Controlling the chaotic dynamics by using approximated system equations obtained by the genetic programming
This paper deals with the control of chaotic dynamics by using the approximated system equations which are obtained by using the Genetic Programming (GP). Well known OGY method utilizes already existing unstable orbits embedded in the chaotic attractor, and use linearlization of system equations and small perturbation for control. However, in the OGY method we need transition time to attain the control, and the noise included in the linealization of equations moves the orbit into unstable region again. In this paper we propose a control method which utilize the estimated system equations obtained by the GP so that the direct nonlinear control is applicable to the unstable orbit at any time. In the GP, the system equations are represented by parse trees and the performance (fitness) of each individual is defined as the inversion of the root mean square error between the observed data and the output of tire system equation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. In the simulation study, the method is applied at first to the artificially generated chaotic dynamics such as the Logistic map and the Henon map. The error of approximation is evaluated based upon the prediction error. The effect of noise included in the time series on the approximation is also discussed. In our control, since the system equations are estimated, we only need to change the input incrementally so that the system moves to the stable region. By assuming the targeted dynamic system f(x(t)) with input u(t) = 0 is estimated by using the GP (denoted (f) over cap (x(t))), their we impose the input u(t) so that x f = (x) over cap (t + 1) = (f) over cap (x(t)) + u(t) where x(f) is the fixed point. Then, the next state x(t + 1) of targeted dynamic system f (x(t)) is replaced by x(t + 1) + u(t). The control method is applied to the approximation and control of chaotic dynamics generating various tune series and even noisy time series by using one dimensional and higher dimensional system. As a result, if the noise level is relatively large, the method of the paper provides better control compared to conventional OGY method.
chaotic dynamics genetic programming control of chaos identification