New globally Lipschitz no-equilibrium fractional order systems and control of hidden memory chaotic attractors

Abstract

In continuous-time no-equilibrium nonlinear fractional order systems, bounded attractors are often termed as hidden memory attractors that can provide very complicated dynamics, and their localization seems crucial. In many engineering applications of interest, a famous problem deals with the identification of globally Lipschitz no-equilibrium nonlinear fractional order systems that can produce hidden memory chaotic attractors that remain unknown. We address two new no-equilibrium 3-state variables nonlinear fractional order systems; one is non-autonomous and another is autonomous, where both systems nonlinearity satisfy the global Lipschitz condition. It has been discovered that such a non-autonomous system gives rise to a globally Lipschitz hidden memory chaotic attractor when system orders ${\delta }_1=0.998,{\delta }_2=0.997,{\delta }_3=0.999$, also when ${\delta }_1=0.999,{\delta }_2=0.999,{\delta }_3=0.999$. The autonomous system produces a globally Lipschitz hidden memory chaotic attractor when system orders become ${\delta }_1=0.997$, ${\delta }_2=0.998$, ${\delta }_3=0.999$ as well as ${\delta }_1=0.997$, ${\delta }_2=0.997$, ${\delta }_3=0.997$. In many applications of interest, it is often needed to have globally Lipschitz hidden memory chaotic attractors and reference control goal dynamics that seem crucial to widen the use of important nonlinear systems. We introduce a novel control strategy to address controlling hidden memory chaotic attractors found in such systems that seem impossible to control in many control design problems. Numerical simulations, including theoretical analysis, illustrate the effectiveness of the proposed control method.