Dynamical Properties and Finite-Time Hybrid Projective Synchronization Using Fractional Nonsingular Sliding Mode Surface in Fractional-Order Two-Stage Colpitts Oscillators

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.