From simple to complex oscillatory behavior in metabolic and genetic control networks

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.     

Synchronization in systems with bimodal dynamics

Considering a prototypic model of a bimodal oscillator we investigate the synchronization of the internal time scales for a system with interacting fast and slow oscillatory modes. Particular emphasis is given to the transition between mode-locked and mode-unlocked chaos. It is shown that this transition involves a homoclinic bifurcation in which the synchronized chaotic attractor loses its band structure. For two coupled bimodal oscillators we illustrate the presence of separate synchronization regions for the fast and the slow modes. The dependence of these regions on the mismatch and coupling parameters is studied.     

Coherent recovery of the degree of polarization of light propagating in random anisotropy media

We report on the study of light polarization behavior in random anisotropy scattering media. It is shown that, in the case of low scattering events, the degree of polarization (DOP) demonstrates oscillatory behavior due to the coherent character of light scattering. A strong increase in the DOP is demonstrated by using electro-optic fine tuning of the refractive index modulation depth of the scattering media.     

Population dynamics on random networks: simulations and analytical models

We study the phase diagram of the standard pair approximation equations for two different models in population dynamics, the susceptible-infective-recovered-susceptible model of infection spread and a predator-prey interaction model, on a network of homogeneous degree k. These models have similar phase diagrams and represent two classes of systems for which noisy oscillations, still largely unexplained, are observed in nature. We show that for a certain range of the parameter k both models exhibit an oscillatory phase in a region of parameter space that corresponds to weak driving. This oscillatory phase, however, disappears when k is large. For k = 3, 4, we compare the phase diagram of the standard pair approximation equations of both models with the results of simulations on regular random graphs of the same degree. We show that for parameter values in the oscillatory phase, and even for large system sizes, the simulations either die out or exhibit damped oscillations, depending on the initial conditions. We discuss this failure of the standard pair approximation model to capture even the qualitative behavior of the simulations on large regular random graphs and the relevance of the oscillatory phase in the pair approximation diagrams to explain the cycling behavior found in real populations.     

Counting spanning trees in a small-world Farey graph

The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.     

Temporal decorrelation of collective oscillations in neural networks with local inhibition and long-range excitation

We consider two neuronal networks coupled by long-range excitatory interactions. Oscillations in the gamma frequency band are generated within each network by local inhibition. When long-range excitation is weak, these oscillations phase lock with a phase shift dependent on the strength of local inhibition. Increasing the strength of long-range excitation induces a transition to chaos via period doubling or quasiperiodic scenarios. In the chaotic regime, oscillatory activity undergoes fast temporal decorrelation. The generality of these dynamical properties is assessed in firing-rate models as well as in large networks of conductance-based neurons.     

基于符号序列描述的一类分段光滑系统中分岔现象与混沌分析

从拓扑序列出发，提出了描述DC/DC变换器一类分段光滑系统中的分岔现象和混沌行为的符号序列方法，根据最大子序列的性态判别分岔的类型，以及检测边界碰撞分岔的发生.例如,当发生倍周期分岔时，最大子序列保持不变；当发生边界碰撞分岔时，最大子序列发生变化；混沌态则没有最大子序列.研究表明，占空比是表征DC/DC变换器一类分段光滑系统动力学行为的一个最本质的量，“饱和非线性”是引起边界碰撞分岔产生的根本原因. 关键词： 符号序列 分岔 混沌 分段光滑系统     

非周期力在混沌控制中的双重功效

探讨了非周期力（有界噪声或混沌驱动力）在非线性动力系统混沌控制中的影响.以一类典型的含有五次非线性项的Duffing-van der Pol系统为范例,通过对系统的轨道、最大Lyapunov指数、功率谱幅值及Poincar截面的分析,发现适当幅值的有界噪声或混沌信号,一方面可以消除系统对初始条件的敏感依赖性,抑制系统的混沌行为,将系统的混沌吸引子转化为奇怪非混沌吸引子;另一方面也可以诱导系统的混沌行为,将系统的周期吸引子转化为混沌吸引子.从而揭示了非周期力在混沌控制中的双重功效：抑制混沌和诱导混沌. 关键词： 混沌控制 有界噪声 混沌驱动力     

一个基于Chen系统的新混沌系统的分析与同步

通过在Chen系统的第一个方程中加入一个可变系数的乘积项, 构造了一个新的三维自治混沌系统.新系统可通过调节其可变系数实现不同系数组合下系统的混沌产生或混沌抑制, 即调节该乘积项的可变系数, 可使不出现混沌的Chen系统产生混沌现象, 同时也可使产生混沌运动的Chen系统不再产生混沌现象.详细分析了新系统的特性, 研究了新系统的混沌同步问题, 并给出了相应的仿真结果.     

The probabilistic symmetry breaking of periodic regimes rapidly passing through a zone of chaos into the transparency window

The problem of transition of a noisy dynamical system to a periodic oscillatory regime through a zone of chaos is considered. Using the noisy logistics map as an example, domains of attraction of energetically equivalent regimes of period three are found for various transition rates and various noise levels. The fine structure of the domains of attraction under the condition of fast transitions is revealed. It is discovered that the settling time of the stable cycle of period three heavily depends on the initial conditions, i.e., on the structure of the domains of attraction. The critical transition rate that separates the region of the probabilistic symmetry of final states from the region of the dynamic behavior of trajectories is estimated.     

Chaos and its implications

Starting with the very definition of chaos, we demonstrate that the study of chaos is not an abstract one but can lead to some useful practical applications. With the advent of some powerful mathematical techniques and with the availability of fast computers, it is now possible to study the fascinating phenomena of chaos — the subject which is truly interdisciplinary. The essential role played by fractals, strange attractors, Poincare maps, etc., in the study of chaotic dynamics, is briefly discussed. Phenomena of self-organization, coherence in chaos and control of chaos in plasmas is highlighted.     

Transport of a passive scalar and Lagrangian chaos in a Hamiltonian hydrodynamic system

An experimental and numerical study is made of the chaotic behavior of Lagrangian trajectories and transport of a passive tracer in a quasi-two-dimensional four-vortex flow with a periodic time dependence of the Euler velocity field. Quantitative measurements are made of tracer transport between isolated vortices in physical space and in “action” variable space. The theory of adiabatic chaos is used to interpret the measurements. The simplest phenomenological models of liquid particle random walks are proposed to describe the anomalous transport in terms of the action.     

Strong and Weak Chaos in Weakly Nonintegrable Many-Body Hamiltonian Systems

We study properties of chaos in generic one-dimensional nonlinear Hamiltonian lattices comprised of weakly coupled nonlinear oscillators by numerical simulations of continuous-time systems and symplectic maps. For small coupling, the measure of chaos is found to be proportional to the coupling strength and lattice length, with the typical maximal Lyapunov exponent being proportional to the square root of coupling. This strong chaos appears as a result of triplet resonances between nearby modes. In addition to strong chaos we observe a weakly chaotic component having much smaller Lyapunov exponent, the measure of which drops approximately as a square of the coupling strength down to smallest couplings we were able to reach. We argue that this weak chaos is linked to the regime of fast Arnold diffusion discussed by Chirikov and Vecheslavov. In disordered lattices of large size we find a subdiffusive spreading of initially localized wave packets over larger and larger number of modes. The relations between the exponent of this spreading and the exponent in the dependence of the fast Arnold diffusion on coupling strength are analyzed. We also trace parallels between the slow spreading of chaos and deterministic rheology.     

基于单驱动变量的混沌广义投影同步及在保密通信中的应用

基于单向耦合提出一种只需传递一个驱动变量实现混沌系统广义投影同步方法.通过改变广义投影同步的比例因子,获得任意比例于原驱动混沌系统输出的混沌信号.由于只需传递一个信号,比起已有的方法具有更高的实用价值,理论推导和数值仿真进一步表明了该方法的有效性.最后,基于统一混沌系统的广义投影同步,给出了一种安全性更好的混沌保密通信方案. 关键词： 混沌系统 单驱动变量 广义投影同步 统一混沌系统     

共轭Chen混沌系统的分岔分析及基于该系统的超混沌生成研究

分析了一个三维自治混沌系统的Hopf分岔现象,该系统的混沌吸引子属于共轭Chen混沌系统.通过引入一个控制器,基于该混沌系统构建了一个四维自治超混沌系统.该超混沌系统含有一个单参数,在一定的参数范围内呈现超混沌现象.通过Lyapunov指数和分岔分析,随着参数的变化该系统轨道呈现周期轨道、准周期轨道、混沌和超混沌的演化过程. 关键词： 混沌 超混沌生成 Hopf分岔 分岔分析     

Curvature fields, topology, and the dynamics of spatiotemporal chaos

The curvature field is measured from tracer-particle trajectories in a two-dimensional fluid flow that exhibits spatiotemporal chaos and is used to extract the hyperbolic and elliptic points of the flow. These special points are pinned to the forcing when the driving is weak, but wander over the domain and interact in pairs at stronger driving, changing the local topology of the flow. Their behavior reveals a two-stage transition to spatiotemporal chaos: a gradual loss of spatial and temporal order followed by an abrupt onset of topological changes.     

Chaos suppression in gas-solid fluidization

Fluidization in granular materials occurs primarily as a result of a dynamic balance between gravitational forces and forces resulting from the flow of a fluid through a bed of discrete particles. For systems where the fluidizing medium and the particles have significantly different densities, density wave instabilities create local pockets of very high void fraction termed bubbles. The fluidization regime is termed the bubbling regime. Such a system is appropriately termed a self-excited nonlinear system. The present study examines chaos suppression resulting from an opposing oscillatory flow in gas-solid fluidization. Time series data representing local, instantaneous pressure were acquired at the surface of a horizontal cylinder submerged in a bubbling fluidized bed. The particles had a weight mean diameter of 345 &mgr;m and a narrow size distribution. The state of fluidization corresponded to the bubbling regime and total air flow rates employed in the present study ranged from 10% to 40% greater than that required for minimum fluidization. The behavior of time-varying local pressure in fluidized beds in the absence of a secondary flow is consistent with deterministic chaos. Kolmogorov entropy estimates from local, instantaneous pressure suggest that the degree of chaotic behavior can be substantially suppressed by the presence of an opposing, oscillatory secondary flow. Pressure signals clearly show a "phase-locking" phenomenon coincident with the imposed frequency. In the present study, the greatest degree of suppression occurred for operating conditions with low primary and secondary flow rates, and a secondary flow oscillation frequency of 15 Hz. (c) 1998 American Institute of Physics.     

Pattern competition as origin of intermittency-type chaos

The one-dimensional sine-Gordon equation, in the presence of driving and damping, is examined numerically. For increasing driver strength the system exhibits an infinite sequence of period doublings leading to chaos. The first bifurcation, however, occurs after a narrow regime characterized by intermittency-type chaos and quasi-periodic oscillations. The origin of the intermittency-type chaos in this system is a competition between two spatial patterns described by the presence of one or two breather-like modes, respectively.