stochastic web is named incompleted. It means that the stochastic web is divided into some different subwebs. Stable and unstable manifolds of every heteroclinic orbit inside subwebs have transversal intersections. The motion on the incompleted stochastic web is located in a subweb and is not chaotic.
The motion on the completed stochastic web is random and spread over the whole web. Another type of stochastic web is named incompleted. It means that the stochastic web is divided into some different subwebs. Stable and unstable manifolds of every heteroclinic orbit inside subwebs have transversal intersections. The motion on the incompleted stochastic web is located in a subweb and is not chaotic.
Reviewer: G.V.Khmelevskaja-Plotnikova (Namur)
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
Keywords:
completed stochastic web; incompleted stochastic web; transversal heteroclinic cycle; degenerated transversal heteroclinic cycleReferences:
| [1] | Chernikov, A. A. et al., Stochastic webs,Physica, D33 (1988), 65. |
| [2] | Chernikov, A. A. et al., Some peculiarities of stochastic layer and stochastic web formation,Physics Letter, A122 (1987) 39. · doi:10.1016/0375-9601(87)90772-9 |
| [3] | Murakami, S. et al., Development of stochastic webs in a wave-driven linear oscillator,Physica, D32 (1988), 269. · Zbl 0653.70026 |
| [4] | Guckenheimer, J. and P. J. Holmes,Nonlinear Osciallation, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag (1983). · Zbl 0515.34001 |
| [5] | Yan, Yin and Qian, Min,KeXutongbao,29 (1985), 961. |
| [6] | Liu Zeng-rong and Cao Yong-luo, Transversal cycle and invariant manifolds. (preprint) |
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