The dynamics and robust finite-time hybrid projective synchronization of a fractional-order four-dimensional nonlinear system based on a two-stage Colpitts oscillator is investigated. The study of the fractional order stability of the equilibrium states of the system is carried out. The bifurcation diagram confirms the occurrence of Hopf bifurcation in the proposed system when the fractional-order passes a sequence of critical values; the Lyapunov exponent shows the different chaotic sequences of the system. Further, a fractional nonsingular terminal sliding surface and an appropriate robust fractional sliding mode control law are proposed for the finite-time hybrid projective synchronization of a fractional-order chaotic two-stage Colpitts oscillator by taking into account the effects of model uncertainties and the external disturbances. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. Finally, some numerical simulations are presented to demonstrate the effectiveness and applicability of the proposed technique. 1. Introduction Fractional calculus has an about 300-year-old history, but its applications to physics and engineering are rather recent [1]. Many systems are known to display fractional-order dynamics, such as viscoelastic systems, dielectric polarization, and electromagnetic waves [2–4], just to name some. For some decades, there is a growing interest in investigating the chaotic behavior and dynamics of fractional-order dynamic systems; this can be understood as it has been found that fractional-order systems possess memory and display more sophisticated dynamics compared to their integral-order counterparts, something that is of great significance in secure communication [5–20]. It has been shown that several chaotic systems can remain chaotic when their models become fractional [5]. A three-dimensional fractional-order modified hybrid optical system is presented in [10] where it was shown that Hopf bifurcation occurs on the proposed system when the fractional order varies and passes a sequence of critical values. Despite these many examples the bifurcation of fractional-order nonlinear system has been studied using solely the Caputo derivative definition and limited to three-dimensional systems. On the other hand, in the past two decades, a new direction of chaos research has emerged to address the more challenging problem of chaos synchronization due to its potential applications in laser physics, chemical reactions, secure communication, biomedicine, and so on [21–23]. The
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