Parametric entrainment of systems governed by ordinary differential equations

ABSTRACT For systems represented by ordinary differential equations of a general form, it is shown that timedependencies for the parameters may be determined to generate new behaviors. These new dynamicsare mathematical solutions determined using a second set of equations. Under many circumstances the system's new driven behavior entrains to these solutions in a stable manner. The method is explored via numerical simulation of a Duffing-like oscillator system. The results of these compittel studies are then applied to an experimental system. A model consisting of a system of ordinary differential equations is determined for the experiment. The parametric driving term is computedand then applied. The response of the system is compared to the response from a sinusoidal drivingforce of similar characteristics and the results discussed. I. ENTRAINMENT CONTROLS Parametric Entrainment [1] (PE) is an extension of Hubler's Model-Based Control [2—8] — a control technique thatcomputes a driving term to the equations governing the behavior of the system. All information is derived from amodel of the system and no feedback is used. The new solutions for the system are specified by a secoiid set of (goal)equations. These are exact mathematical solutions by construction. This method requires a model of the ysteiii inthe form of differential equations or maps and an exact goal or target behavior.In contrast with Hubler's original formulation of Model-Based Control, this tecimique uses a palainetlic iatherthan an inhomogeneous driving term. This extension applies to systems that cannot be externally forced but containone or more parameters that can be varied in a time dependent manner in the spirit of traditional control methods.Unfortunately, changing from an inhomogeneous to a parametric forcing limits the space of accessible goal solutionsthereby reducing the possible range of motions the system can entrain to. The advantages of parametric rather thaninhomogeneous driving include a wider class of applicable systems and control at a potentially lower energy cost.These control techniques differ significantly from the OGY [9,10] method of control in several aspects. Thoughboth can be used to control chaos, no feedback is used in this method. All information comes from the model of theexperiment in the form of equations. The initial conditions for these equations are simply ignored in the application.Another important difference is that this method (as it now stands) is formulated in flow space rather than map spaceyielding far more flexibility in the choice of goal behaviors. Simply put, the technique is not limited to goals of periodone, period two, etc. in Poincare space but can entrain say, a nonlinear oscillator to the motion of a simple harmonicoscillator. It should be noted that the OGY method entrains a system to one of its uiistable orbits while PE caim inprinciple entrain to any trajectory except those that are already solutions to the original equations of motion.This reports describes tests of PE in a physical system — that of a parametrically driven magnetostrictive ribboim[11]. A global model of the ribbon was constructed and studied with computer simulations. The driving forces were