A new 4-D hyper chaotic system generated from the 3-D Rösslor chaotic system, dynamical analysis, chaos stabilization via an optimized linear feedback control, it’s fractional order model and chaos synchronization using optimized fractional order sliding mode control

Abstract

This research article reports the finding of a new twelve term hyper-chaotic system with three nonlinearities in a fourth dimensional order, obtained by adding a nonlinear state feedback to the first equation and a cross-product nonlinear term to the third equation of the 3 dimensional famous Rösslor continuous-time chaotic system with two new optimized additive control parameters. Firstly various basic dynamic properties have been analyzed. Then the stabilization of the new hyper-chaotic motion trajectories via genetically optimized linear feedback controllers to zero, as the direct Lyapunov stability analysis of the robust controller and the Lipschitz condition to get an appropriate negative define Lyapunov function which contain non linear third order terms is difficult, the stability of the resulting error system in a given finite time is proved. The fractional order model and it’s synchronization are also included, based on an optimized fractional order sliding mode controller, not solved analytically but using genetic algorithm which can give a smart solution with an objective function and some additive unknown parameters. Numerical simulations of phase portraits (integer and fractional), bifurcation diagrams and spectrum of Lyapunov exponents are presented, chaos stabilization and synchronization are depicted and show the use of controllers scheme.