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This paper presents a novel approach to analyze the dynamic effect of the fractional-order derivative of the two mutually coupled van der Pol oscillators. The stability analysis is presented by two complementary phase diagrams: the isospike diagrams and the two Lyapunov exponent spectra. These diagrams reveal precisely the Hubs, spirals bifurcation and chaos when the derivative order is fixed at q = 0.95. In addition, when the fractional-order is set as a control parameter, various methods for detecting chaos including bifurcation diagrams, spectrum of largest Lyapunov exponent are exploited to establish the connection between the system parameters and various complicated dynamics. A transition was also observed between a desynchronized state and a multistability situation. These diagrams displayed the coexistence of four disconnected attractors (two symmetric). We study the basins of attraction of the system in the multistability regime which thereby reveal the coexistence of attractors in the systems when the fractional-order derivative is taken as a function of initials conditions. Based on the parametric control, we have controlled this striking phenomenon in the system. Finally, the hardware circuit is implemented and the results are found to be in good agreement with the numerical investigations.

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 addresses the dynamic analysis and two different control strategies for the synchronization of new topology of Colpitts oscillator submitted to uncertainties and external disturbances. The diagrams obtained reveal precisely spirals bifurcation and chaos when for a specific values of the system parameters. Based on the relevant control, the authors have controlled this striking phenomenon in the system. The first (control) deals with the sliding mode control (SMC) method. Some important aspects of the design and implementation are considered to reach a suitable controller for the applications. The second presents an adaptive robust tracking control strategy based on a modified polynomial observer which tends to follow exponentially the chaotic Colpitts circuits brought back to a topology of the Chua oscillator with perturbations. To highlight the contribution, they also present some simulation results with the purpose to compare the proposed method to the classical polynomial observer.