Number of periodic trajectories for analytic diffeomorphisms of the circle

Yakobson, M V

doi:10.1007/bf01086040

Cited by 4 publications

(3 citation statements)

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“…We can, however, obtain some control by utilizing some results from complex dynamics. The following lemma was communicated to us by R. L. Devaney, N. Fagella and G. R. Hall; the first published version we are aware of (with essentially the same proof) appears in Yakobson [1985].…”

Section: Flames In P/q Resonance Hornsmentioning

confidence: 99%

Arnold Flames and Resonance Surface Folds

McGehee

Peckham

1996

Int. J. Bifurcation Chaos

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Periodically forced planar oscillators are often studied by varying the two parameters of forcing amplitude and forcing frequency. For low forcing amplitudes, the study of the essential oscillator dynamics can be reduced to the study of families of circle maps. The primary features of the resulting parameter plane bifurcation diagrams are "(Arnold) resonance horns" emanating from zero forcing amplitude. Each horn is characterized by the existence of a periodic orbit with a certain period and rotation number. In this paper we investigate divisions of these horns into subregions -different subregions corresponding to maps having different numbers of periodic orbits. The existence of subregions having more than the "usual" one pair of attracting and repelling periodic orbits implies the existence of "extra folds" in the corresponding surface of periodic points in the cartesian product of the phase and parameter spaces. The existence of more than one attracting and one repelling periodic orbit is shown to be generic. For some of the families we create, the resulting parameter plane bifurcation pictures appear in shapes we call "Arnold flames." Results apply both to circle maps and forced oscillator maps.

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Section: Flames In P/q Resonance Hornsmentioning

confidence: 99%

Arnold Flames and Resonance Surface Folds

McGehee

Peckham

1996

Int. J. Bifurcation Chaos

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“…We will see that in many cases a "typical" family will have infinite cyclicity: this means that for any N ∈ N there exist parameter values r and a such that the map F r,a has more than N attracting periodic trajectories. On the other hand, families which have finite cyclicity certainly exist: if f is a trigonometric polynomial of degree d, then the number of attracting periodic trajectories of the map F r,a is bounded by d, see [Yak85].…”

Section: Introductionmentioning

confidence: 99%

Cyclicity in families of circle maps

Kozlovski

2013

Ergod. Th. Dynam. Sys.

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“…12 (1978) theory of (non)oscillations in linear Hamiltonian systems, 23 various attempts to prove this conjecture (including a series of papers by Petrov-Tan'kin on Abelian integrals over elliptic curves, my own application of Sturm's theory to nonoscillation of hyperelliptic integrals, and more recent estimates of Grigoriev, Novikov-Yakovenko), and further work by Horozov, Khovansky, Ilyashenko and others. Yet another modification of the problem (a discrete one-dimensional analogue) suggested by Arnold led to a beautiful and nontrivial theorem of Yakobson in the theory of dynamical systems [39].…”

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confidence: 99%