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Josephson’s junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets. (English) Zbl 0582.58020

We investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to D. G. Aronson, M. A. Chory, G. R. Hall and R. P. McGehee [Commun. Math. Phys. 83, 303-354 (1982; Zbl 0499.70034)]: 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 S. Smale [Differential and Combinat. Topology, Sympos. Marston Morse, Princeton, 63-80 (1965; Zbl 0142.411)] 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” [G. D. Birkhoff, Bull. Soc. Math. Fr. 60, 1-26 (1932; Zbl 0005.22002)].

MSC:

37C70 Attractors and repellers of smooth dynamical systems and their topological structure
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