Projective synchronization for a new class of chaotic/hyperchaotic systems with and without parametric uncertainty

The term "chaotic system" refers to a kind of dynamical system that is characterized by its extreme sensitivity to its initial conditions. The ability of such dynamical systems to maintain synchronization is one of the extremely crucial features of these systems. There is an increasing fascination in chaotic systems with real-state variables, and they are being discovered as well as explored in greater depth in a variety of areas of nonlinear science. In this manuscript, a mixture of active nonlinear control scheme as well as adaptive control scheme has been recommended to investigate the full-order and reduced-order projective synchronization for a unique class of chaotic and hyperchaotic systems with as well as without uncertain parameters. Active nonlinear control technique is an effective method, which has been extensively used to synchronize two chaotic systems in master-slave configuration when parameters of the systems under consideration are fully known in advance. However, in practical scenarios, system parameters are not exactly known in advance. This often occurs due to the factors such as limitations associated with the experimental conditions, mutual interference among system components or due to the influence of external environmental factors. Therefore, in such cases, the designer has to go for the adaptive control technique to develop the adaptation control laws for unknown system parameters. Adaptive active nonlinear controller and parameter adaptation laws have been introduced here using Lyapunov stability criteria to confirm that the error dynamics of the synchronizing chaotic systems become asymptotic stable. Furthermore, numerical simulations on Lorenz hyperchaotic fourth-order system and Pehlivan chaotic third-order system are shown to demonstrate that the proposed synchronization scheme is working effectively.