PERIODIC OSCILLATIONS IN LINEAR CONTINUOUS MEDIA COUPLED WITH NONLINEAR DISCRETE-SYSTEMS

A general derivation of partial differential equations with boundary c onditions in the form of ordinary differential equations is obtained u sing the principle of stationary action for a Lagrangian function comp osed of continuous plus discrete parts in interaction across the bound aries of a 1-dimensional medium. This approach leads directly to the t heorem of energy conservation. For linear continuous medium, homogeneo us Dirichelet condition at one boundary, and nonlinear oscillator at t he other boundary, the entire differential problem reduces to a nonlin ear differential-difference equation of neutral type and of the second order. The lag parameter is tau = l/c, where c is the phase speed, l the length of the continuum. We investigate the problem of the occurre nce of periodic solutions of period integer multiple of the lag (super harmonic solutions) in the case of zero inertia of the boundary system . The problem for such oscillations is shown to reduce to systems of o rdinary differential equations with matching conditions in a phase spa ce of lower dimensionality. Phase-plane techniques are used to determi ne solutions of period 4 tau, 8 tau and 6 tau.