Noise-induced spatial dynamics in the presence of memory loss

We study the spatial dynamics of noise-induced waves in two-dimensional excitable media in dependence on the duration of the artificially imposed refractory time that is introduced to each constitutive system unit after an excitation. Due to the introduction of refractory times, a randomly induced spatial wave is temporarily unable to transmit information to the opposite site of its propagation direction. Thus, once the wave leaves the absorbing boundaries of the spatial grid the system has little or no recollection, depending on the duration of the refractory time, of its existence. We show that even in the presence of such memory loss, self-organization of excitatory events leads to noise-induced spatial periodicity in the media. We present a simple analytical treatment of a two-unit system to capture and explain the essence of the observed phenomenon. Since refractory times are widespread in biological systems, our results provide interesting insights into functioning of real-life organisms at the cellular as well as tissue level.     

Polymorphic and regular localized activity structures in a two-dimensional two-component reaction-diffusion lattice with complex threshold excitation

Space-time dynamics of the system modeling collective behaviour of electrically coupled nonlinear units is investigated. The dynamics of a local cell is described by the FitzHugh-Nagumo system with complex threshold excitation. It is shown that such a system supports formation of two distinct kinds of stable two-dimensional spatially localized moving structures without any external stabilizing actions. These are regular and polymorphic structures. The regular structures preserve their shape and velocity under propagation while the shape and velocity as well as other integral characteristics of polymorphic structures show rather complex temporal behaviour. Both kinds of structures represent novel sorts of spatially temporal patterns which have not been observed before in typical two-component reaction-diffusion type systems. It is demonstrated that there exist two types of regular structures: single and bound states and three types of polymorphic structures: periodic, quasiperiodic and even chaotic ones. The partition of the parameter plane into regions corresponding to the existence of these different types of structures is carried out. High multistability of regular structures is indicated. The interaction of regular structures is investigated. The correspondence between the structures and trajectories in multidimensional phase space associated with the system is given. Bifurcation mechanisms leading to the loss of stability of regular structures as well as to a transition from one type of polymorphic structure to another are indicated. The mechanisms of formation of regular and polymorphic structures are discussed.     

Exploiting the Hamiltonian structure of a neural field model

We study the unexpected disappearance of stable homoclinic orbits in regions of parameter space in a neural field model with one spatial dimension. The usual approach of using numerical continuation techniques and local bifurcation theory is insufficient to explain the qualitative change in the model’s behaviour. The lack of robustness of the model to small perturbations in parameters is surprising, and the phenomenon may be of broader significance than just our model. By exploiting the Hamiltonian structure of the time-independent system, we develop a numerical technique with which we discover that a small, separate solution curve exists for a range of parameter values. As the firing rate function steepens, the small curve causes the main curve to break and stable homoclinic orbits are destroyed in a region of parameter space. Numerically, we use level set analysis to find that a codimension-one heteroclinic bifurcation occurs at the terminating ends of the solution curves. By replacing the firing rate function with a step function, we show analytically that the bifurcation is related to the value of the firing threshold. We also show the existence of heteroclinic orbits at the breakpoints using a travelling front analysis in the time-dependent system.     

Quasi-periodic breathers in Hamiltonian networks of long-range coupling

This work is concerned with Hamiltonian networks of weakly and long-range coupled oscillators with either variable or constant on-site frequencies. We derive an infinite dimensional KAM-like theorem by which we establish that, given any N-sites of the lattice, there is a positive measure set of small amplitude, quasi-periodic breathers (solutions of the Hamiltonian network that are quasi-periodic in time and exponentially localized in space) having N-frequencies which are only slightly deformed from the on-site frequencies.     

Noise induced patterning in periodic systems with conserved dynamics

We study pattern formation in periodic systems with conserved dynamics driven by both thermally sustained flux and external (athermal) flux. Assuming the stochastic nature of each kind of flux we discuss noise induced patterning with competing stochastic dynamics in the framework of the modulated phase field method. Analytical results obtained within the mean field theory are compared with computer simulations.     

Noise-induced quasiperiodicity in a ring of unidirectionally-coupled nonidentical maps

Transitions from equilibrium to quasiperiodicity and from a two-cycle to a quasiperiodic regime are studied in a ring of unidirectionally-coupled nonidentical logistic maps. The former scenario is realized through a “soft” (Neimark–Sacker) bifurcation, while the latter through a “hard” (saddle-node) bifurcation. Special attention is paid on a noise-induced transition through “hard” bifurcation, where a phenomenon of structural stabilization of the quasiperiodic system near the bifurcation point is observed and analyzed in detail.     

Propagation of travelling waves in sub-excitable systems driven by noise and periodic forcing

It has been reported that traveling waves propagate periodically and stably in sub-excitable systems driven by noise [Phys. Rev. Lett. 88, 138301 (2002)]. As a further investigation, here we observe different types of traveling waves under different noises and periodic forces, using a simplified Oregonator model. Depending on different noises and periodic forces, we have observed different types of wave propagation (or their disappearance). Moreover, reversal phenomena are observed in this system based on the numerical experiments in the one-dimensional space. We explain this as an effect of periodic forces. Thus, we give qualitative explanations for how stable reversal phenomena appear, which seem to arise from the mixing function of the periodic force and the noise. The output period and three velocities (normal, positive and negative) of the travelling waves are defined and their relationship with the periodic forces, along with the types of waves, are also studied in sub-excitable system under a fixed noise intensity. Electronic supplementary material Supplementary Online Material     

耦合系统的混沌相位同步

通过引入混沌运动的相位定义分析了线性和非线性耦合参数对两个主共振子系统之间的混沌相位同步的影响.讨论了在近似于主共振条件下,两子系统不同步、不完全相位同步和完全相位同步之间的演化过程,揭示了不同状态相互转化与Lyapunov指数变化之间的关系,指出随着线性耦合力的增加,相位同步效应增强,然而随着非线性耦合力的增加,相位同步效应减弱. 关键词： 相位同步 Rssler振子 耦合 Lyapunov指数     

高双折射光纤中模式耦合空间分布的干涉测量法

本文应用干涉技术测量高双折射偏振保持光纤中模式耦合点的空间分布，测量系统由采用宽带来半导体激光器作为光源的调制迈克尔逊干涉仪组成，通过光程扫描方法探测沿高双折射光纤分布的模式的耦合点。     

Self-replication of mesa patterns in reaction-diffusion systems

Certain two-component reaction-diffusion systems on a finite interval are known to possess mesa (box-like) steady-state patterns in the singularly perturbed limit of small diffusivity for one of the two solution components. As the diffusivity D of the second component is decreased below some critical value D_c, with D_c=O(1), the existence of a steady-state mesa pattern is lost, triggering the onset of a mesa self-replication event that ultimately leads to the creation of additional mesas. The initiation of this phenomena is studied in detail for a particular scaling limit of the Brusselator model. Near the existence threshold D_c of a single steady-state mesa, it is shown that an internal layer forms in the centre of the mesa. The structure of the solution within this internal layer is shown to be governed by a certain core problem, comprised of a single nonautonomous second-order ODE. By analysing this core problem using rigorous and formal asymptotic methods, and by using the Singular Limit Eigenvalue Problem (SLEP) method to asymptotically calculate small eigenvalues, an analytical verification of the conditions of Nishiura and Ueyama [Y. Nishiura, D. Ueyama, A skeleton structure of self-replicating dynamics, Physica D 130 (1) (1999) 73-104], believed to be responsible for self-replication, is given. These conditions include: (1) The existence of a saddle-node threshold at which the steady-state mesa pattern disappears; (2) the dimple-shaped eigenfunction at the threshold, believed to be responsible for the initiation of the replication process; and (3) the stability of the mesa pattern above the existence threshold. Finally, we show that the core problem is universal in the sense that it pertains to a class of reaction-diffusion systems, including the Gierer-Meinhardt model with saturation, where mesa self-replication also occurs.     

Reliability of linear coupling synchronization of hyperchaotic systems with unknown parameters

Complete synchronization could be reached between some chaotic and/or hyperchaotic systems under linear coupling. More generally, the conditional Lyapunov exponents are often calculated to confirm the stability of synchronization and reliability of linear controllers. In this paper, detailed proof and measurement of the reliability of linear controllers are given by constructing a Lyapunov function in the exponential form. It is confirmed that two hyperchaotic systems can reach complete synchronization when two linear controllers are imposed on the driven system unidirectionally and the unknown parameters in the driving systems are estimated completely. Finally, it gives the general guidance to reach complete synchronization under linear coupling for other chaotic and hyperchaotic systems with unknown parameters.     

二次耦合光力学系统的一类高维可控自持振荡行为

光力学系统通常的耦合是光压耦合, 是光场强度和纳米振子位移的一次耦合, 但在光场很强和振子振幅较大的光力学系统中, 非线性的耦合效应会变得非常明显和重要, 而且其所产生的非线性效应对制造具有特殊功能的光力学器件具有重要意义. 本文在二次耦合模型的基础上研究了光腔和振子之间通过二次耦合作用达到能 量平衡状态时系统所产生的自持振荡现象, 给出了二次耦合光力学系统的一般模型, 并通过数值方法研究了系统的定态行为和远离定态的极限环动力学行为, 标定了系统定态响应的稳定区域到极限环行为的分岔点. 发现在调节输入场参数(改变耦合系数)以及光腔和振子的弛豫系数时, 系统的相空间会出现一些稳定的高维自持振荡极限环. 通过数值分析发现该四维极限环在三维相空间的投影都趋于稳定的三维周期轨道, 并且该极限环轨道会随外部调控参数的改变发生扭动, 出现类似二维李萨如图样的稳定纽结结构. 该现象表明: 通过光场与振子的能量耦合, 利用一定强度的外部驱动可以有效控制振子的定态响应和振动, 可以让微振子锁定在具有一定振幅和频率的自发振动上, 为开发物理器件提供了可靠的光力学控制系统. 关键词： 光力系统 二次耦合 自持振荡 极限环     

管束穿孔板的管腔耦合共振吸声机理研究

为了揭示管束穿孔板共振吸声结构的吸声机理,利用热黏性条件下基于有限元算法的管束穿孔板仿真模型,研究了平面声波正入射条件下,管束穿孔板内部声场分布特征,并利用阻抗管对吸声系数的理论仿真结果进行了试验验证.结果表明,管束穿孔板在低频主要靠腔体共振吸声,在高频主要靠管共振吸声,管束穿孔板整体呈现出较为明显的管腔耦合共振吸声特征。管束穿孔板共振时管中声强和质点法向振速较大,高频次吸声峰频点处管中和腔中均有驻波形成,频率越高驻波数量越多.管束穿孔板的耦合共振受到管长、腔深、穿孔率和管内径等参数变化的影响,管长对高频耦合共振的影响最大,管长增大使高频主吸声峰频点移向低频,并使相邻主吸声峰之间的间距减小.     

Finite-size scaling in systems with long-range interaction

The finite-size critical properties of the (n) vector ϕ⁴ model, with long-range interaction decaying algebraically with the interparticle distance r like r ^{-d - σ}, are investigated. The system is confined to a finite geometry subject to periodic boundary condition. Special attention is paid to the finite-size correction to the bulk susceptibility above the critical temperature T _c. We show that this correction has a power-law nature in the case of pure long-range interaction i.e. 0 < σ < 2 and it turns out to be exponential in case of short-range interaction i.e.σ = 2. The results are valid for arbitrary dimension d, between the lower ( d _< = σ) and the upper ( d _> = 2σ) critical dimensions. Received 2 July 2001 and Received in final form 4 Septembre 2001     

双层非线性耦合反应扩散系统中复杂Turing斑图

采用双层耦合的Brusselator模型, 研究了两个子系统非线性耦合时Turing 模对斑图的影响, 发现两子系统Turing 模的波数比和耦合系数的大小对斑图的形成起着重要作用. 模拟结果表明: 斑图类型随波数比值的增加, 从简单斑图发展到复杂斑图; 非线性耦合项系数在0–0.1时, 系统1中短波模在系统2失稳模的影响下不仅可形成简单六边形、四边形和条纹斑图, 两模共振耦合还可以形成蜂窝六边形、超六边形和复杂的黑眼斑图等超点阵图形, 首次在一定范围内调整控制参量观察到由简单正四边形向超六边形斑图的转化过程; 耦合系数在0.1–1时, 系统1中短波模与系统2失稳模未发生共振耦合仅观察到与系统2相同形状的简单六边形、四边形和条纹斑图. 关键词： Brusselator模型 非线性耦合 Turing模     

单向耦合网络连接的Lorenz系统的追踪控制

基于Lyapunov稳定性理论，将单个Lorenz系统的一种追踪控制方案用于单向耦合网络连接的Lorenz系统. 只对网络系统的一个节点加入控制器，就实现了单向耦合网络连接的Lorenz系统的单个输出变量对任意给定参考信号的追踪. 数值仿真表明，当网络的耦合强度足够大时，可以实现包括同结构和异结构系统信号之间的有效追踪. 关键词： Lorenz混沌系统 复杂动态网络 追踪控制 单向耦合