An explicit description of the global attractor of the damped and driven sine-Gordon equation (original) (raw)
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Physica D Nonlinear Phenomena, 1992
In this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two-degree-of-freedom Hamiltonian systems. Our methods are fundamentally different than other global perturbation methods (e.g. standard Melnikov theory) in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation. Our methods combine the higher dimensional Melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds. We apply our methods to a modified model of the forced and damped sine-Gordon equation developed by Bishop et al. We give explicit conditions (in terms of the system parameters) for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle-focus type; chaotic dynamics follow from a theorem of Silnikov. This provides a mechanism for chaotic dynamics geometrically similar to that observed by Bishop et al.; namely, a random "jumping" between two spatially dependent states with an intermediate passage through a spatially independent state. However, in order for this type of Silnikov dynamics to exist we require a different, and unphysical, type of damping compared to that used by Bishop et al.
Mel'nikov Analysis of Homoclinic Chaos in a Perturbed sine-Gordon Equation
1999
We describe and characterize rigorously the chaotic behavior of the sine-Gordon equation. The existence of invariant manifolds and the persistence of homoclinic orbits for a perturbed sine--Gordon equation are established. We apply a geometric method based on Mel'nikov's analysis to derive conditions for the transversal intersection of invariant manifolds of a hyperbolic point of the perturbed Poincare map.
The center manifold and bifurcations of damped and driven sine-Gordon breathers
Physica D: Nonlinear Phenomena, 1992
The generic bifurcations of breathers in the damped and driven sine-Gordon equation are investigated both numerically and analytically. The linear stability analysis and information from periodic spectral theory suggest that three modes are relevant for the system. They correspond to frequency and (temporal) phase changes and to the fiat pendulum. Using t~/ese modes (nonautonomous) amplitude equations are derived and compared with numerical simulations of the pertu/bed sine-Gordon equation.
Parametrically driven one-dimensional sine-Gordon equation and chaos
1988
The chaotic behaviour of the parametrically driven one-dimensional sine-Gordon equation with periodic boundary conditions is studied. The initial condition is u (x, 0) = / (x), u, (x, 0) = 0 where/ is the breather solution of the one-dimensional sine-Gordon equation at t = 0. We vary the amplitude of the driving force, the frequency of the driving force and the damping constant. For appropriate values of the driving force, frequency and damping constant chaotic behaviour with respect to the time-evolution of u(x = fixed, f) can be found. The space structure u(t = fixed, .x) changes with increasing driving force from a zero mode structure to a breather-like structure consisting of a few modes.
Strange Attractors Across the Boundary¶of Hyperbolic Systems
Communications in Mathematical Physics, 2000
We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors.
Strange Attractors Across the Boundary¶of Hyperbolic Systems
Communications in Mathematical Physics, 2000
We study a bifurcation of Axiom A (hyperbolic) vector fields in dimension three leading to robust strange attractors with singularities. The Axiom A vector fields involved in the bifurcation exhibit a basic set equivalent to the suspension of a three symbol subshift. The attractors arising from this kind of bifurcation are not equivalent to the geometric Lorenz attractors.
Uniform exponential attractors for a singularly perturbed damped wave equation
Discrete and Continuous Dynamical Systems, 2003
Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
Finite dimensional exponential attractors for semilinear wave equations with damping
Journal of Mathematical Analysis and Applications
We consider the initial value problem for a class of second order evolution equations that includes, among others, the 3D sine-Gordon equation with damping and the 3D Klein-Gordon type equations with damping. We show the existence of a set with finite fractal dimension that contains the global attractor and attracts all smooth solutions at an exponential rate.
The explosion of singular-hyperbolic attractors
Ergodic Theory and Dynamical Systems, 2004
A {\em singular hyperbolic attractor} for flows is a partially hyperbolic attractor with singularities (hyperbolic ones) and volume expanding central direction \cite{mpp1}. The geometric Lorenz attractor \cite{gw} is an example of a singular hyperbolic attractor. In this paper we study the perturbations of singular hyperbolic attractors for three-dimensional flows. It is proved that any attractor obtained from such perturbations contains
“Horseshoe chaos” in the space-independent double sine-Gordon system
Wave Motion, 1986
For the driven, damped space-independent double sine-Gordon equation threshold curves for horseshoe chaos of the Smale type are derived by the Melnikov technique. Different qualitative behaviour of the solutions is found in different regions of parameter space.