Internal perturbations of homoclinic classes:non-domination, cycles, and self-replication

Details Internal perturbations of homoclinic classes:non-domination, cycles, and self-replication Internal perturbations of homoclinic classes:non-domination, cycles, and self-replication

2010-11-12

arxiv.org/abs/1011.2935v1

Mathematics

Dynamical Systems

42 pages, 1 figure

Scientific paper

Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes have the property of generating copies of themselves producing wild dynamics (systems with infinitely many homoclinic classes with some persistence). Such wild dynamics also exhibits uncountably many aperiodic chain recurrence classes. A scenario (related with non-dominated dynamics) is presented where viral homoclinic classes occur. A key ingredient are adapted perturbations of a diffeomorphism along a periodic orbit. Such perturbations preserve certain homoclinic relations and prescribed dynamical properties of a homoclinic class.

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