Regions of stability phases discovered in a general class of Genesio−Tesi chaotic oscillators are proposed. In a relatively large region of two-parameter space, the system has coexisting point attractors and limit cycles. The variation of two parameters is used to characterize the multistability by plotting the isospike diagrams for two nonsymmetric initial conditions. The parameters window in which the jerk system exhibits the unusual and striking feature of multiple attractors (e.g., coexistence of six disconnected periodic chaotic attractors and three-point attraction) is investigated. The second aspect of this study presents the synchronization of systems that act as mediators between two dynamical units that, in turn, show function projective synchronization (FPS) with each other. These are the so-called relay systems. In a wide range of operating parameters; this setup leads to synchronization between the outer circuits, while the relaying element remains unsynchronized. The results show that the coupled systems can achieve function projective synchronization in a determined time despite the unpredictability of the scaling function. In the coupling path, the outer dynamical systems show finite-time synchronization of their outputs, that is, displaying the same dynamics at exactly the same moment. Further, this effect is rather general and it has a wide range of applications where sustained oscillations should be retained for proper functioning of the systems.

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This paper deals with a new approach to explore the precise dynamic response of the maglev system train and its control. Magnetic-suspension systems are characterized by high nonlinearity and open-loop instability which are the core components of maglev vehicles. Firstly, we use the electromagnetics and mechanics laws to derive the mathematical expressions of the proposed maglev system. Analytical investigation and theoretical calculation show that for the specific values of the control system parameters, the maglev system train can be significantly improved. It points out that the inherent nonlinearity, the inner coupling, misalignments between the sensors and actuators, and external disturbances are the main issues that should be considered for maglev engineering. Secondly, a control strategy based on the precise model of a nonsing ular robust sliding mode control is designed to reduce the upper bound of both the uncertainty and interference of the sliding mode controller. This approach presents an added value compared to the new sliding control methods in terms of overshoot and speed of convergence which is designed to control the vertical position of the proposed system. By using rigorous mathematical transformation associated with the adaptation laws in the frequency domain, a sufficient condition is drawn for the stability of the dynamical error based on the Lyapunov theory. This allows us a great possibility for interpreting the operation of the maglev train system. Numerical results are presented to show the effectiveness of our proposed control scheme.

This chapter announces a new four-dimensional hyperchaotic system having two positive Lyapunov exponents, a zero Lyapunov exponent, and a negative Lyapunov exponent. Since the sum of the Lyapunov exponents of the new hyperchaotic system is shown to be negative, it is a dissipative system. The phase portraits of the new hyperchaotic system are displayed with both two-dimensional and three-dimensional phase portraits. Next, the qualitative properties of the new hyperchaotic system are dealt with in detail. It is shown that the new hyperchaotic system has three unstable equilibrium points. Explicitly, it is shown that the equilibrium at the origin is a saddle-point, while the other two equilibrium points are saddle-focus equilibrium points. Thus, it is shown that all three equilibrium points of the new hyperchaotic system are unstable. Numerical simulations with MATLAB have been shown to validate and demonstrate all the new results derived in this chapter. Finally, a circuit design of the new hyperchaotic system is implemented in MultiSim to validate the theoretical model.

In this chapter, the dynamics of a particular topology of Colpitts oscillator with fractional order dynamics is presented. The first part is devoted to the dynamics of the model using standard nonlinear analysis techniques including time series, bifurcation diagrams, phase space trajectories plots, and Lyapunov exponents. One of the major results of this innovative work is the numerical finding of a parameter region in which the fractional order Colpitts oscillator's circuit experiences multiple attractors' behavior. This phenomenon was not reported previously in the Colpitts circuit (despite the huge amount of related research works) and thus represents an enriching contribution to the understanding of the dynamics of Chua's oscillator. The second part of this chapter deals with the synchronization of fractional order system. Based on fractional-order Lyapunov stability theory, this chapter provides a novel method to achieve generalized and phase synchronization of two and network fractional-order chaotic Colpitts oscillators, respectively.

A new scheme for secure information transmission is proposed using the generalized modified projective synchronization (GMPS) method. ). This circuit is employed to encrypt the information signal. In the receiver end, by designing the controllers and the parameter update rule, GMPS between the transmitter and receiver systems is achieved and the unknown parameters are estimated simultaneously. Based on the Lyapunov stability theory, the controllers and corresponding parameters update rule are constructed to achieve generalized modified projective synchronization between the transmitter and receiver system with uncertain parameters. The original information signal can be recovered successfully through some simple operations by the estimated parameter. The message signal can be finally recovered by the identified parameter and the corresponding demodulation method. Numerical simulations are performed to show the validity and feasibility of the presented secure communication scheme.

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