Spike-burst synchronization in an ensemble of electrically coupled discrete model neurons

In this paper, we study the dynamics of a system of two model neurons interacting via the electrical synapse. Each neuron is described by a two-dimensional discontinuous map. A chaotic relaxational-type attractor, which corresponds to the spiking-bursting chaotic oscillations of neurons is shown to exist in a four-dimensional phase space. It is found that the dynamical mechanism of formation of chaotic bursts is based on a new phenomenon of generation of transient chaotic oscillations. It is demonstrated that transition from the chaotic-burst generation to the state of relative rest occurs with a certain time delay. A new characteristic which estimates the degree of synchronization of the spiking-bursting oscillations is introduced. The dependence of the synchronization degree on the strength of coupling of the ensemble elements is studied.     

Emergence of chaotic attractor and anti-synchronization for two coupled monostable neurons

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincare section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.     

Autonomous coupled oscillators with hyperbolic strange attractors

We propose several examples of smooth low-order autonomous dynamical systems which have apparently uniformly hyperbolic attractors. The general idea is based on the use of coupled self-sustained oscillators where, due to certain amplitude nonlinearities, successive epochs of damped and excited oscillations alternate. Because of additional, phase sensitive coupling terms in the equations, the transfer of excitation from one oscillator to another is accompanied by a phase transformation corresponding to some chaotic map (in particular, an expanding circle map or Anosov map of a torus). The first example we construct is a minimal model possessing an attractor of the Smale-Williams type. It is a four-dimensional system composed of two oscillators. The underlying amplitude equations are similar to those of the predator-pray model. The other three examples are systems of three coupled oscillators with a heteroclinic cycle. This scheme presents more variability for the phase manipulations: in the six-dimensional system not only the Smale-Williams attractor, but also an attractor with Arnold cat map dynamics near a two-dimensional toral surface, and a hyperchaotic attractor with two positive Lyapunov exponents, are realized.     

Boundary crisis and transient in a dissipative relativistic standard map

Some dynamical properties for a problem concerning the acceleration of particles in a wave packet are studied. The model is described in terms of a two-dimensional nonlinear map obtained from a Hamiltonian which describes the motion of a relativistic standard map. The phase space is mixed in the sense that there are regular and chaotic regions coexisting. When dissipation is introduced, the property of area preservation is broken and attractors emerge. We have shown that a tiny increase of the dissipation causes a change in the phase space. A chaotic attractor as well as its basin of attraction are destroyed thereby leading the system to experience a boundary crisis. We have characterized such a boundary crisis via a collision of the chaotic attractor with the stable manifold of a saddle fixed point. Once the chaotic attractor is destroyed, a chaotic transient described by a power law with exponent −1 is observed.     

一两维平面映射系统奇怪动力学行为

通过对一平面二维映射系统非线性动力学行为的分析,发现该系统状态随参数变化,经过稳定焦点、极限环(不变环)、倍周期分岔、收缩到低维流形上的混沌吸引子(具有一个正Lyapunov指数)、最后到在有界区域弥散开来的混沌吸引子(具有两个正Lyapunov指数)的过程.通过对该系统不动点的分析揭示了吸引子的吸引域边界结构,即不稳定第二类结点与不稳定偶数周期点在吸引域边界上的相间排列. 关键词：     

Observation of a continuous interior crisis in the Hindmarsh-Rose neuron model

Interior crises are understood as discontinuous changes of the size of a chaotic attractor that occur when an unstable periodic orbit collides with the chaotic attractor. We present here numerical evidence and theoretical reasoning which prove the existence of a chaos-chaos transition in which the change of the attractor size is sudden but continuous. This occurs in the Hindmarsh-Rose model of a neuron, at the transition point between the bursting and spiking dynamics, which are two different dynamic behaviors that this system is able to present. Moreover, besides the change in attractor size, other significant properties of the system undergoing the transitions do change in a relevant qualitative way. The mechanism for such transition is understood in terms of a simple one-dimensional map whose dynamics undergoes a crossover between two different universal behaviors.     

Phase locked periodic solutions and synchronous chaos in a model of two coupled molecular lasers

We study a rate-equation model for two coupled molecular lasers with a saturable absorber. A numerical bifurcation study shows the existence of isolas for in-phase periodic solutions as physical parameters change. In addition there are other non-isola families of in-phase, anti-phase and intermediate-phase periodic oscillations. In this model the unstable periodic orbits belonging to the in-phase isolas constitute a skeleton of the attractor, when chaotic synchronization sets in for a set of physically relevant control parameters.     

Numerical study of the oscillatory convergence to the attractor at the edge of chaos

This paper compares three different types of “onset of chaos” in the logistic and generalized logistic map: the Feigenbaum attractor at the end of the period doubling bifurcations; the tangent bifurcation at the border of the period three window; the transition to chaos in the generalized logistic with inflection 1/2 (x_n+1 = 1-μx_n^1/2), in which the main bifurcation cascade, as well as the bifurcations generated by the periodic windows in the chaotic region, collapse in a single point. The occupation number and the Tsallis entropy are studied. The different regimes of convergence to the attractor, starting from two kinds of far-from-equilibrium initial conditions, are distinguished by the presence or absence of log-log oscillations, by different power-law scalings and by a gap in the saturation levels. We show that the escort distribution implicit in the Tsallis entropy may tune the log-log oscillations or the crossover times.     

Conversion of a chaotic attractor into a strange nonchaotic attractor in an one dimensional map and BVP oscillator

In this paper we investigate numerically the possibility of conversion of a chaotic attractor into a nonchaotic but strange attractor in both a discrete system (an one dimensional map) and in a continuous dynamical system — Bonhoeffer—van der Pol oscillator. In these systems we show suppression of chaotic property, namely, the sensitive dependence on initial states, by adding appropriate i) chaotic signal and ii) Gaussian white noise. The controlled orbit is found to be strange but nonchaotic with largest Lyapunov exponent negative and noninteger correlation dimension. Return map and power spectrum are also used to characterize the strange nonchaotic attractor.     

Chaotic oscillations in a coupled system of bistable oscillators

The influence of the asymmetry of the nonlinear element characteristic on the chaotic oscillations of Chua’s bistable oscillator is studied. It is shown that such asymmetry causes asymmetry of a chaotic attractor that maps the switching of motions between two basins of attraction up to the concentration of oscillations in one basin. Oscillation control in a bistable chaotic self-oscillating system (two coupled Chua’s oscillators) is considered. It is demonstrated that oscillations excited in two basins of attraction may pass to one of them and that oscillations may build up in two basins when they are autonomously excited in different basins. It is also found that chaotic oscillations in a coupled system may be excited at parameter values for which the autonomous chaotic oscillations of partial oscillators are absent. The influence of external noiselike oscillations is investigated.     

Soliton as strange attractor: nonlinear synchronization and chaos

We show that dissipative solitons can have dynamics similar to that of a strange attractor in low-dimensional systems. Using a model of a passively mode-locked fiber laser as an example, we show that soliton pulsations with periods equal to several round-trips of the cavity can be chaotic, even though they are synchronized with the round-trip time. The chaotic part of this motion is quantified using a two-dimensional map and estimating the Lyapunov exponent. We found a specific route to chaotic motion that occurs through the creation, increase, and overlap of "islands" of chaos rather than through multiplication of frequencies.     

Oscillations and chaos in CO+O(2) combustion

The gas-phase reaction between carbon monoxide and oxygen (in the presence of small amounts of hydrogen) shows bistability and oscillatory behavior. Typically, the oscillatory ignition has a period-1 relaxation waveform. The limit cycle is born at a saddle-node loop and terminates via a supercritical Hopf bifurcation. For a mean residence time of 8 s there is a period-doubling to a period-2 solution followed by period-halving to quasisinusoidal period-1 oscillations. At longer residence times, more period-doublings forming a full cascade to chaos with subsequent periodic windows are observed. The chaotic attractor has an underlying single-humped next maximum map.     

用数字有限脉冲响应滤波器控制混沌

利用数字有限脉冲响应滤波器稳定微分动力系统和二维离散映象混沌吸引子中不稳定周期轨道的方法，实现了高周期轨道的稳定控制.分别研究了Lorenz系统和Henon映象，给出了初步的分析和数值模拟结果.这种方法的主要特点是不需要获取混沌系统中不稳定周期轨道的任何信息，控制参数的选择与被控混沌系统无关. 关键词：     

一个具有完全四翼形式的四维混沌

本文通过引进一个非线性状态反馈控制器, 提出了一个新的四维混沌系统, 该混沌吸引子能在任何方向上都表现出四翼形式. 由于存在一个大的正李雅普诺夫指数, 混沌系统具有一些非常有趣和复杂的动力学行为. 对系统的一些基本动力学特性进行了数值模拟和理论分析, 如平衡点、耗散性、Poincaré映射、频谱、时域谱和混沌行为等. 通过对Lyapunov指数谱和分岔图的分析, 进一步研究了混沌行为的系统参数敏感性. 最后, 设计了一个实现四翼混沌系统的振荡电路, EWB观察结果与数值模拟结果具有良好的一致性.     

基于参数切换算法的混沌系统吸引子近似及其电路设计

基于参数切换算法和离散混沌系统, 设计一种新的混沌系统参数切换算法, 给出了两算法的原理. 采用混沌吸引子相图观测法, 研究了不同算法下统一混沌系统和Rössler混沌系统参数切换结果, 最后引入方波发生器, 设计了Rössler混沌系统参数切换电路. 结果表明, 采用参数切换算法可以近似出指定参数下的系统, 其吸引子与该参数下吸引子一致; 基于离散系统的参数切换结果更为复杂, 当离散序列分布均匀时, 只可近似得到指定参数下的系统; 相比传统切换混沌电路, 参数切换电路不用修改原有系统电路结构, 设计更为简单, 输出结果受方波频率影响, 通过加入合适频率的方波发生器, 数值仿真与电路仿真结果一致.     

Anomalous biennial oscillations in a Fisher equation with a discretized verhulst term

The dynamics of a biological population governed by a modified Fisher equation is studied by means of Monte Carlo simulations. Reproduction of the population occurs at discrete times, while transport caused by diffusion and conduction takes place on shorter time scales. The discrete reproduction, modeled with a set of coupled logistic maps, exhibits phenomena which are not evident in the usual continuum version of the Fisher equation. Several mechanisms for biennial oscillations of the total population are investigated. One of these shows an ordered coupling between random diffusive motion and the chaotic attractor of the logistic map.     

A three-scroll chaotic attractor

This Letter introduces a new chaotic member to the three-dimensional smooth autonomous quadratic system family, which derived from the classical Lorenz system but exhibits a three-scroll chaotic attractor. Interestingly, the two other scrolls are symmetry related with respect to the z-axis as for the Lorenz attractor, but the third scroll of this three-scroll chaotic attractor is around the z-axis. Some basic dynamical properties, such as Lyapunov exponents, fractal dimension, Poincaré map and chaotic dynamical behaviors of the new chaotic system are investigated, either numerically or analytically. The obtained results clearly show this is a new chaotic system and deserves further detailed investigation.     

非线性系统混沌运动的神经网络控制

设计前馈反传神经网络控制非线性系统混沌运动的新方法.根据扰动参数模型输入输出数据,按照非线性学习算法训练网络产生系统稳定所需的小扰动控制信号,去镇定混沌运动,使嵌入在混沌吸引子中的不稳定周期轨道回到稳定不动点上.Hnon映射数值仿真结果表明,这种方法控制非线性混沌系统响应速度快、控制精度高 关键词： 混沌控制 神经网络 吸引子 非线性     

Towards a quasi-periodic mean field theory for globally coupled oscillators

We show how a quasi-periodic mean field theory may be used to understand the chaotic dynamics and geometry of globally coupled complex Ginzburg-Landau equations. The Poincaré map of the mean field equations appears to have saddlenode-homoclinic bifurcations leading to chaotic motion, and the attractor has the characteristic ρ shape identified by numerical experiments on the full equations.     

常微分方程系统中内部激变现象的研究

应用广义胞映射图论方法研究常微分方程系统的激变.揭示了边界激变是由于混沌吸引子与 在其吸引域边界上的周期鞍碰撞产生的,在这种情况下,当系统参数通过激变临界值时,混 沌吸引子连同它的吸引域突然消失,在相空间原混沌吸引子的位置上留下了一个混沌鞍.研 究混沌吸引子大小(尺寸和形状)的突然变化,即内部激变.发现这种混沌吸引子大小的突然 变化是由于混沌吸引子与在其吸引域内部的混沌鞍碰撞产生的,这个混沌鞍是相空间非吸引 的不变集,代表内部激变后混沌吸引子新增的一部分.同时研究了这个混沌鞍的形成与演化. 给出了对永久自循环胞集和瞬态自循环胞集进行局部细化的方法. 关键词： 广义胞映射 有向图 激变 混沌鞍