Analysis and hyper-chaos control of a new 4-D hyper-chaotic system by using optimal and adaptive control design

In this manuscript,a new 4-D hyper-chaotic system is proposed. Followed by optimal and adaptive control theory. Firstly, we analyze the chaotic properties of new 4-D hyper-chaotic system such as dissipation, equilibrium, stability, time series, phase portraits, Lyapunov exponents, bifurcation and Poincaré maps. Next, we study the optimal control for the new 4-D hyper-chaotic system which is based on the Pontryagin minimum principle. Further, Lyapunov stability theory is used for adaptive control approach and a parameter estimation update law is given for the new 4-D hyper-chaotic system with completely unknown parameters. Finally, to demonstrate the effectiveness of the proposed method we use MATLAB bvp4c and ode45 for numerical simulation which illustrate the stabilized behaviour of states and control functions for different equilibrium points. The plots displaying the time history of states functions and the parameters estimates have been drawn for the different values of equilibrium points. © 2016, Springer-Verlag Berlin Heidelberg.