Unstable attractors induce perpetual synchronization and desynchronization

Common experience suggests that attracting invariant sets in nonlinear dynamical systems are generally stable. Contrary to this intuition, we present a dynamical system, a network of pulse-coupled oscillators, in which unstable attractors arise naturally. From random initial conditions, groups of synchronized oscillators (clusters) are formed that send pulses alternately, resulting in a periodic dynamics of the network. Under the influence of arbitrarily weak noise, this synchronization is followed by a desynchronization of clusters, a phenomenon induced by attractors that are unstable. Perpetual synchronization and desynchronization lead to a switching among attractors. This is explained by the geometrical fact, that these unstable attractors are surrounded by basins of attraction of other attractors, whereas the full measure of their own basin is located remote from the attractor. Unstable attractors do not only exist in these systems, but moreover dominate the dynamics for large networks and a wide range of parameters.     

Strange attractors are classified by bounding Tori

There is at present a doubly discrete classification for strange attractors of low dimension, d(L)<3. A branched manifold describes the stretching and squeezing processes that generate the strange attractor, and a basis set of orbits describes the complete set of unstable periodic orbits in the attractor. To this we add a third discrete classification level. Strange attractors are organized by the boundary of an open set surrounding their branched manifold. The boundary is a torus with g holes that is dressed by a surface flow with 2(g-1) singular points. All known strange attractors in R3 are classified by genus, g, and flow type.     

Stochastic switching in systems with rare and hidden attractors

Complex biochemical networks are commonly characterised by the coexistence of multiple stable attractors. This endows living systems with plasticity in responses under changing external conditions, thereby enhancing their probability for survival. However, the type of such attractors as well as their positioning can hinder the likelihood to randomly visit these areas in phase space, thereby effectively decreasing the level of multistability in the system. Using a model based on the Hodgkin–Huxley formalism with bistability between a silent state, which is a rare attractor, and oscillatory bursting attractor, we demonstrate that the noise-induced switching between these two stable attractors depends on the structure of the phase space and the disposition of the coexisting attractors to each other.     

Coexistence of attractors in autonomous Van der Pol–Duffing jerk oscillator: Analysis,chaos control and synchronisation in its fractional-order form

Dynamics at infinity and a Hopf bifurcation for a Sprott E system with a very small perturbation constant are studied in this paper. By using Poincaré compactification of polynomial vector fields in \(R^3\), the dynamics near infinity of the singularities is obtained. Furthermore, in accordance with the centre manifold theorem, the subcritical Hopf bifurcation is analysed and obtained. Numerical simulations demonstrate the correctness of the dynamical and bifurcation analyses. Moreover, by choosing appropriate parameters, this perturbed system can exhibit chaotic, quasiperiodic and periodic dynamics, as well as some coexisting attractors, such as a chaotic attractor coexisting with a periodic attractor for \(a>0\), and a chaotic attractor coexisting with a quasiperiodic attractor for \(a=0\). Coexisting attractors are not associated with an unstable equilibrium and thus often go undiscovered because they may occur in a small region of parameter space, with a small basin of attraction in the space of initial conditions.     

Symmetry-increasing bifurcation of chaotic attractors

Bifurcation in symmetric is typically associated with spontaneous symmetry breaking. That is, bifurcation is associated with new solution having less symmetry.In this paper we show that symmetry-increasing bifurcation in the discrete dynamics of symmetric mappings is possible (and is perhaps generic). The reason for these bifurcations may be understood as follows. The existence of one attractor in a system with symmetry gives rise to a family of conjugate attractors all related by symmetry. Typically, in computer experiments, what we see is a sequence of symmetry-breaking bifurcations leading to the existence of conjugate chaotic attractors. As the bifurcation parameter is varied these attractors grow in size and merge leading to a single attractor having greater symmetry.We prove a theorem suggesting why this new attractor should have greater symmetry and present a number of striking examples of the symmetric patterns that can be formed by iterating the simplest mappings on the plane with the symmetry of the regular m-gon. In the last section we discuss period-doubling in the presence of symmetry.     

Attractor selection in chaotic dynamics

For different settings of a control parameter, a chaotic system can go from a region with two separate stable attractors (generalized bistability) to a crisis where a chaotic attractor expands, colliding with an unstable orbit. In the bistable regime jumps between independent attractors are mediated by external perturbations; above the crisis, the dynamics includes visits to regions formerly belonging to the unstable orbits and this appears as random bursts of amplitude jumps. We introduce a control method which suppresses the jumps in both cases by filtering the specific frequency content of one of the two dynamical objects. The method is tested both in a model and in a real experiment with a CO2 laser.     

Analysis of grazing bifurcations of quasiperiodic system attractors

This paper presents the first application of the discontinuity-mapping approach to the study of near-grazing bifurcations of originally quasiperiodic, co-dimension-two system attractors. The paper establishes an exact formulation for the discontinuity-mapping methodology under the assumption that a Poincaré section can be found that is everywhere transversal to the grazing attractor. In particular, it is shown that, while a reduced formulation may be employed successfully in the case of co-dimension-one attractors, it fails to capture dynamics in directions transversal to the original quasiperiodic attractor. This shortcoming necessitates the full machinery presented here. The generality of the proposed approach is illustrated through numerical analysis of two nonlinear dynamical systems of dimension three and four.     

On determining the dimension of chaotic flows

We describe a method for determining the approximate fractal dimension of an attractor. Our technique fits linear subspaces of appropriate dimension to sets of points on the attractor. The deviation between points on the attractor and this local linear subspace is analyzed through standard multilinear regression techniques. We show how the local dimension of attractors underlying physical phenomena can be measured even when only a single time-varying quantity is available for analysis. These methods are applied to several dissipative dynamical systems.     

Localization of hidden Chua?s attractors

The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria. In the present Letter for localization of hidden attractors of Chua?s circuit it is suggested to use a special analytical-numerical algorithm.     

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.     

Homoclinic tangency and chaotic attractor disappearance in a dripping faucet experiment

A sequence of attractors, reconstructed from interdrops time series data of a leaky faucet experiment, is analyzed as a function of the mean dripping rate. We established the presence of a saddle point and its manifolds in the attractors and we explained the dynamical changes in the system using the evolution of the manifolds of the saddle point, as suggested by the orbits traced in first return maps. The sequence starts at a fixed point and evolves to an invariant torus of increasing diameter (a Hopf bifurcation) that pushes the unstable manifold towards the stable one. The torus breaks up and the system shows a chaotic attractor bounded by the unstable manifold of the saddle. With the attractor expansion the unstable manifold becomes tangential to the stable one, giving rise to the sudden disappearance of the chaotic attractor, which is an experimental observation of a so called chaotic blue sky catastrophe.     

Periodic occurrence of chaotic behavior of homoclinic tangles

In this article, we illustrate, through numerical simulations, some important aspects of the dynamics of the periodically perturbed homoclinic solutions for a dissipative saddle. More explicitly, we demonstrate that, when homoclinic tangles are created, three different dynamical phenomena, namely, horseshoes, periodic sinks, and attractors with Sinai-Ruelle-Bowen measures, manifest themselves periodically with respect to the magnitude of the forcing function. In addition, when the stable and the unstable manifolds are pulled apart so as not to intersect, first, rank 1 attractors, then quasi-periodic attractors are added to the dynamical scene.     

Multistability and the control of complexity

We show how multistability arises in nonlinear dynamics and discuss the properties of such a behavior. In particular, we show that most attractors are periodic in multistable systems, meaning that chaotic attractors are rare in such systems. After arguing that multistable systems have the general traits expected from a complex system, we pass to control them. Our controlling complexity ideas allow for both the stabilization and destabilization of any one of the coexisting states. The control of complexity differs from the standard control of chaos approach, an approach that makes use of the unstable periodic orbits embedded in an extended chaotic attractor. (c) 1997 American Institute of Physics.     

A special type of attractor and transitions in a delayed system

We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through T² torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark-Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues.     

一个四翼混沌吸引子

在新的四维混沌系统中数值观察到四翼混沌吸引子，然而，通过进一步分析发现，该四翼吸引子并非真实的，实际上它是上、下两个共存的双翼混沌吸引子，他们各自有独立的混沌吸引域，由于其位置靠得太近和数值误差产生的一种假象.通过引入一个线性状态反馈控制项，系统的一些相似性被破坏，受控系统能产生穿越上下吸引域界限的对角双翼混沌吸引子，进一步，随着动力学模态的演化，上下混沌吸引子与对角混沌吸引子融合成一个真正的四翼混沌吸引子.最后，通过比较该四翼混沌吸引子的系统、Lorenz系统、Chua氏电路等混沌信号的频谱发现，四翼混沌吸引子的系统信号具有极宽的频谱带宽，该特性在通讯加密等工程应用中具有重要价值. 关键词： 四维混沌系统 双翼吸引子 四翼吸引子 频谱分析     

Dynamics of unidirectionally coupled bistable Hénon maps

We study dynamics of two bistable Hénon maps coupled in a master-slave configuration. In the case of coexistence of two periodic orbits, the slave map evolves into the master map state after transients, which duration determines synchronization time and obeys a −1/2 power law with respect to the coupling strength. This scaling law is almost independent of the map parameter. In the case of coexistence of chaotic and periodic attractors, very complex dynamics is observed, including the emergence of new attractors as the coupling strength is increased. The attractor of the master map always exists in the slave map independently of the coupling strength. For a high coupling strength, complete synchronization can be achieved only for the attractor similar to that of the master map.     

环域反馈下复Ginzburg-Landau系统的螺旋波动力学行为

针对自激振荡系统的复金兹伯格-朗道(Complex Ginzbury-Landau, 简称CGL)方程, 研究圆形环域与方形环域两种反馈控制下的螺旋波动力学。结果表明: 圆形环域反馈控制下, 螺旋波波头通常经过一段过渡漂移后进入圆形吸引子, 圆形吸引子的半径以及反馈刚启动时波头的漂移方向随环域参数呈周期性变化, 过渡漂移阶段波头轨道的平缓程度与复反馈信号模的时间函数中钟形部分的陡度有关, 且反馈增益的正负与大小也会影响受控螺旋波的动力学行为。方形环域反馈控制下的螺旋波波头的吸引子更为丰富, 主要包括方形吸引子、小的极限环吸引子、菱形吸引子以及点吸引子, 点吸引子通常位于方形环域的两条对角线上, 且波头运动随环域控制参数呈现规律性变换。     

Chaotic itinerancy, temporal segmentation and spatio-temporal combinatorial codes

We study a deterministic dynamics with two time scales in a continuous state attractor network. To the usual (fast) relaxation dynamics towards point attractors (“patterns”) we add a slow coupling dynamics that makes the visited patterns lose stability, leading to an itinerant behavior in the form of punctuated equilibria. One finds that the transition frequency matrix for transitions between patterns shows non-trivial statistical properties in the chaotic itinerant regime. We show that mixture input patterns can be temporally segmented by the itinerant dynamics. The viability of a combinatorial spatio-temporal neural code is also demonstrated.     

Structure and evolution of strange attractors in non-elastic triangular billiards

We study non-elastic billiard dynamics in an equilateral triangular table. In such dynamics, collisions with the walls of the table are not elastic, as in standard billiards; rather, the outgoing angle of the trajectory with the normal vector to the boundary at the point of collision is a uniform factor λ < 1 smaller than the incoming angle. This leads to contraction in phase space for the discrete-time dynamics between consecutive collisions, and hence to attractors of zero Lebesgue measure, which are almost always fractal strange attractors with chaotic dynamics, due to the presence of an expansion mechanism. We study the structure of these strange attractors and their evolution as the contraction parameter λ is varied. For λ∈(0,1/3), we prove rigorously that the attractor has the structure of a Cantor set times an interval, whereas for larger values of λ gaps arise in the Cantor structure. For λ close to 1, the attractor splits into three transitive components, whose basins of attraction have fractal boundaries.     

Vulnerability to paroxysmal oscillations in delayed neural networks: a basis for nocturnal frontal lobe epilepsy?

Resonance can occur in bistable dynamical systems due to the interplay between noise and delay (τ) in the absence of a periodic input. We investigate resonance in a two-neuron model with mutual time-delayed inhibitory feedback. For appropriate choices of the parameters and inputs three fixed-point attractors co-exist: two are stable and one is unstable. In the absence of noise, delay-induced transient oscillations (referred to herein as DITOs) arise whenever the initial function is tuned sufficiently close to the unstable fixed-point. In the presence of noisy perturbations, DITOs arise spontaneously. Since the correlation time for the stationary dynamics is ～τ, we approximated a higher order Markov process by a three-state Markov chain model by rescaling time as t?→?2sτ, identifying the states based on whether the sub-intervals were completely confined to one basin of attraction (the two stable attractors) or straddled the separatrix, and then determining the transition probability matrix empirically. The resultant Markov chain model captured the switching behaviors including the statistical properties of the DITOs. Our observations indicate that time-delayed and noisy bistable dynamical systems are prone to generate DITOs as switches between the two attractors occur. Bistable systems arise transiently in situations when one attractor is gradually replaced by another. This may explain, for example, why seizures in certain epileptic syndromes tend to occur as sleep stages change.