Parametric Control to Avoid Bifurcation Based on Maximum Local Lyapunov Exponent

Abstract

This chapter presents a parametric controller to avoid bifurcations of stable periodic points in nonlinear discrete-time dynamical systems. The parameter regulation in the controller can be theoretically derived from the optimization of the maximum local Lyapunov exponent (MLLE) that is closely related to the stability index of stable fixed and periodic points. Differently from the stability index, the MLLE is differentiable with respect to system parameters in general and can be computed in real time without finding the exact position of fixed and periodic points. The computation of parameter updating to avoid bifurcations can be also realized along the passage of time. Therefore, the parametric controller we propose can detect the approach of parameter values to bifurcation points by monitoring the MLLE and avoid the bifurcation points by suppressing the MLLE below a prescribed negative value even when unexpected parameter variation causing bifurcations occurs. The outline of our controller and experimental results to evaluate whether our controller is effective for avoiding bifurcations are presented.