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  • Ergodic Theory and Dynamical Systems, Volume 6, Issue 2
  • June 1986, pp. 205-239

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

  • Kevin Hockett (a1) and Philip Holmes (a1)
  • DOI: http://dx.doi.org/10.1017/S0143385700003412
  • Published online: 01 September 2008
Abstract
Abstract

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].

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[1]A. A. Abidi & L. O. Chua . On the dynamics of Josephson junction circuits. Electronic Circuits and Systems 3 (1979), 186200.

[3]D. G. Aronson , M. A. Chory , G. R. Hall & R. P. McGeehee . Bifurcations from an invariant circle for two parameter families of maps of the plane: a computer assisted study. Comm. Math Phys. 83 (1982), 303354.

[6]V. N. Belykh , N. F. Pedersen & O. H. Soerensen . Shunted Josephson Junction model, I—The autonomous case and II—the non-autonomous case. Phys. Rev. B. 16 (1977), 48534871.

[7]G. D. Birkhoff . Proof of Poincaré's geometric theorem. Trans. Amer. Math. Soc. 14 (1913), 1422.

[14]P. F. Byrd & M. D. Friedman . Handbook of Elliptic Integrals for Scientists and Engineers. Springer-Verlag: New York, Heidelberg, Berlin, 1971.

[21]B. V. Chirikov . A universal instability of many dimensional oscillator systems. Physics Reports 52, (1979), 263379.

[22]S. N. Chow , J. K. Hale & J. Mallet-Paret . An example of bifurcation to homoclinic orbits. J. Dig. Eqns. 37 (1980), 351373.

[24]B. D. Greenspan & P. J. Holmes . Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators. SIAM J. on Math. Analysis 15 (1984), 6997.

[25]J. Guckenheimer & P. J. Holmes . Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer Verlag: New York, 1983.

[26]J. Guckenheimer & R. F. Williams . Structural stability of Lorenz attractors. Publ. Math. IHES 50 (1979), 5972.

[29]G. A. Hedlund . Sturmian minimal sets. Amer. J. Math. 66 (1944), 605620.

[32]P. J. Holmes . The dynamics of repeated impacts with a sinusoidally vibrating table. J. Sound Vibration 84 (1982), 173189.

[33]P. J. Holmes & J. E. Marsden . Horseshoes in perturbations of Hamiltonian systems with two degrees of freedom. Comm. Math. Phys. 82 (1982), 523544.

[35]B. A. Huberman , J. P. Crutchfield & N. H. Packard . Noise phenomena in Josephson junctions. Appl. Phys. Lett. 37 (1980), 750752.

[38]A. Katok . Some remarks on Birkhoff and Mather twist map theorems. Ergod. Th. & Dynam. Sys. 2 (1982), 185194.

[44]N. Levinson . A second order differential equation with singular solutions. Annals of Math. 50 (1949), 127153.

[45]J. N. Mather . Existence of quasi-periodic orbits for twist homeomorphisms of the annulus. Topology 21 (1982), 457467.

[46]J. N. Mather . Non-uniqueness of solutions of Percival's Euler-Lagrange Equation. Comm. Math. Phys. 86 (1982), 465476.

[48]J. Matisoo . Josephson-type superconductive tunnel junctions and applications. IEEE Transactions on Magnetics 5 (1969), 848873.

[51]S. E. Newhouse . Lectures on dynamical systems. In ‘Dynamical Systems’ ed. J. K. Moser . Birkhauser: Boston, 1980.

[52]S. Newhouse , J. Palis & F. Takens . Bifurcations and stability of families of diffeomorphisms. Publ Math. I.H.E.S. 57 (1983), 572.

[53]M. Odyniec & L. O. Chua . Josephson junction circuit analysis via integral manifolds. IEEE Transactions on Circuits and Systems CAS-30 (1983).

[54]J. Palis . On Morse Smale dynamical systems. Topology 8 (1969), 385405.

[62]S. Smale . Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747817.

[63]S. Smale . The Mathematics of Time: Essays on Dynamical Systems, Economic Processes and Related Topics. Springer-Verlag: New York, Heidelberg, Berlin, 1980.

[66]J. A. York & K. T. Alligood . Cascades of period-doubling bifurcations: a prerequisite for horseshoes. Bull. Amer. Math. Soc. 9 (1983), 319322.

[67]E. Zehnder . Homoclinic points near elliptic fixed points. Comm. Pure. Appl. Math. 26 (1973), 141182.

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