2015 Volume 6 Issue 1 Pages 2-14
We propose a novel controlling methodology for avoiding local bifurcations of stable, hyperbolic fixed and periodic points in nonlinear discrete-time dynamical systems with parameter variation. Dynamical systems may not work as expected owing to bifurcations of desired behavior caused by parameter variation. Controlling parameters to avoid bifurcations enables us to construct robust dynamical systems against unexpected parameter variation. Assuming that desired behavior is a hyperbolic fixed or periodic point, as a control strategy, optimizing the degree of its stability (i.e. the maximum modulus of its characteristic multipliers) is considerable. However, it cannot be optimized by simple gradient methods, and off-line calculation to find the exact positions of fixed and periodic points and their characteristic multipliers is also needed. In contrast, the method we propose is to control the maximum local Lyapunov exponent (MLLE) that is defined in finite time and is closely related to the degree of stability on hyperbolic fixed and periodic points. This method can predict occurrence of local bifurcations for parameter variation by monitoring the MLLE along the passage of time and adjust parameter values on line to control the MLLE. This leads that occurrence of local bifurcations can be avoided without directly analyzing bifurcations in advance. The parameter regulation to avoid local bifurcations is theoretically derived from the minimization of an objective function with respect to the MLLE. Our experimental results applied to the Kawakami and Hénon maps demonstrate that the proposed controller could be used to avoid local bifurcations.
