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On moduli of systems with a structurally unstable homoclinic Poincaré curve. (English. Russian original) Zbl 0801.58034

Russ. Acad. Sci., Izv., Math. 41, No. 3, 417-445 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 6, 1165-1197 (1992).
For higher-dimensional dynamical systems with a structurally unstable homoclinic Poincaré curve the authors find moduli of topological and \(\Omega\)-equivalence (that is functionals on the space of dynamical systems that have equal values for topologically equivalent systems). Part of these results can be considered as an essential generalization of the classical results by S. Newhouse, J. Palis and F. Takens [Publ. Math., Inst. Hautes Étud. Sci. 57, 5-71 (1983; Zbl 0518.58031)]. Using these results, some sufficient conditions for \(\Omega\)-equivalence on small neighborhoods of homoclinic curves are obtained for systems with a nontrivial structure. These conditions are given in terms of invariants, that depend on multipliers for a saddle periodic orbit.

MSC:

37D99 Dynamical systems with hyperbolic behavior
37G99 Local and nonlocal bifurcation theory for dynamical systems
37C75 Stability theory for smooth dynamical systems

Citations:

Zbl 0518.58031
Full Text: DOI