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About ergodicity in the family of limaçon billiards

Holger R Dullin1 and Arnd Bäcker2

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Recommended by T Prosen

By continuation from the hyperbolic limit of the cardioid billiard we show that there is an abundance of bifurcations in the family of limaçon billiards. The statistics of these bifurcation shows that the size of the stable intervals decreases with approximately the same rate as their number increases with the period. In particular, we give numerical evidence that arbitrarily close to the cardioid there are elliptic islands due to orbits created in saddle-node bifurcations. This shows explicitly that if in this one-parameter family of maps ergodicity occurs for more than one parameter, the set of these parameter values has a complicated structure.


PACS

05.45.Mt Quantum chaos; semiclassical methods

02.30.Oz Bifurcation theory

MSC

37Axx Ergodic theory (See also 28Dxx)

37G15 Bifurcations of limit cycles and periodic orbits

37E10 Maps of the circle

37D50 Hyperbolic systems with singularities (billiards, etc.)

Subjects

Mathematical physics

Statistical physics and nonlinear systems

Dates

Issue 6 (November 2001)

Received 4 April 2001

Published 22 October 2001



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