Abstract
We prove that the size of the finite-dimensional attractor of the damped and driven sine-Gordon equation stays small as the damping and driving amplitude become small. A decomposition of finite-dimensional attractors in Banach space is found, into a partℬ that attracts all of phase space, except sets whose finitedimensional projections have Lebesgue measure zero, and a partC that only attracts sets whose finite-dimensional projections have Lebesgue measure zero. We describe the components of the ℬ-attractor andC, which is called the “hyperbolic” structure, for the damped and driven sine-Gordon equation. ℬ is low-dimensional but the dimension ofC, which is associated with transients, is much larger. We verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the ℬ-attractor in this small parameter region.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A] Arnold, V.I.: Ordinary differential equations. Boston: MIT Press, 1973
[ABB] Armbruster, D., Birnir, B., Buys, M.: The damped and driven sine-Gordon equation and basic turbulence. UCSB preprint (1993)
[B1] Birnir, B.: Complex Hill's equation and the complex periodic Korteweg-de Vries equation. Comm. Pure Applied Math.39, 1–49 (1986)
[B2] Birnir, B.: Singularities of the complex KdV flows. Comm. Pure and Applied Math.39, 283–305 (1986)
[B3] Birnir, B.: An example of finite-time blow-up of the complex Korteweg-de Vries equation and existence beyond blow-up. SIAM J. Appl. Math.47, 710–725 (1987)
[B4] Birnir, B.: Dimension estimates of attractors of nonlinear dissipative wave equations with applications to the damped and driven sine-Gordon attraction. In preparation
[BF1] Bishop, A.R., Forest, M.G., McLaughlin, D.W., Overman, E.A. II: A quasi-periodic route to chaos in a near integrable PDE. Physica D23, 293–328 (1986)
[BF2] Bishop, A.R., Forest, M.G., McLaughlin, D.W., Overman, E.A. II: Quasi-periodic route to chaos in a near integrable PDE, homoclinic crossings. Phys. Lett. A127, 335–340 (1988)
[BF3] Bishop, A.R., Forest, M.G., McLaughlin, D.W., Overman, E.A. II: A modal representation of chaotic attractors for the driven damped pendulum chain. Phys. Lett. A144, 17–25 (1990)
[BF4] Bishop, A.R., Flesch, R., Forest, M.G., McLaughlin, D.W., Overman, E.A. II: Correlations between chaos in a perturbed sine-Gordon equation and a truncated model system. SIAM J. Math. Anal. 1511–1536 (1990)
[BJ] Bates, P., Jones, C.K.R.T.: Invariant manifolds for semilinear partial differential equations. Dynamics Reported2, 1–39 (1989)
[BMW] Birnir, B., McKean, H., Weinstein, A.: Nonexistence of breathers. CPAM, to appear
[BN] Birnir, B., Nelson, K.: Global attractors of periodically forced nonlinear dissipative wave equations. In preparation
[BS] Birnir, B., Smereka, P.: Existence theory and invariant manifolds of driven bubble clouds. Comm. Pure. Appl. Math.43, 363–413 (1990)
[BV] Babin, A.V. Vishik, M.I.: Attractors of evolution equations. Studies in Appl. Math. and its Applic. Vol.25. Amsterdam: North-Holland, 1992
[C] Carr, J.: Applications of center manifold theory. Berlin, Heidelberg, New York: Springer 1981
[CH] Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory. Berlin, Heidelberg, New York: Springer, 1982
[CL] Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill, 1955
[D] Date, E.: Multisoliton solutions and quasi-periodic solutions of nonlinear equations of sine-Gordon type. Osaka J. Math.19, 125–128 (1982)
[GT] Ghidaglia, J.M., Temam, R.: Attractors for damped nonlinear hyperbolic equations. J. Math pures and appl.66, 273–319 (1987)
[GB] Grauer, R., Birnir, B.: The center manifold and bifurcations of the sine-Gordon breather. Physica D56, 165–184 (1992)
[H] Hartman, P.: Ordinary differential equations. New York: Wiley, 1964
[Ha] Hale, J.: Asymptotic behavior of dissipative systems. Providence: AMS, 1988
[K] Kuratowski, K.: Topology, Vol.1. New York: Academic Press 1966
[Ko] Kovacic, G.: Chaos in a model of the forced and damped sine-Gordon equation. Ph.D. thesis, California Institute of Technology, 1989
[KW] Kovacic, G., Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped sine-Gordon equation. Physica D, to appear
[L] Lions, J.L.: Quelques methods de résolution des problème aux limites nonlinéaires. Paris: Dunod, 1969
[LS] Lomdahl, P.S., Samuelsen, M.R.: Persistent breather excitations in an AC-driven sine-Gordon system with loss. Phys. Rev. Lett.34, 664 (1986)
[M] Mañé, R.: Dynam. Syst. and turbulence. Warwick, ed. Rand, D.A. and Young, L.-S., Berlin, Heidelberg, New York: Springer, 1980
[McK] McKean, H.: The sine-Gordon and sinh-Gordon equation on the circle. Comm. Pure Appl. Math.34, 197–243 (1981)
[Mi] Milnor, J.: On the concept of attractor. Commun. Math. Phys.99, 177–195 (1985)
[MB] Mazor, A., Bishop, A.R., McLaughlin, D.W.: Phase-puling and space-time complexity in an ac driven damped one-dimensional sine-Gordon system. Phys. Lett.A 119, 273–279 (1986)
[MO] McLaughlin, D.W., Overman, E.A., II: Whiskered tori for integrable PDEs and chaotic behavior in near integrable PDEs, Surveys in Appl. Math.1 (1992)
[MS] McLaughlin, D.W., Scott, A.C.: Perturbation analysis of fluxon dynamics. Phys. Rev.A 28, 1652–1680 (1978)
[MW] Magnus, W., Wrinkler, W.: Hill's equation. New York: Willey-Interscience, 1966
[NM] Nayfeh, A.H., Mook, D.T.: Nonlinear oscillation. New York: Willey 1979
[O] Overman, E., Xiong, C., Berliner, M.: Low mode truncation methods in the sine-Gordon equation. Preprint, Ohio State University, 1991
[P] Pedersen, N.F.: Advances in superconductivity. ed. Deaver and Ruvolds. New York: Plenum, 1982
[PV] du Val, P.: Elliptic functions and elliptic curves. Cambridge: Cambridge Univ. Press, 1973
[R] Reed, M.: Abstract non-linear wave eqs. Lecture Notes in Math. 507. Berlin, Heidelberg, New York: Springer, 1976
[RF] Reinish, G., Fernandez, J.C., Goupil, M.J., Legrand, O.: Relativistic dynamics of sine-Gordon solitons trapped in confining potentials. Phys. Rev. B.34, 6207 (1986)
[S] Siegel, C.L.: Topics in Complex Function Theory. Vol.1–3, New York: Wiley-Interscience, 1971
[SK] Segur, H., Kruskal, M.D.: Nonexistence of small amplitude breather solutions inφ 4 theory. Phys. Rev. Lett.58, 747–750 (1987)
[SYC] Sauer, T., York, J.A., Casdagli, M.: Embedology. Stat. Phys.65, 3–4, 579–617 (1991)
[T] Teman, R.: Infinite-dimensional dynamical systems in mechanics and physics. Berlin, Heidelberg, New York: Springer 1988)
[Ta] Taki, M., Spatschek, K.H., Fernandez, J.C., Grauer, R., Reinisch, G.: Breather dynamics in the nonlinear Schrödinger regime of perturbed sine-Gordon systems. Physica D40, 65–84 (1989)
[TF] Takhatajan, L.A., Faddeev, L.D.: Essentially nonlinear one-dimensional model of classical field theory. Theor. Math. Phys.21, 1046–1156
Author information
Authors and Affiliations
Additional information
Communicated by T. Spencer
Partially supported by NSF grants DMS89-05770 and DMS89-03012
Partially supported by an INCOR grant and a grant from the German Science Foundation
Rights and permissions
About this article
Cite this article
Birnir, B., Grauer, R. An explicit description of the global attractor of the damped and driven sine-Gordon equation. Commun.Math. Phys. 162, 539–590 (1994). https://doi.org/10.1007/BF02101747
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02101747