Sine-Gordon方程中混沌跳跃行为的机制

MECHANISM OF CHAOTIC JUMPING BEHAVIOR IN THE SINE-GORDON EQUATION

定性研究了ｓｉｎｅ－Ｇｏｒｄｏｎ方程在其广义近似惯性流形上的一个四维常微分方程．通过奇 异扰动理论得到Ｓｉｌｎｉｋｏｖ同宿轨道存在的一个解析判定，然而在以往数值模拟中使用过的参 数条件下，上面的判定不完全成立．于是进一步利用能量差方法，在这些参数条件下证明了至 少存在四条结构稳定的脉冲轨道，它们正好反映了在ｓｉｎｅ－Ｇｏｒｄｏｎ方程的数值模拟中观察到 的混沌跳跃行为．

Nonlinear dynamics of the damped driven sine-Gordon equation where 0 < ε≤1, α, Γ > 0, with even spatial symmetry and periodic boundary conditions has been studied extensively by numerical and analytical methods, because it is a typical model for the study of chaotic attractors in infinite dimensional dynamical systems. However, there has been not any rigorous proof on some interesting dynamics of this model, such as chaotic jumping behavior in time with the spatial coherence. In this paper, the dynamics mechanism of the phenomenon is investigated analytically. Firstly, based on the idea of generalized asymptotic inertial manifold (GAIM) of the dissipative partial differential equcition (PDE), a four-dimensional first-order ordinary differential equation (ODE) is derived by restricting the above PDE to its GAIM, which is a perturbed near-integrable Hamiltonian system with a certain resonance. Then the ODE is studied qualitatively by the singular perturbation-theory. Without any artificial condition, an analytical criterion for the existence of Silnikov-type orbits homoclinic to the resonant band is obtained. In comparison with the previous results, these results are more coincident with the physical facts. However, such criterion does not hold completely under the parametric values specified in the previous numerical experiments, but it is indicated that other orbits homoclinic (or heteroclinic) to the resonant band probably exist. Then under these parametric values, it is shown that there exist at least four stable structurally pulse-orbits connecting a saddle-saddle-type equilibrium to a saddle-focus-type equilibrium in the resonant band by applying the method proposed by Haller and Wiggins, which are numerically observable, and thus just explain the chaotic jumping behavior observed by the numerical experiments.