Linear superposition of chaotic and orderly vibrations on two serially connected strings with a Van der Pol joint

Two identical vibrating strings are serially coupled end-to-end with nonlinear joints that behave like a Van der Pol oscillator. This coupled PDE system has an infinite dimensional center manifold of orderly periodic solutions of vibration for which the nonlinearity at the joint is not excited. Based upon the authors' study of chaotic vibration on a single vibrating string in [Chen et al., 1995], we analyze the special structure of the nonlinear reflection operator by decoupling. The decoupled operator has a linear part which is idempotent, and this linear part does not interact with the iterates of the nonlinear part. Using this, we show that chaotic vibrations and orderly periodic vibrations coexist and satisfy a certain rule of linear superposition, namely, if u is a linear periodic solution and if U is a general nonlinear solution, then U + Cu is also a solution for any constant C. Numerical simulations and computer graphics are also included to illustrate the superposition effect.