Multistability in nonlinearly coupled ring of Duffing systems

In this paper we consider dynamics of three unidirectionally coupled Duffing oscillators with nonlinear coupling function in the form of third degree polynomial. We focus on the influence of the coupling on the occurrence of different bifurcation’s scenarios. The stability of equilibria, using Routh-Hurwitz criterion, is investigated. Moreover, we check how coefficients of the nonlinear coupling influence an appearance of different types of periodic solutions. The stable periodic solutions are computed using path-following. Finally, we show the two parameters’ bifurcation diagrams with marked areas where one can observe the coexistence of solutions.     

On Algebraic Integrability of the Deformed Elliptic Calogero-Moser Problem

Abstract

We discuss stationary solutions of the discrete nonlinear Schrödinger equation (DNSE) with a potential of the ? ⁴ type which is generically applicable to several quantum spin, electron and classical lattice systems. We show that there may arise chaotic spatial structures in the form of incommensurate or irregular quantum states. As a first (typical) example we consider a single electron which is strongly coupled with phonons on a 1D chain of atoms — the (Rashba)–Holstein polaron model. In the adiabatic approximation this system is conventionally described by the DNSE. Another relevant example is that of superconducting states in layered superconductors described by the same DNSE. Amongst many other applications the typical example for a classical lattice is a system of coupled nonlinear oscillators. We present the exact energy spectrum of this model in the strong coupling limit and the corresponding wave function. Using this as a starting point we go on to calculate the wave function for moderate coupling and find that the energy eigenvalue of these structures of the wave function is in exquisite agreement with the exact strong coupling result. This procedure allows us to obtain (numerically) exact solutions of the DNSE directly. When applied to our typical example we find that the wave function of an electron on a deformable lattice (and other quantum or classical discrete systems) may exhibit incommensurate and irregular structures. These states are analogous to the periodic, quasiperiodic and chaotic structures found in classical chaotic dynamics.     

Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling

In this paper, we study a system of three coupled van der Pol oscillators that are coupled through the damping terms. Hopf bifurcations and amplitude death induced by the coupling time delay are first investigated by analyzing the related characteristic equation. Then the oscillation patterns of these bifurcating periodic oscillations are determined and we find that there are two kinds of critical values of the coupling time delay: one is related to the synchronous periodic oscillations, the other is related to eight branches of asynchronous periodic solutions bifurcating simultaneously from the zero solution. The stability of these bifurcating periodic solutions are also explicitly determined by calculating the normal forms on center manifolds, and the stable synchronous and stable phase-locked periodic solutions are found. Finally, some numerical simulations are employed to illustrate and extend our obtained theoretical results and numerical studies also describe the switches of stable synchronous and phase-locked periodic oscillations.     

Taylor与Agrawal模型的解析求解与模型比较

采用解析方法对双导传输线Taylor模型和Agrawal模型进行研究,获得了这两个模型基于平面电磁波激励下的终端负载响应的频域解析解,并证明它们的解析解是相同的,从而得到双导传输线基于平面电磁波激励的终端负载响应的频域解析解公式。Taylor模型和Agrawal模型其实是对同一个解的不同描述,由于Agrawal模型比Taylor模型相对简单,因此在实际应用时可以采用Agrawal模型或频域解析解公式进行求解。应用频域解析解公式和Fourier变换与逆变换技术,可以求得终端电压或电流的瞬时响应解。     

厄尔尼诺-南方涛动时滞海气振子耦合模型的周期解

运用重合度理论探讨了一类非线性问题的周期解的存在性.然后将其应用于一个厄尔尼诺-南方涛动时滞海气振子耦合模型周期解问题的研究,获得到了该模型存在周期解的新结果. 关键词： 非线性 厄尔尼诺-南方涛动 周期解     

Taylor与Agrawal模型的解析求解与模型比较

 采用解析方法对双导传输线Taylor模型和Agrawal模型进行研究,获得了这两个模型基于平面电磁波激励下的终端负载响应的频域解析解,并证明它们的解析解是相同的,从而得到双导传输线基于平面电磁波激励的终端负载响应的频域解析解公式。Taylor模型和Agrawal模型其实是对同一个解的不同描述,由于Agrawal模型比Taylor模型相对简单,因此在实际应用时可以采用Agrawal模型或频域解析解公式进行求解。应用频域解析解公式和Fourier变换与逆变换技术,可以求得终端电压或电流的瞬时响应解。     

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.     

On the solutions of 3-particle spin elliptic Calogero-Moser systems

We describe a class of the singular solutions to the multicomponent analogs of the Lamé equation, arising as equations of motion of the elliptic Calogero-Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.     

Soliton switching in nonlinear couplers

We present a theoretical overview of soliton switching phenomena in two-mode nonlinear couplers. By complementing numerical studies with perturbative or exact solitary wave solutions, one finds that nonlinear Schrödinger or sine-Gordon solitons tend to maintain their identity in the coupled systems. Moreover, the coupling itself may originate novel vector solitary waves, such as gap solitons in periodic media. The switching dynamics in the presence of dissipative perturbations such as linear gain or intrapulse Raman scattering is also discussed.     

分数阶时滞反馈对Duffing振子动力学特性的影响

研究了含分数阶时滞耦合反馈的Duffing自治系统, 通过平均法得到了系统周期解的一阶近似解析形式, 定义了以反馈系数、分数阶阶次、时滞参数表示的等效刚度和等效阻尼系数, 发现分数阶时滞耦合反馈同时具有速度时滞反馈和位移时滞反馈的作用. 比较了三种参数条件下近似解析解与数值积分的结果, 二者的吻合精度都很高, 证明了近似解析解的正确性和准确性. 分析了反馈系数、分数阶阶次和非线性刚度系数等参数对系统分岔点、周期解稳定性、周期解的存在范围、零解的稳定性以及稳定性切换次数等系统动力学特性的影响.     

Symmetry breaking and high-frequency periodic oscillations in mutually coupled laser diodes

We investigate the dynamical behavior of two laser diodes coupled through mutual injection of their optical fields when placed face to face with a small separation between them. We report symmetry breaking in periodic solutions at low coupling rates. In addition, we demonstrate that at higher coupling rates both lasers exhibit very fast periodic oscillations. The system is of practical interest, since it constitutes a tunable all-optical source of microwave oscillations.     

Variational Approach to Uniform Rigid Rotation of Ginzburg—Landau Vortices

Time periodic solutions for the hyperbolic gauged Ginzburg–Landau system, with spatial domain the unit disc, are shown to exist. Time periodic solutions representing bound states of vortices rotating about one another have been previously obtained in the near self-dual limit, using perturbative techniques. In contrast, we here take a variational approach, the solutions being obtained as critical points of an indefinite functional. We consider a special class of solutions which map out, uniformly in time, an orbit of the rotation group SO(2). It is shown that in the limit of large coupling constant the solutions have nontrivial time dependence or, as is shown to be equivalent, are not radially symmetric in any gauge.     

A Borel Transform Method for Locating Singularities of Taylor and Fourier Series

Given a Taylor series with a finite radius of convergence, its Borel transform defines an entire function. A theorem of Pólya relates the large distance behavior of the Borel transform in different directions to singularities of the original function. With the help of the new asymptotic interpolation method of van der Hoeven, we show that from the knowledge of a large number of Taylor coefficients we can identify precisely the location of such singularities, as well as their type when they are isolated. There is no risk of getting artefacts with this method, which also gives us access to some of the singularities beyond the convergence disk. The method can also be applied to Fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the Burgers equation. Large precision on scaling exponents (up to twenty accurate digits) can be achieved.     

Applications of Jacobi Elliptic Function Expansion Method for Nonlinear Differential-Difference Equations

The Jacobi elliptic function expansion method is extended to derive the explicit periodic wave solutions for nonlinear differential-difference equations. Three well-known examples are chosen to illustrate the application of the Jacobi elliptic function expansion method. As a result, three types of periodic wave solutions including Jacobi elliptic sine function, Jacobi elliptic cosine function and the third elliptic function solutions are obtained. It is shown that the shock wave solutions and solitary wave solutions can be obtained at their limit condition.     

周期场聚焦电子束

引言 自从发现“强聚焦效应”可以应用在粒子加速器内以后,人们知道利用周期磁场能把离子束控制得更细,于是可以减少磁铁的需要量。因此近年来为了改建或新设计各种类型粒子加速器,对周期场聚焦离子束或电子束方面曾进行了很多研究工作。另一方面人们也开始研究用周期电场和磁场聚焦各种电子管内的电子束,例如行波管内电子束的聚焦等。聚焦电子束的主要特点在於必要考虑空间电荷。     

Suppression of chaos and other dynamical transitions induced by intercellular coupling in a model for cyclic AMP signaling in Dictyostelium cells

The effect of intercellular coupling on the switching between periodic behavior and chaos is investigated in a model for cAMP oscillations in Dictyostelium cells. We first analyze the dynamic behavior of a homogeneous cell population which is governed by a three-variable differential system for which bifurcation diagrams are obtained as a function of two control parameters. We then consider the mixing of two populations behaving in a chaotic and periodic manner, respectively. Cells are coupled through the sharing of a common chemical intermediate, extracellular cAMP, which controls its production and release by the cells into the extracellular medium; the dynamics of the mixed suspension is governed by a five-variable differential system. When the two cell populations differ by the value of a single parameter which measures the activity of the enzyme that degrades extracellular cAMP, the bifurcation diagram established for the three-variable homogeneous population can be used to predict the dynamic behavior of the mixed suspension. The analysis shows that a small proportion of periodic cells can suppress chaos in the mixed suspension. Such a fragility of chaos originates from the relative smallness of the domain of aperiodic oscillations in parameter space. The bifurcation diagram is used to obtain the minimum fraction of periodic cells suppressing chaos. These results are related to the suppression of chaos by the small-amplitude periodic forcing of a strange attractor. Numerical simulations further show how the coupling of periodic cells with chaotic cells can produce chaos, bursting, simple periodic oscillations, or a stable steady state; the coupling between two populations at steady state can produce similar modes of dynamic behavior.     

Non-minimal coupling of torsion–matter satisfying null energy condition for wormhole solutions

We explore wormhole solutions in a non-minimal torsion–matter coupled gravity by taking an explicit non-minimal coupling between the matter Lagrangian density and an arbitrary function of the torsion scalar. This coupling describes the transfer of energy and momentum between matter and torsion scalar terms. The violation of the null energy condition occurred through an effective energy-momentum tensor incorporating the torsion–matter non-minimal coupling, while normal matter is responsible for supporting the respective wormhole geometries. We consider the energy density in the form of non-monotonically decreasing function along with two types of models. The first model is analogous to the curvature–matter coupling scenario, that is, the torsion scalar with T-matter coupling, while the second one involves a quadratic torsion term. In both cases, we obtain wormhole solutions satisfying the null energy condition. Also, we find that the increasing value of the coupling constant minimizes or vanishes on the violation of the null energy condition through matter.     

On generation of coherent structures induced by modulational instability in linearly coupled cubic-quintic Ginzburg-Landau equations

We study the properties of the coherent structures induced by the modulational instability (MI) of the two linearly coupled complex Ginzburg-Landau equations with both cubic and quintic terms, which in nonlinear optics can model ring lasers based on dual-core fibers. We obtain new stationary solutions different from the previous result and the analytic gain formula as function of the linear coupling constant and the model parameters. The fact that the system can be modulationally unstable for the vast region of the parameters space is demonstrated. The effects of the linear coupling constant on the evolution of a continuous wave under the MI are numerically investigated in the presence of the linear loss or gain. It is found that doubly asymmetric stable solitary pulses and stable breathers can be formed from the perturbed continuous waves state by the MI. The conditions for generating the periodic stable solitary pulses and fronts by the MI are identified by varying the linear coupling constant.     

Dynamics of Finger Formation in Laplacian Growth Without Surface Tension

We study the dynamics of “finger” formation in Laplacian growth without surface tension in a channel geometry (the Saffman–Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic “finger” whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.