Self-organized stable pacemakers near the onset of birhythmicity

General amplitude equations are derived for reaction-diffusion systems near the soft onset of birhythmicity described by a supercritical pitchfork-Hopf bifurcation. Using these equations and applying singular perturbation theory, we show that stable autonomous pacemakers represent a generic kind of spatiotemporal patterns in such systems. This is verified by numerical simulations, which also show the existence of breathing and swinging pacemaker solutions. The drift of self-organized pacemakers in media with spatial parameter gradients is analytically and numerically investigated.     

Oscillations vs. chaotic waves: Attractor selection in bistable stochastic reaction–diffusion systems

In bistable systems, the long-term behavior of solutions depends on the location of the initial conditions. In a deterministic setting, where the initial condition is kept fixed in one particular basin of attraction, repeated numerical simulations will always lead to the same long-term behavior. The other possible asymptotic solution type will never be observed. This clear distinction does not hold anymore if the system is forced by random fluctuations. In this case, both asymptotic solutions can be attained, and the relative frequency of different long-term behaviors observed in many repeated simulation runs will follow a certain probability distribution. We present a simple reaction–diffusion model of spatial predator–prey interaction, where depending on the initial spatial distribution of the two populations either spatially homogeneous or spatiotemporal irregular oscillations may be observed. We show by repeated stochastic simulations that, when starting in the basin of attraction of the spatiotemporal irregular solution, in the randomly forced system the probability to observe spatially homogeneous oscillations instead of spatiotemporally irregular oscillations follows a non-trivial bimodal distribution.     

Spatial bistability: a source of complex dynamics. From spatiotemporal reaction-diffusion patterns to chemomechanical structures

We show experimentally and theoretically that reaction systems characterized by a slow induction period followed by a fast evolution to equilibrium can readily generate "spatial bistability" when operated in thin gel reactors diffusively fed from one side. This phenomenon which corresponds to the coexistence of two different stable steady states, not breaking the symmetry of the boundary conditions, can be at the origin of diverse reaction-diffusion instabilities. Using different chemical reactions, we show how stationary pulses, labyrinthine patterns or spatiotemporal oscillations can be generated. Beyond simple reaction-diffusion instabilities, we also demonstrate that the cross coupling of spatial bistability with the size responsiveness of a chemosensitive gel can give rise to autonomous spatiotemporal shape patterns, referred to as chemomechanical structures.     

Noise-induced synchronization of spatiotemporal chaos in the Ginzburg-Landau equation

We have studied noise-induced synchronization in a distributed autooscillatory system described by the Ginzburg-Landau equations, which occur in a regime of chaotic spatiotemporal oscillations. A new regime of synchronous behavior, called incomplete noise-induced synchronization (INIS), is revealed, which can arise only in spatially distributed systems. The mechanism leading to the development of INIS in a distributed medium under the action of a distributed source of noise is analytically described. Good coincidence of analytical and numerical results is demonstrated.     

Specific external forcing of spatiotemporal dynamics in reaction-diffusion systems

Self-organization behavior and in particular pattern forming spatiotemporal dynamics play an important role in far from equilibrium chemical and biochemical systems. Specific external forcing and control of self-organizing processes might be of great benefit in various applications ranging from technical systems to modern biomedical research. We demonstrate that in a cellular chemotaxis system modeled by one-dimensional reaction-diffusion equations particular forms of spatiotemporal dynamics can be induced and stabilized by controlling spatially distributed influx patterns of a chemical species as a function of time. In our model study we show that a propagating wave with certain shape and velocity and static symmetrical and asymmetrical patterns can be forced and manipulated by numerically computing open-loop optimal influx controls.     

Chaos out of internal noise in the collective dynamics of diffusively coupled cells

We study the transition from stochasticity to determinism in calcium oscillations via diffusive coupling of individual cells that are modeled by stochastic simulations of the governing reaction-diffusion equations. As expected, the stochastic solutions gradually converge to their deterministic limit as the number of coupled cells increases. Remarkably however, although the strict deterministic limit dictates a fully periodic behavior, the stochastic solution remains chaotic even for large numbers of coupled cells if the system is set close to an inherently chaotic regime. On the other hand, the lack of proximity to a chaotic regime leads to an expected convergence to the fully periodic behavior, thus suggesting that near-chaotic states are presently a crucial predisposition for the observation of noise-induced chaos. Our results suggest that chaos may exist in real biological systems due to intrinsic fluctuations and uncertainties characterizing their functioning on small scales.     

不确定自催化反应扩散时空混沌系统的延时同步

设计了一种延迟同步控制器实现了时空混沌系统之间的同步控制.基于Lyapunov稳定性定理,确定了延迟同步控制器的结构以及系统状态变量之间的误差方程.以自催化反应扩散时空混沌系统为例,仿真模拟验证了该控制器的有效性.进一步设计了参量辨识器,对不确定自催化反应扩散时空混沌系统中的参量进行了有效辨识.通过研究有界噪声影响下系统的同步效果,表明该同步方法具有较好的抗干扰能力. 关键词： 时空混沌 延时同步 参量辨识 有界噪声     

Analysis of chaotic saddles in high-dimensional dynamical systems: the Kuramoto-Sivashinsky equation

This paper presents a methodology to study the role played by nonattracting chaotic sets called chaotic saddles in chaotic transitions of high-dimensional dynamical systems. Our methodology is applied to the Kuramoto-Sivashinsky equation, a reaction-diffusion partial differential equation. The paper describes a novel technique that uses the stable manifold of a chaotic saddle to characterize the homoclinic tangency responsible for an interior crisis, a chaotic transition that results in the enlargement of a chaotic attractor. The numerical techniques explained here are important to improve the understanding of the connection between low-dimensional chaotic systems and spatiotemporal systems which exhibit temporal chaos and spatial coherence.     

Dynamical evolution of spatiotemporal patterns in mammalian middle cortex

The spatiotemporal structure of brain oscillations are important in understanding neural function. We analyze oscillatory episodes from isotropic preparations from the middle layers of a mammalian cortex which display irregular and chaotic spatiotemporal wave activity, within which spontaneously emerge spiral and plane waves. The dimensionality of these dynamics shows a consistent decrease during the middle of these episodes, regardless of the presence of simple spiral or plane waves. It is important to define the relevant biological order parameters which govern these dynamical bifurcations.     

Human electroencephalogram induces transient coherence in excitable spatiotemporal chaos

A time series from a human electroencephalogram (EEG) is used as a local perturbation to a reaction-diffusion model with spatiotemporal chaos. For certain finite ranges of amplitude and frequency it is observed that the strongly irregular perturbations can induce transient coherence in the chaotic system. This could be interpreted as "on-line" detection of an inherently correlated pattern embedded in the EEG.     

Emergence of unstable modes in an expanding domain for energy-conserving wave equations

Motivated by recent work on instabilities in expanding domains in reaction-diffusion settings, we propose an analog of such mechanisms in energy-conserving wave equations. In particular, we consider a nonlinear Schrödinger equation in a finite domain and show how the expansion or contraction of the domain, under appropriate conditions, can destabilize its originally stable solutions through the modulational instability mechanism. Using both real and Fourier space diagnostics, we monitor and control the crossing of the instability threshold and, hence, the activation of the instability. We also consider how the manifestation of this mechanism is modified in a spatially inhomogeneous setting, namely in the presence of an external parabolic potential, which is relevant to trapped Bose-Einstein condensates.     

The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems

Many oscillatory biological systems show periodic travelling waves. These are often modelled using coupled reaction-diffusion equations. However, the effects of different movement rates (diffusion coefficients) of the interacting components on the predictions of these equations are largely unknown. Here we investigate the ways in which varying the diffusion coefficients in such equations alters the wave speed, time period, wavelength, amplitude and stability of periodic wave solutions. We focus on two sets of kinetics that are commonly used in ecological applications: lambda-omega equations, which are the normal form of an oscillatory coupled reaction-diffusion system close to a supercritical Hopf bifurcation, and a standard predator-prey model. Our results show that changing the ratio of the diffusion coefficients can significantly alter the shape of the one-parameter family of periodic travelling wave solutions. The position of the boundary between stable and unstable waves also depends on the ratio of the diffusion coefficients: in all cases, stability changes through an Eckhaus (‘sideband’) instability. These effects are always symmetrical in the two diffusion coefficients for the lambda-omega equations, but are asymmetric in the predator-prey equations, especially when the limit cycle of the kinetics is of large amplitude. In particular, there are two separate regions of stable waves in the travelling wave family for some parameter values in the predator-prey scenario. Our results also show the existence of a one-parameter family of travelling waves, but not necessarily a Hopf bifurcation, for all values of the diffusion coefficients. Simulations of the full partial differential equations reveals that varying the ratio of the diffusion coefficients can significantly change the properties of periodic travelling waves that arise from particular wave generation mechanisms, and our analysis of the travelling wave families assists in the understanding of these effects.     

Spontaneous resistance oscillations inp-germanium at low temperatures and their spatial correlation

Spontaneous oscillations and chaotic behavior of the electric resistance can be observed inp-doped germanium during avalanche breakdown at liquid-He temperatures. By a series of different ohmic contacts attached to the specimen surface, the spatial coupling between the resistance oscillations in different sections of the same single-crystalline Ge sample can be studied. Using a geometry where coupling via charge carriers can be ignored, we have investigated the case where the lattice heat conductivity of the Ge crystal represents the dominant coupling mechanism. If two coupled sample sections were operated in the post-breakdown regime, a distinct phase shift developed between the current oscillations in both sections eventually reaching exact phase reversal. These results show a striking similarity to the dynamics of a coupled Rashevsky-Turing reaction-diffusion system.     

Fluctuation-regularized front propagation dynamics in reaction-diffusion systems

We introduce and study a new class of fronts in finite particle-number reaction-diffusion systems, corresponding to propagating up a reaction-rate gradient. We show that these systems have no traditional mean-field limit, as the nature of the long-time front solution in the stochastic process differs essentially from that obtained by solving the mean-field deterministic reaction-diffusion equations. Instead, one can incorporate some aspects of the fluctuations via introducing a density cutoff. Using this method, we derive analytic expressions for the front velocity dependence on bulk particle density and show self-consistently why this cutoff approach can get the correct leading-order physics.     

Synchronization of spatiotemporal chaos in a class of complex dynamical networks

This paper studies the synchronization of complex dynamical networks constructed by spatiotemporal chaotic systems with unknown parameters. The state variables in the systems with uncertain parameters are used to construct the parameter recognizers, and the unknown parameters are identified. Uncertain spatiotemporal chaotic systems are taken as the nodes of complex dynamical networks, connection among the nodes of all the spatiotemporal chaotic systems is of nonlinear coupling. The structure of the coupling functions between the connected nodes and the control gain are obtained based on Lyapunov stability theory. It is seen that stable chaos synchronization exists in the whole network when the control gain is in a certain range. The Gray--Scott models which have spatiotemporal chaotic behaviour are taken as examples for simulation and the results show that the method is very effective.     

Self-sustained spatiotemporal oscillations induced by membrane-bulk coupling

We propose a novel mechanism leading to spatiotemporal oscillations in extended systems that does not rely on local bulk instabilities. Instead, oscillations arise from the interaction of two subsystems of different spatial dimensionality. Specifically, we show that coupling a passive diffusive bulk of dimension d with an excitable membrane of dimension d-1 produces a self-sustained oscillatory behavior. An analytical explanation of the phenomenon is provided for d=1. Moreover, in-phase and antiphase synchronization of oscillations are found numerically in one and two dimensions. This novel dynamic instability could be used by biological systems such as cells, where the dynamics on the cellular membrane is necessarily different from that of the cytoplasmic bulk.     

周期切换下Rayleigh振子的振荡行为及机理

本文研究了自治与非自治电路系统在周期切换连接下的动力学行为及机理.基于自治子系统平衡点和极限环的相应稳定性分析和切换系统李雅普诺夫指数的理论推导及数值计算.讨论了两子系统在不同参数下的稳态解在周期切换连接下的复合系统的各种周期振荡行为,进而给出了切换系统随参数变化下的最大李雅普诺夫指数图及相应的分岔图,得到了切换系统在不同参数下呈现出周期振荡,概周期振荡和混沌振荡相互交替出现的复杂动力学行为并分析了其振荡机理.给出了切换系统通过倍周期分岔,鞍结分岔以及环面分岔到达混沌的不同动力学演化过程.     

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.