Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.     

Coupling sensitivity of chaos and the Lyapunov dimension: The case of coupled two-dimensional maps

We numerically investigate the response of spectra of the Lyapunov exponents in chaotic two-dimensional (2-d) maps to perturbations generated by coupling two such maps. The results reveal the coupling sensitivity of chaos, which was discovered previously in coupled 1-d maps, with a number of features some of which are inherent in higher-dimensional systems. In particular, the Lyapunov dimension of a strange attractor is also found to be strongly sensitive to coupling perturbations. Our results suggest a new quantity characterizing chaos, χ_coup, which measures the strength of the coupling sensitivity.     

Monostability,bistability, periodicity and chaos in gene regulatory network

This letter gives a general review on the monostability, bistability, periodicity and chaos in gene regulatory network. Some simple motifs that generate monostability, bistability, periodicity and chaos are analytically and numerically reported. Further research directions of the nonlinear dynamics of gene regulatory network are discussed.     

Effects of randomness on chaos and order of coupled logistic maps

Natural systems are essentially nonlinear being neither completely ordered nor completely random. These nonlinearities are responsible for a great variety of possibilities that includes chaos. On this basis, the effect of randomness on chaos and order of nonlinear dynamical systems is an important feature to be understood. This Letter considers randomness as fluctuations and uncertainties due to noise and investigates its influence in the nonlinear dynamical behavior of coupled logistic maps. The noise effect is included by adding random variations either to parameters or to state variables. Besides, the coupling uncertainty is investigated by assuming tinny values for the connection parameters, representing the idea that all Nature is, in some sense, weakly connected. Results from numerical simulations show situations where noise alters the system nonlinear dynamics.     

Space-time renormalization at the onset of spatio-temporal chaos in coupled maps

The transition regime to spatio-temporal chaos via the quasiperiodic route as well as the period-doubling route is examined for coupled-map lattices. Space-time renormalization-group analysis is carried out and the scaling exponents for the coherence length, the Lyapunov exponent, and the size of the phase fluctuations are determined. Universality classes for the different types of coupling at various routes to chaos are identified.     

Anomalous spatio-temporal chaos in a two-dimensional system of nonlocally coupled oscillators

A two-dimensional system of nonlocally coupled complex Ginzburg-Landau oscillators is investigated numerically for the first time. As previously shown for the one-dimensional case, this two-dimensional system exhibits anomalous spatio-temporal chaos characterized by power-law spatial correlations. In this chaotic regime, the amplitude difference between neighboring elements displays temporal noisy on-off intermittency. The system is also spatially intermittent in this regime, as revealed by multiscaling analysis: The amplitude field is multiaffine and the difference field is multifractal. Correspondingly, the probability distribution function of the measure defined for each field is strongly non-Gaussian, exhibiting scale-dependent deviations in the tail due to intermittency. (c) 1999 American Institute of Physics.     

全局耦合网络的参量辨识与时空混沌同步

研究了参量未知的离散型时空混沌系统构成全局耦合网络的参量辨识与同步问题.首先将Milosavljevic所设计的控制律加以推广.利用推广后的控制律进行了网络节点时空混沌系统中未知参量的有效辨识,并完成了该网络的完全同步.进一步以物理中具有时空混沌行为的一维对流方程的离散形式作为实例进行了仿真分析. 关键词： 同步 参量辨识 全局耦合网络 时空混沌     

最近邻耦合网络的时空混沌同步研究

本文进行了最近邻网络的时空混沌同步研究.以时空混沌系统作为网络的节点,基于Lyapunov稳定性定理,通过确定网络的最大Lyapunov指数,得到了实现网络完全同步的条件.采用Fisher-Kolmogorov时空混沌系统作为网络节点实例进行了仿真模拟,获得了理想的同步效果.进一步研究了有界噪声影响下网络的同步性能,结果显示它具有较强的抗干扰能力.     

Generating macroscopic chaos in a network of globally coupled phase oscillators

We consider an infinite network of globally coupled phase oscillators in which the natural frequencies of the oscillators are drawn from a symmetric bimodal distribution. We demonstrate that macroscopic chaos can occur in this system when the coupling strength varies periodically in time. We identify period-doubling cascades to chaos, attractor crises, and horseshoe dynamics for the macroscopic mean field. Based on recent work that clarified the bifurcation structure of the static bimodal Kuramoto system, we qualitatively describe the mechanism for the generation of such complicated behavior in the time varying case.     

A neural network model as a globally coupled map and applications based on chaos

First, a neural network model as the globally coupled map (GCM) is proposed. The model is obtained by modification of a Hopfield network model that has a negative self-feedback connection. Second, information processed by this model is interpreted in terms of the variety of the maps acting on the network elements, and a new, dynamic information processing model is described. The search for information using vague keywords, and solution of the traveling salesman problem (TSP) are introduced as applications.     

Sterilization of E. coli bacterium with an atmospheric pressure surface barrier discharge

The atmospheric pressure surface barrier discharge (APSBD) in air has been used in killing Escherichia coli (E.coli), There is almost no bacterial colony in the sample after treatment by discharge plasma for 2rain, A diagnostic technique based on mass spectrum has been applied to the discharge gas and the mechanism of killing is discussed. Ozone and monatomic oxide are considered to be the major antimicrobial active species. There is almost no harmful by-product. The experiment proves that APSBD plasma is a very simple, effective and innocuous tool for sterilization.     

Statistical properties of chaos in Chebyshev maps

Statistical properties of fully developed chaotic maps in the form of Chebyshev polynomials are calculated exactly. We derive analytic expressions for characteristic functions, moments, and moment functions and mention a number of other properties. We also determine higher-order moment functions, which are important for a characterization of the non-gaussian processes exhibited by many maps.     

Discrete chaos in fractional sine and standard maps

Fractional standard and sine maps are proposed by using the discrete fractional calculus. The chaos behaviors are then numerically discussed when the difference order is a fractional one. The bifurcation diagrams and the phase portraits are presented, respectively.     

Properties of fully developed chaos in one-dimensional maps

We consider single-humped symmetric one-dimensional maps generating fully developed chaotic iterations specified by the property that on the attractor the mapping is everywhere two to one. To calculate the probability distribution function, and in turn the Lyapunov exponent and the correlation function, a perturbation expansion is developed for the invariant measure. Besides deriving some general results, we treat several examples in detail and compare our theoretical results with recent numerical ones. Furthermore, a necessary condition is deduced for the probability distribution function to be symmetric and an effective functional iteration method for the measure is introduced for numerical purposes.     

Stability of functions in Boolean models of gene regulatory networks

Boolean networks are used to model large nonlinear systems such as gene regulatory networks. We will present results that can be used to understand how the choice of functions affects the network dynamics. The so called bias-map and its fixed points depict much of the function's dynamical role in the network. We define the concept of stabilizing functions and show that many Post and canalizing functions are also stabilizing functions. Boolean networks constructed using the same type of stabilizing functions are always stable regardless of the average in-degree of network functions. We derive the number of all stabilizing functions and find it to be much larger than the number of Post and canalizing functions. We also discuss the implementation of functions and apply the presented results to biological data that give an approximation of the distribution of regulatory functions in eucaryotic cells. We find that the obtained theoretical results on the number of active genes are biologically plausible. Finally, based on the presented results, we discuss why canalizing and Post regulatory functions seem to be common in cells.     

Jerk系统耦合的分岔和混沌行为

通过分析耦合的Jerk系统的平衡点及其稳定性,给出了参数空间中不同的分岔集,进而将参数空间划分为对应于各种动力学行为的不同区域.探讨了耦合系统随不同参数变化的动力学演化过程,重点分析了系统耦合强度变化对其动力学行为的影响.揭示了多种运动模式共存及倍周期分岔等各种非线性现象的产生机理.