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Abstract: Vibration phenomena of the fractionally damped systems have attracted increasing attentions in recent years. In this paper, dynamics of the fractionally damped Duffing equation is examined. The fractionally damped Duffing equation is transformed into a set of fractional integral equations solved by a predictor–corrector method. The effect of fractional order of damping on the dynamic behaviors of the motion is the main subject of the study. In this work, bifurcation of the parameter-dependent system is drawn numerically. The time evolutions of the nonlinear dynamical system responses are also described in phase portraits and the Poincaré map technique. In addition, the occurrence and the nature of chaotic attractors are verified by evaluating the largest Lyapunov exponents. Results obtained from this study illustrates that the fractional order of damping has a significant effect on the dynamic behaviors of the motion. The size of the attractor trends to enlarge when fractional order α increases. Regular motions (including period-3 motion) and chaotic motions are examined. Moreover, a period doubling route to chaos is also found. Many period-3 windows are also observed in bifurcation diagram. [Copyright &y& Elsevier]

Abstract: The dynamics of fractional-order systems have attracted a great deal of attentions in recent years. With fractional order, the dynamics of a system which includes comprehensive dynamical behaviors, such as fixed point, periodic motion, chaotic motion, and transient chaos is studied numerically in this paper. It is known that chaos exists in the fractional-order system with order less than 3. In this study, the lowest order found for this system to yield chaos is 2.43. The results are validated by the existence of a positive Lyapunov exponent. Period doubling routes to chaos in the fractional-order system are also obtained. Moreover, generation of a four-scroll chaotic attractor by the system is observed. [Copyright &y& Elsevier]

Abstract: The dynamic behaviors of a SMIB power system are studied in this paper. A single modal equation is used to analyze the qualitative behaviors of the system. The famous equation of motion is called “swing equation”. The Lyapunov direct method is applied to obtain conditions of stability of the equilibrium points of the system. The bifurcation of the parameter dependent system is studied numerically. Besides, the phase portraits, the Poincaré maps, and the Lyapunov exponents are presented to observe periodic and chaotic motions. Further, the addition of periodic force and the feedback control are used to control chaos effectively. Finally, the chaotification problem of the SMIB power system is also issued. [Copyright &y& Elsevier]

Abstract: Nonlinear control is an effective method for making two identical chaotic systems or two different chaotic systems be synchronized. However, this method assumes that the Lyapunov function of error dynamic (e) of synchronization is always formed as V(e)=1/2eTe. In this paper, modification based on Lyapunov stability theory to design a controller is proposed in order to overcome this limitation. The method has been applied successfully to make two identical new systems and two different chaotic systems (new system and Lorenz system) globally asymptotically synchronized. Since the Lyapunov exponents are not required for the calculation, this method is effective and convenient to synchronize two identical systems and two different chaotic systems. Numerical simulations are also given to validate the proposed synchronization approach. [Copyright &y& Elsevier]