Dynamical Time delay occurs in many dynamical systems

Dynamical systems with time delay exhibit a rich variety of different behaviors, including the possibility of high dimensional chaotic behavior.

A system is said to have a delay when the rate of variation in the system state depends on past states. Such a system is called a time delay system. Time delay occurs in many dynamical systems such as biological systems, chemical systems, network systems and engineering systems. They are often a source of instability and greatly increase the difficulty of stability analysis and control design. The aim of this thesis is to study and investigate the effect of the delay in chaotic and hyperchaotic complex systems.

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These systems with time delay exhibit more complex and adequate dynamic behavior than those without time delay such as, it has higher dimensional chaotic behavior that cannot be anticipated by a low dimensional system (ordinary differential equations). Understanding and exploring the behavior of these systems are important both from the academic and engineering perspective. Additionally, the infinite dimensionality of delay systems offers a great opportunity to the researchers to harness the richness of hyperchaos, having multiple positive Lyapunov exponents (LEs). It has already been established that communication with a low dimensional chaos (having a single positive LE) is not fully secure because an eavesdropper can reconstruct the chaotic attractor and retrieve the hidden message. Moreover, its characteristic equation is transcendental due to the exponential functions associated with time delay.

This transcendental function brings an infinite number of characteristic roots, which are cumbersome to handle. The stability analysis of fixed points is stated in details. Numerically the range of parameters values of these systems at which chaotic and hyperchaotic attractors exist is calculated. Lyapunov exponents are computed to classify the dynamics of these systems. Also, an especially powerful control scheme was introduced by Pyragas which is called time delayed feedback control or time delay autosynchronization is used to control some chaotic systems. On the other hand, several kinds of synchronization of time delay systems are studied and introduced since they appear in many applications in applied sciences.

Such as, generalized complex modified hybrid function projective lag synchronization (GCMHFPLS) and phase and anti-phase combination synchronization.