Two-dimensional anisotropic vortex–bright soliton and its dynamics in dipolar Bose–Einstein condensates in optical lattice

We study construction and dynamics of two-dimensional (2D) anisotropic vortex–bright (VB) soliton in spinor dipolar Bose–Einstein condensates confined in a 2D optical lattice (OL), with two localized components linearly mixed by the spin–orbit coupling and long-range dipole–dipole interaction (DDI). It is found that the OL and DDI can support stable anisotropic VB soliton in the present setting for arbitrarily small value of norm N. We then present a new method via examining the mean square error of norm share of bright component to implement stability analysis. It is revealed that one can control the stability of anisotropic VB soliton only by adjusting OL depth for a fixed DDI. In addition, the dynamics of the anisotropic VB soliton was studied by applying the kick to them. The mobility of the single kicked VB soliton is Rabbi-like oscillation. However, for the collision dynamics of two kicked anisotropic VB solitons, their properties mainly depend on their initial distance and OL, and they can realize the transition from the bright component to the vortex component. Our work may provide a convenient way to prepare and manipulate anisotropic VB soliton in high-dimensional space.

     

Solitons and their collisions in the spinor Bose–Einstein condensates

Under investigation in this paper is a system of three-component Gross?CPitaevskii equations, which can describe the dynamics of F=1 spinor Bose?CEinstein condensates in one dimension. By employing the Hirota method and symbolic computation, we obtain the explicit bright one- and two-soliton solutions for the system in the integrable case, which is associated with the attractive mean-field collision and ferromagnetic spin-exchange collision. According to the spin states, we classify the one-soliton solutions into two types: the ferromagnetic and polar solitons. Ferromagnetic solitons in three components share the same pulse shape. Polar solitons in three components have the one- or two-peak profiles, and the separated distance between two peaks is related to the polarization parameters. Based on the asymptotic analysis, collisions between two solitons are discussed in the polar?Cpolar, polar?Cferromagnetic, and ferromagnetic?Cferromagnetic cases, respectively.     

Lax pair,binary Darboux transformation and dark solitons for the three-component Gross–Pitaevskii system in the spinor Bose–Einstein condensate

Nonlinear Dynamics - In this paper, we investigate the dark solitons for the three-component Gross–Pitaevskii system, which describes the $$F=1$$ spinor Bose–Einstein condensate, with F...     

Multipulse Phases in k-Mixtures of Bose–Einstein Condensates

For the system

$-\Delta U_i+ U_i=U_i^3-\beta U_i\sum_{j\neq i}U_j^2,\quad i=1,\dots,k,$

(with k ≧ 3), we prove the existence for β large of positive radial solutions on \({\mathbb R^N}\) . We show that as β →  + ∞, the profile of each component U _i separates, in many pulses, from the others. Moreover, we can prescribe the location of such pulses in terms of the oscillations of the changing-sign solutions of the scalar equation  ? ΔW  +  W  =  W³. Within an Hartree–Fock approximation, this provides a theoretical indication of phase separation into many nodal domains for the k-mixtures of Bose–Einstein condensates.

     

Bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances

The bifurcation and chaos of a cable–beam coupled system under simultaneous internal and external resonances are investigated. The combined effects of the nonlinear term due to the cable’s geometric and coupled behavior between the modes of the beam and the cable are considered. The nonlinear partial-differential equations are derived by the Hamiltonian principle. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. The bifurcation diagrams in three separate loading cases, namely, excitation acting on the cable, on the beam and simultaneously on the beam and cable, are analyzed with changing forcing amplitude. Based on careful numerical simulations, bifurcations and possible chaotic motions are represented to reveal the combined effects of nonlinearities on the dynamics of the beam and the cable when they act as an overall structure.     

Scattering of solitons in binary Bose–Einstein condensates with spin-orbit and Rabi couplings

Nonlinear Dynamics - In this paper, we study the pricing decisions stability when a retailer adopts bundling strategy for complementary products. We firstly explore whether or not the decision...     

Bifurcations and chaos in a discrete predator–prey model with Crowley–Martin functional response

In this paper, a discrete-time predator–prey model with Crowley–Martin functional response is investigated based on the center manifold theorem and bifurcation theory. It is shown that the system undergoes flip bifurcation and Neimark–Sacker bifurcation. An explicit approximate expression of the invariant curve, caused by Neimark–Sacker bifurcation, is given. The fractal dimension of a strange attractor and Feigenbaum’s constant of the model are calculated. Moreover, numerical simulations using AUTO and MATLAB are presented to support theoretical results, such as a cascade of period doubling with period-2, 4, 6, 8, 16, 32 orbits, period-10, 20, 19, 38 orbits, invariant curves, codimension-2 bifurcation and chaotic attractor. Chaos in the sense of Marotto is also proved by both analytical and numerical methods. Analyses are displayed to illustrate the effect of magnitude of interference among predators on dynamic behaviors of this model. Further the chaotic orbit is controlled to be a fixed point by using feedback control method.     

Bifurcation of periodic solution in a Predator–Prey type of systems with non-monotonic response function and periodic perturbation

A Predator–Prey type of dynamical systems with non-monotonic response function and time-periodic perturbation is considered in this paper. We present a proof for the number of equilibria in the unperturbed system at some parts of the parameter space. The perturbed system is a dynamical system defined by a periodic vector field. We present an alternative proof for a classical result on the period of the periodic solution. By using a numerical continuation method AUTO (Doedel et al., 1986 [9]), we present a bifurcation analysis for periodic solution of the perturbed system where we found fold, cusp and Swallowtail bifurcations.     

On the response spectrum of Euler–Bernoulli beams with a moving mass and horizontal support excitation

In this study, the response spectrum of a time-varying system such as a beam subjected to moving masses under a harmonic and earthquake support excitations is explored. The excitations are supposed to act on the horizontal directions of the beam axis. The inertial effect of the moving masses on the natural frequencies of the beam for different cases of loading is investigated and a critical value of a so called parameter “mass staying time” is presented to avoid dynamic instability of the system. Finally, some 3D response spectra for different supports excitations as well as the beam natural frequencies are depicted.     

Effects of the dynamic wheel–rail interaction on the ground vibration generated by a moving train

Based on Biot’s fully dynamic poroelastic theory, the dynamic responses of the poroelastic half-space soil medium due to quasi-static and dynamic loads from a moving train are investigated semi-analytically. The dynamic loads are assumed to be generated from the rail surface irregularities. The vehicle dynamics model is used to simulate the axle loads (quasi-static loads) and the dynamic loads from a moving train. The compatibility of the displacements at wheel–rail contact points couple the vehicle and the track–ground subsystem, and yield equations for the dynamic wheel–rail loads. A linearized Hertzian contact spring between the wheel and rail is introduced to calculate the dynamic loads. Using the Fourier transform, the governing equations for the poroelastic half-space are then solved in the frequency–wavenumber domain. The time domain responses are evaluated by the fast inverse Fourier transform. Numerical results show that the dynamic loads can make important contribution to dynamic response of the poroelastic half-space for different train speed, and the dynamically induced responses lie in a higher frequency range. The ground vibrations caused by the moving train can be intensified as the primary suspension stiffness of the vehicle increases.     

On fallacies in the decision between the Caputo and Riemann–Liouville fractional derivatives for the analysis of the dynamic response of a nonlinear viscoelastic oscillator

Recently Dal [Dal, F., 2011. Multiple time scale solution of an equation with quadratic and cubic nonlinearities having fractional-order derivative. Mathematical and Computational Applications 16 (1), 301–308] presented ‘a new analytical scheme’ to calculate the dynamic response of a fractionally damped nonlinear oscillator possessing both quadratic and cubic nonlinearities via the method of multiple time scales. It has been claimed that damping features are modeled via the Caputo fractional derivative. In the present paper, it is shown that both the scheme and the object of investigation are not new, and moreover, the above mentioned author's statement is inconsistent, since under the assumptions made in the paper under consideration these two fractional-order derivatives coincide. Besides, the utilized procedure was inconsequential. It has been proved that the investigation of the dynamic response of a nonlinear viscoelastic oscillator presents the case that, with some minimal restrictions, the Riemann–Liouville and Caputo definitions produce completely equivalent mathematical models of the nonlinear viscoelastic phenomenon.     

Bifurcation analysis in a predator–prey system with a functional response increasing in both predator and prey densities

Poor dispersion characteristics of rockets, due to the orientation of the launcher for multiple launch rocket system (MLRS) departing from that intended, have always restricted the MLRS development for several decades. Orienting control is a key technique to improve the dispersion characteristics of rockets. The purpose of this paper is to propose an orienting control method for launcher of the MLRS in a salvo firing. Because the MLRS is a typical nonlinear system, the major difficulty in designing the orienting controller lies in the nonlinearity. To deal with the nonlinearity, the concept of computed torque control is introduced. The MLRS equation of motion is established using Lagrange method. The inner loop feedforward and the outer loop feedback are adopted to design the controllers for the azimuth and elevation axes of MLRS. By combining the inner and outer control loops together, the PID-computed torque controller is designed. The numerical simulation is implemented to show the control performance, and then, the effectiveness and applicability of the proposed controller are demonstrated by the firing experiment of a salvo of three rockets.     

Relaxation oscillations,subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment

We study the dynamic interactions between traveling waves propagating in a linear lattice and a lightweight, essentially nonlinear and damped local attachment. Correct to leading order, we reduce the dynamics to a strongly nonlinear damped oscillator forced by two harmonic terms. One of the excitation frequencies is characteristic of the traveling wave that impedes to the attachment, whereas the other accounts for local lattice dynamics. These two frequencies are energy-independent; a third energy-dependent frequency is present in the problem, characterizing the nonlinear oscillation of the attachment when forced by the traveling wave. We study this three-frequency strongly nonlinear problem through slow-fast partitions of the dynamics and resort to action-angle coordinates and Melnikov analysis. For damping below a critical threshold, we prove the existence of relaxation oscillations of the attachment; these oscillations are associated with enhanced targeted energy transfer from the traveling wave to the attachment. Moreover, in the limit of weak or no damping, we prove the existence of subharmonic oscillations of arbitrarily large periods, and of chaotic motions. The analytical results are supported by numerical simulations of the reduced order model.     

Soliton dynamics and interaction in the Bose–Einstein condensates with harmonic trapping potential and time-varying interatomic interaction

Investigated in this paper is the quasi-one-dimensional Gross–Pitaevskii equation, which describes the dynamics of the Bose–Einstein condensates with the harmonic trapping potential and time-varying interatomic interaction. Via the Horita method and symbolic computation, analytic bright N-soliton solution is obtained. One-, two- and three-soliton solutions are analyzed graphically. Based on the limit analysis on the one- and two-soliton solutions, the modulation on the speed of the matter-wave bright solitons is realized. Via the parameters, the interaction between the matter-wave solitons are adjustable. Furthermore, an approach to construct the interference between the matter-wave solitons has been proposed. Finally, investigation on the three-soliton solution verifies our conclusions drawn from the one and two solitons. Our conclusions might be useful in the fields of the control on the matter-wave solitons, atom lasers, and atomic accelerators.     

Monotonicity and 1-Dimensional Symmetry for Solutions of an Elliptic System Arising in Bose–Einstein Condensation

We study monotonicity and 1-dimensional symmetry for positive solutions with algebraic growth of the following elliptic system: $$\left\{\begin{array}{ll} -\Delta u = -u \upsilon^2 &\quad {\rm in}\, \mathbb{R}^N\\ -\Delta \upsilon= -u^2 \upsilon &\quad {{\rm in}\, \mathbb{R}^N},\end{array}\right.$$ for every dimension ${N \geqq 2}$ . In particular, we prove a Gibbons-type conjecture proposed by Berestycki et al.     

Bifurcation analysis of a diffusive predator–prey system with nonmonotonic functional response

Nonlinear Dynamics - In this paper, a diffusive predator–prey model with nonmonotonic functional response is investigated. The stability of the positive spatially homogeneous steady states...     

Bifurcation of a predator–prey model with disease in the prey

In this paper, a predator–prey model with disease in the prey is considered. Assume that the predator eats only the infected prey, and the incidence rate is nonlinear. We study the dynamics of the model in terms of local analysis of equilibria and bifurcation analysis of a boundary equilibrium and a positive equilibrium. We discuss the Bogdanov–Takens bifurcation near the boundary equilibrium and the Hopf bifurcation near the positive equilibrium; numerical simulation results are given to support the theoretical predictions.     

Bifurcations and chaos in a two-dimensional discrete Hindmarsh–Rose model

In this paper, the dynamics of a two-dimensional discrete Hindmarsh–Rose model is discussed. It is shown that the system undergoes flip bifurcation, Neimark–Sacker bifurcation, and 1:1 resonance by using a center manifold theorem and bifurcation theory. Furthermore, we present the numerical simulations not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, including orbits of period 3, 6, 15, cascades of period-doubling bifurcation in orbits of period 2, 4, 8, 16, quasiperiodic orbits, and chaotic sets. These results obtained in this paper show far richer dynamics of the discrete Hindmarsh–Rose model compared with the corresponding continuous model.