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[en] The multi-coupled nonlinear factors existing in the giant magnetostrictive actuator (GMA) have a serious impact on its output characteristics. If the structural parameters are not properly designed, it is easy to fall into the nonlinear instability, which has seriously hindered its application in many important fields. The electric–magnetic-machine coupled dynamic mathematical model for GMA is established according to J-A dynamic hysteresis model, ampere circuit law, nonlinear quadratic domain model and structure dynamics equation. Nonlinear dynamic analysis method is applied to study the nonlinear dynamic behaviour of the key structure parameters to reveal their influence on the system stability. The design principle of structural parameters is obtained by studying stability of GMA, which provides theoretical basis and technical support for the structural stability design.
[en] The Boussinesq equation can describe wave motions in media with damping mechanism, e.g., the propagation of long waves in shallow water and the oscillations of nonlinear elastic strings. To study the propagation of gravity waves on the surface of water, a second spatial variable (say, y) is weakly dependent, and an alternative form of generalized two-dimensional Boussinesq equation is investigated in this paper. Four families of lump solutions are derived by searching for positive quadratic function solutions to the associated bilinear equation. To guarantee the analyticity and rational localization of the lumps, some conditions are posed on both the lump parameters and the coefficients of the generalized two-dimensional Boussinesq equation. Localized structures and energy distribution of the lumps are analyzed as well.
[en] A mathematical model of a contact interaction between two plates made from materials with different elasticity modulus is derived taking into account physical and design nonlinearities. In order to study the stress–strain state of this complex mechanical structure, the method of variational iteration has been employed allowing for reduction of partial differential equations to ordinary differential equations (ODEs). The theorem regarding convergence of this method is formulated for the class of similar-like problems. The convergence of the proposed iterational procedure used for obtaining a solution to contact problems of two plates is proved. In the studied case, the physical nonlinearity is introduced with the help of variable parameters associated with plate stiffness. The work is supplemented with a few numerical examples. Both Fourier and Morlet power spectra are employed to detect and analyse regular and chaotic vibrations of two interacting plates.
[en] Deformation rogue wave as exact solution of the (2+1)-dimensional Korteweg–de Vries (KdV) equation is obtained via the bilinear method. It is localized in both time and space and is derived by the interaction between lump soliton and a pair of resonance stripe solitons. In contrast to the general method to get the rogue wave, we mainly combine the positive quadratic function and the hyperbolic cosine function, and then the lump soliton can be evolved rogue wave. Under the small perturbation of parameter, rich dynamic phenomena are depicted both theoretically and graphically so as to understand the property of (2+1)-dimensional KdV equation deeply. In general terms, these deformations mainly have three types: two rogue waves, one rogue wave or no rogue wave.
[en] Application of feedback perturbation is a very well-known method of controlling the chaotic dynamics of a system. All the existing feedback perturbation methods in the literature suggest applying the feedback perturbation uninterruptedly to the original system to stabilize any unstable periodic orbit embedded in a chaotic attractor. In this paper, a new time-dependent feedback perturbation method has been proposed in order to control the chaotic dynamics of any finite-dimensional map. The novelty of this proposed method is that we can apply the suggested feedback perturbation at some constant time gaps, preassigned according to our own choice.
[en] This study investigates the collective stochastic resonance (SR) behavior of globally coupled fractional Langevin equations with multiplicative noise and external signal. We define the mean field S(t) and derive the steady-state output amplitude of the first moment by using the stochastic average method. We characterize the effects of fractional order, intrinsic frequency, noise correlation rate, and driving frequency on the steady-state output amplitude as a function of noise intensity. We observe that the collective SR phenomenon occurs in a fractional coupled stochastic dynamic system. We also demonstrate that collective SR behavior versus noise intensity can ensue when system parameters satisfy the necessary and sufficient conditions; this notion means that we can control the collective SR of our fractional dynamic model by properly adjusting the system parameters within a certain range. This study verifies the reliability and effectiveness of the theoretical results by various numerical simulations. Our results on SR in a globally coupled fractional harmonic oscillator provide useful information in modern science.
[en] A study of high-order solitons in three nonlocal nonlinear Schrödinger equations is presented. These include the -symmetric, reverse-time, and reverse-space-time nonlocal nonlinear Schrödinger equations. General high-order solitons in three different equations are derived from the same Riemann–Hilbert solutions of the AKNS hierarchy, except for the difference in the corresponding symmetry relations on the “perturbed” scattering data. Dynamics of general high-order solitons in these equations is further analyzed. It is shown that the high-order fundamental-soliton is moving on several different trajectories in nearly equal velocities, and they can be nonsingular or repeatedly collapsing, depending on the choices of the parameters. It is also shown that the high-order multi-solitons could have more complicated wave structures and behave very differently from high-order fundamental-solitons. More interestingly, via the combinations of different size of block matrix in the Riemann–Hilbert solutions, high-order hybrid-pattern solitons are found, which describe the nonlinear interaction between several types of solitons.
[en] In this paper, we suggest a new measure for testing reversibility of time series which combines two different tools: the visibility algorithm and the inversion number. First, the visibility algorithm maps the time series to the network according to a geometric criterion. After that, the degree of irreversibility of the time series can be estimated by the relative asynchronous index (RAI), based on the inverse number, between out and degree sequences of the network (out and represent the outgoing sequence of forward time series and reverse time series, respectively). This method does not need to rely on additional parameters, so it can avoid the error caused by parameter estimation. In addition, we also study the multiscale RAI and find that the optimal scale selection for detection time irreversibility is 1–4. Different types of time series are used to confirm the validity of this metric. Finally, we apply the method to financial time series and find that the financial crisis can be detected by RAI.