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Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets

Published online by Cambridge University Press:  19 September 2008

Kevin Hockett  and
Philip Holmes
Kevin Hockett
Affiliation:
Center for Applied Mathematics and Departments of Theoretical and Applied Mechanics and Mathematics, Cornell University, Ithaca NY 18453, USA
Philip Holmes
Affiliation:
Center for Applied Mathematics and Departments of Theoretical and Applied Mechanics and Mathematics, Cornell University, Ithaca NY 18453, USA
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Abstract

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We investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronson et al. [3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 2 , June 1986 , pp. 205 - 239
Copyright
Copyright © Cambridge University Press 1986

References

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