Monotonicity And Dynamics In Coupled Pendulum Type Equations

We establish the strong monotonicity for the coupled pendulum type equations under the overdamped condition by the monotonicity theory of cooperative systems. Employing the monotonicity approach and dissipation property, we show that under the overdamped condition, the Poincare map P^T of the system has an invariant curve, on which P^T behaves like an orientation preserving circle homeomorphism.For non-automous cases, we show that the average velocity V for each solution of (1.1)(1.2) exists, and V is unique. Moreover, V is continuous with respect to the system parameters and V(F) is a nondecreasing function of F,which is average of the external force.Moreover, there exists F₀ with 0 < F₀≤1 such that V(F) = 0 for 0 < F≤F₀ and V(F) > 0 if F > F₀. Ifρ= VT = p/q is a rational number, then there exists a running periodic solution of (p, q) type with period T.In particular,if V = 0,there is a periodic solution. For the autoumous systems,there exist either equilibria, or running periodic solutions.