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Chaos synchronization of the Duffing, Lorenz and Rössler systems with fractional orders are studied theoretically and numerically. Three methods are applied in this paper: combination of active-passive decomposition (APD) and one-way coupling methods, Pecora–Carroll method, bidirectional coupling method. The sufficient conditions of achieving synchronization between two identical fractional systems are derived by using the Laplace transform theory. Numerical simulations demonstrate the effectiveness of the proposed synchronization schemes for these fractional systems.

Stochastic bifurcation of a Duffing system subject to a combination of a deterministic harmonic excitation and a white noise excitation is studied in detail by the generalized cell mapping method using digraph. It is found that under certain conditions there exist two stable invariant sets in the phase space, associated with the randomly perturbed steady-state motions, which may be called stochastic attractors. Each attractor owns its attractive basin, and the attractive basins are separated by boundaries. Along with attractors there also exists an unstable invariant set, which might be called a stochastic saddle as well, and stochastic bifurcation always occurs when a stochastic attractor collides with a stochastic saddle. As an alternative definition, stochastic bifurcation may be defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value. This definition applies equally well either to randomly perturbed motions, or to purely deterministic motions. Our study reveals that the generalized cell mapping method with digraph is also a powerful tool for global analysis of stochastic bifurcation. By this global analysis the mechanism of development, occurrence and evolution of stochastic bifurcation can be explored clearly and vividly.

In this paper, a modification of homotopy analysis method (HAM) is applied to study the two-degree-of-freedom coupled Duffing system. Firstly, the process of calculating the two-degree-of-freedom coupled Duffing system is presented. Secondly, the single periodic solutions and double periodic solutions are obtained by solving the constructed nonlinear algebraic equations. Finally, comparing the periodic solutions obtained by the multi-frequency homotopy analysis method (MFHAM) and the fourth-order Runge–Kutta method, it is found that the approximate solution agrees well with the numerical solution.

This paper focuses on the classification of the bifurcation modes of a Duffing system under the combined excitations of constant force and harmonic excitation. The Harmonic Balance method combined with the arc-length continuation is used to obtain the periodic solutions of the system, and the Floquet theory is employed to analyze the stability of the corresponding solutions. Accordingly, the frequency-response curves affected respectively by the constant force and the magnitude of the harmonic excitation are analyzed to show the basic dynamical properties of the system. Afterwards, the bifurcation investigations are carried out with the aid of the two-state variable singularity method. It is derived that there are a total of six different types of bifurcation modes due to the effects of the constant force and the magnitude of the harmonic excitation. At last, the effects of the nonlinearity parameter and the damping ratio on the bifurcation modes of the system are also discussed. The results obtained in this paper extend the findings in reference that the system can have markedly three types of frequency-response curves: with only one solution, or with maximum three or five solutions for a certain excitation frequency, and contribute to a better understanding of the significant influence of the constant force.

Two or more dynamical systems can achieve external synchronization motions with an especially designed controller, in which one dynamical system is considered as the master, and the other as the slave. In this paper, a general strategy of feedback control to carry out the external synchronization is firstly introduced. The master system consists of a hysteretic device to account for the friction phenomenon, and the slave system comprises a hardening stiffness component that can generate the third order displacement as a Duffing system. The slave system can keep in synchronization with the movements of the master via a feedback control strategy. A stability analysis of the synchronization errors is conducted. Different synchronization motions under different system parameters are simulated, based on which discussions are made for the effects of the controller parameters.