Nonlinear control strategies for chaotic and hyperchaotic synchronization of first order systems of ordinary differential equations

Chaos synchronization is a process that can establish similar responses of two or more chaotic or hyperchaotic systems for different initial conditions using feedback controllers. Past study developed controllers that use larger feedback gains to increase synchronization speed. Even though large feedback gains can reduce errors and convergence time, they cause oscillations in the system state variables when the synchronization errors converge to zero. These oscillations take longer to plateau and give rise to erroneous system behavior, thus slowing the rate of convergence. Most feedback controllers eliminate nonlinear terms in the closed-loop system which jeopardizes robust performance. In this thesis, the main objective is to construct nonlinear controllers that can produce oscillation-free, faster, and robust synchronization of chaotic and hyperchaotic systems of first order differential equations, while retaining the nonlinear terms in the systems. Using Lyapunov stability theory, the controllers are designed to achieve synchronization for systems with known and unknown parameters. A comparison study is conducted to verify the performance of the controllers in reducing convergence time and oscillation. Therefore, three novel controllers embedded with hyperbolic tangent functions are proposed: nonlinear and finite-time synchronization controllers for systems with known parameters, and adaptive synchronization controllers for systems with unknown parameters. The findings showed that the proposed controllers performed better than past controllers in reducing the convergence time and oscillations. Furthermore, the controllers are robust due to their ability to preserve the nonlinear terms in the systems. Hence, the proposed controllers can potentially be used as a basis for secure communication.