Correlations between chaos in a perturbed sine-Gordon equation and a truncated model system

Journal Article · Thu Nov 01 04:00:00 EST 1990 · SIAM Journal of Mathematical Analysis (Society for Industrial and Applied Mathematics); (United States)

DOI:https://doi.org/10.1137/0521083· OSTI ID:7140466

Bishop, A R ^[1]; Flesch, R ^[2]; Forests, M G; Overman, E A ^[3]; McLaughlin, D W ^[4]

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1. Los Alamos National Labs., NM (United States)
2. L.A.M.F. Technical Univ. of Denmark, Lyngby (Denmark)
3. Ohio State Univ., Columbus, OH (United States). Dept. of Mathematics
4. Univ. of Arizona, Tucson, AZ (United States)

The purpose of this paper is to present a first step toward providing coordinates and associated dynamics for low-dimensional attractors in nearly integrable partial differential equations (pdes), in particular, where the truncated system reflects salient geometric properties of the pde. This is achieved by correlating: (1) numerical results on the bifurcations to temporal chaos with spatial coherence of the damped, periodically forced sine-Gordon equation with periodic boundary conditions; (2) an interpretation of the spatial and temporal bifurcation structures of this perturbed integrable system with regard to the exact structure of the sine-Gordon phase space; (3) a model dynamical systems problem, which is itself a perturbed integrable Hamiltonian system, derived from the perturbed sine-Gordon equation by a finite mode Fourier truncation in the nonlinear Schroedinger limit; and (4) the bifurcations to chaos in the truncated phase space. In particular, a potential source of chaos in both the pde and the model ordinary differential equation systems is focused on: the existence of homoclinic orbits in the unperturbed integrable phase space and their continuation in the perturbed problem. The evidence presented here supports the thesis that the chaotic attractors of the weakly perturbed periodic sine-Gordon system consists of low-dimensional metastable attacking states together with intermediate states that are O(1) unstable and correspond to homoclinic states in the integrable phase space. It is surmised that the chaotic dynamics on these attractors is due to the perturbation of these homocline integrable configurations.

• OSTI ID: 7140466
• Journal Information: SIAM Journal of Mathematical Analysis (Society for Industrial and Applied Mathematics); (United States), Journal Name: SIAM Journal of Mathematical Analysis (Society for Industrial and Applied Mathematics); (United States) Vol. 21:6; ISSN 0036-1410; ISSN SJMAAH
• Country of Publication: United States
• Language: English

Related Subjects

662110* -- General Theory of Particles & Fields-- Theory of Fields & Strings-- (1992-)
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
99 GENERAL AND MISCELLANEOUS
990200 -- Mathematics & Computers
ATTRACTORS
DIFFERENTIAL EQUATIONS
EQUATIONS
FIELD EQUATIONS
MATHEMATICS
NONLINEAR PROBLEMS
NUMERICAL ANALYSIS
PARTIAL DIFFERENTIAL EQUATIONS
PERTURBATION THEORY
SINE-GORDON EQUATION