Invariant Curves Of The Overdamped Pendulum Type Equations And Travelling Waves In A Chain Of Coupled Oscillators With Strong Damping

This paper is composed of two parts. We firstly discuss the damped pendulum type equationwhereε> 0, / and p are smooth functions, 1-periodic on x and T-periodic on t, i.e., f(x +1) = f(x), p(x + 1,t) = p(x, t) = p(x, t + T). Moreover, we assume the nonlinear damping coefficient f(x) is positive, i.e., f(x)≥γ> 0; |p'(x,t)| = |(?)p(x,t)/(?)x|≤m. The main conclusions are as follows:Assume 0 <ε<γ²/4m, i.e., the overdamped condition is satisfied. Then the system is strongly monotone and the Poincare map P^T has invariant curve, which is also the global attractor. P^T is actrually an orientation preserving circle homeomorphism on the invariant curve, so the rotation number p and the average velocity v exist independently of the initial points. Furthermore, we have v =ρ/T. In addition, if f is even function and p(x, t + T/2) = -p(-x,t), thenρ= 0. The invariant curve is C¹ smooth if the assumption 0 <ε<2γ²/9m is satisfied. In this case, ifρis irrational, then P^T is ergodic on the invariant curve.In the second part, we study a chain of coupled mechanical oscillators with strong dampingwhich satifies periodic boundary conditionwhere M > 0, N > 0 and j are integers, g is periodic function, satisfying g(x +2Ï€) =g(x),∫₀²Ï€(x)dx = 0, and |g'(x)|≤g, F > 0 indicates the external driving force,Γ> 0 is damping coefficient,α> 0 andβ> 0 measure the coupling strengths through position and velocity respectively. What we are concerned with is the existence and global stability of travelling waves of the formwhere f is the waveform function: R→R, satisfying for some T > 0, T is the periodicity of the waveform function. Our main conclusions are as follows:For any T > 0, there exists a travelling wave solution for an appropriate F > 0. For any F > 1, there exists a travelling wave solution. Fixing any F > 0,α> 0, there is aΓ₀ > 0, such that if 0 <Γ<Γ₀, the system has a travelling wave solution, for all F≥F, 0 <α≤α. AssumeΓ>α/β+ gβ/α. Then the system is strongly monotone. Furthermore, the travelling wave of the system is globally stable.