An explicit description of the global attractor of the damped and driven sine-Gordon equation (original) (raw)

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 part 3B that attracts all of phase space, except sets whose finitedimensional projections have Lebesgue measure zero, and a part W that only attracts sets whose finite-dimensional projections have Lebesgue measure zero. We describe the components of the JF-attractor and W, which is called the "hyperbolic" structure, for the damped and driven sine-Gordon equation. 33 is low-dimensional but the dimension of W, 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 ^-attractor in this small parameter region. Contents 1. Introduction 539 2. Energy Estimates 542 3. The Attractor of the Poincare Map 551 4. Periodic Orbits 559 5. The B-Attractor and the Hyperbolic Structure 572 Appendix. The Center Manifold of the Flappers 583 References 588