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Abstract
Chaos is a very interesting complex nonlinear phenomenon that has been intensively studied in the last four decades within the science, mathematics and engineering communities. Recently, chaos has been found to be very useful and has great potential in many technological disciplines, such as information and computer sciences, power systems protection, biomedical systems analysis, flow dynamics and liquid mixing, encryption and communications, and so on. The research aim: The aim of this thesis is to study the dynamical behaviour of some present and proposed chaotic circuits. The cases of chaotic, hyperchaotic, integer order, and fractional order systems are investigated. We use various analytical, numerical, and simulation methods in the study of these systems. Also, some chaos control and synchronization techniques are developed and applied to output of some identical and different chaotic systems.
Steps of study: First, we study the dynamics of MADVP circuit and provide new theoretical findings that are verified using numerical simulations. Next, the MADVP circuit is rebuilt using the memristor circuit element. The dynamics of the modified circuit are examined then we introduce two new four-dimensional hyperchaotic systems and their circuit implementations. Continuous dependence on initial conditions of the two systems solution and some stability conditions of equilibrium points are studied. The existence of pitchfork bifurcations is demonstrated by using the center manifold theorem and the local bifurcation theory.
Second, the problems of robust exponential generalized and exponential Q-S chaos synchronization are investigated between different dimensional chaotic systems when unknown time varying parameters with uncertainties, environmental disturbances, and nonlinearity of input control signals are present.
Third, a new fractional order hyperchaotic system is introduced. Existence and uniqueness of the proposed system solution are proved. Stability of system’s equilibrium points and continuous dependence on initial conditions of the system’s solution are studied. Dynamical behaviour of the system is explored, then the circuit implementation of the fractional order system is proposed