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.

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.

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