Abdulahi Ndzi Njah

This paper introduces a new type of synchronization scheme, referred to as difference synchronization scheme, wherein the difference between the state variables of two master [slave] systems synchronizes with the state variable of a single slave [master] system. Using the Lyapunov stability theory and the active backstepping technique, controllers are derived to achieve the difference synchronization of three identical hyperchaotic Liu systems evolving from different initial conditions, as well as the difference synchronization of three nonidentical systems of different orders, comprising the 3D Lorenz chaotic system, 3D Chen chaotic system, and the 4D hyperchaotic Liu system. Numerical simulations are presented to demonstrate the validity and feasibility of the theoretical analysis. The development of difference synchronization scheme has increases the number of existing chaos synchronization scheme.

The dynamics and response of particles in an asymmetric potential of the ratchet type is considered. An approximate asymmetric ratchet potential, which allows us to apply an analytical approach to investigate the systems’ response, is proposed based on the power series interpolation scheme. The method of multiple scales (MMS) is then employed to obtain an approximate solution at the principal resonance. Comparison between results obtained from the approximated and exact potentials show excellent agreement. The frequency response curves (FRCs) for different forcing are presented. The numerically computed FRCs were compared with the theoretical result to show the extent of agreement. Furthermore, by using perturbation methods, the stable and unstable domains of the solution were obtained.

In this paper active controllers based on the Lyapunov stability theory and the Routh-Hurwitz criteria, are designed to completely synchronize two parametrically and externally excited φ⁶ Van der Pol oscillators, φ⁶ Duffing oscillators, and a φ⁶ Van der Pol oscillator with a φ⁶ Duffing oscillator in the triple-well configuration of the φ⁶ potential. The coefficient matrix of the error dynamics between each pair of synchronized systems is chosen such that the number of active control functions reduces from two to one, thereby significantly reducing controller complexity in the design. The designed controllers enable the state variables of the response system to synchronize with those of the master system in both the identical and nonidentical cases. The results are validated using numerical simulations. Application to secure communications is computationally demonstrated.

In this paper, we investigate the synchronization of chaotic systems consisting of non-identical parametrically excited oscillators. The backstepping design, which is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller is generalized and employed so as to achieve global chaos synchronization between a parametrically excited gyroscope and each of the parametrically excited pendulum and Duffing oscillator. Numerical simulations are implemented to verify the results.