Partially unstable attractors in networks of forced integrate-and-fire oscillators

The asymptotic attractors of a nonlinear dynamical system play a key role in the long-term physically observable behaviors of the system. The study of attractors and the search for distinct types of attractor have been a central task in nonlinear dynamics. In smooth dynamical systems, an attractor is often enclosed completely in its basin of attraction with a finite distance from the basin boundary. Recent works have uncovered that, in neuronal networks, unstable attractors with a remote basin can arise, where almost every point on the attractor is locally transversely repelling. Herewith we report our discovery of a class of attractors: partially unstable attractors, in pulse-coupled integrate-and-fire networks subject to a periodic forcing. The defining feature of such an attractor is that it can simultaneously possess locally stable and unstable sets, both of positive measure. Exploiting the structure of the key dynamical events in the network, we develop a symbolic analysis that can fully explain the emergence of the partially unstable attractors. To our knowledge, such exotic attractors have not been reported previously, and we expect them to arise commonly in biological networks whose dynamics are governed by pulse (or spike) generation.     

On properties of hyperchaos: Case study

Some properties of hyperchaos are exploited by studying both uncoupled and coupled CML. In addition to usual properties of chaotic strange attractors, there are other interesting properties, such as: the number of unstable periodic points embedded in the strange attractor increases dramatically increasing and a large number of low-dimensional chaotic invariant sets are contained in the strange attractor. These properties may be useful for regarding the edge of chaos as the origin of complexity of dynamical systems. The project supported by the National Natural Science Foundation of China     

COMPLEX RELAXATION OSCILLATION TRIGGERED BY BOUNDARY CRISIS IN THE DISCRETE DUFFING MAP

多时间尺度问题具有广泛的工程与科学研究背景，慢变参数则是多时间尺度问题的典型标志之一.然而现有文献所报道的慢变参数问题，其展现出的振荡形式及内部分岔结构，大多较为单一，此外少有文献涉及到混沌激变的现象.本文以含慢变周期激励的达芬映射为例，探讨了一类具有复杂分岔结构的张弛振荡.快子系统的分岔表现为S形不动点曲线，其上、下稳定支可经由倍周期分岔通向混沌.而在一定的参数条件下，存在着导致混沌吸引子突然消失的一对临界参数值.当分岔参数达到此临界值时，混沌吸引子可能与不稳定不动点相接触，也可能与之相距一定距离.对快子系统吸引域分布的模拟，表明存在着导致边界激变（boundary crisis）的临界值，在这些值附近，经由延迟倍周期分岔演化而来的混沌吸引子可与2ⁿ（n=0，1，2，…）周期轨道乃至混沌吸引子共存.当慢变量周期地穿过临界点后，双稳态的消失导致原本处于混沌轨道的轨线对称地向此前共存的吸引子转迁，从而使系统出现了不同吸引子之间的滞后行为，由此产生了由边界激变所诱发的多种对称式张弛振荡.本文的结果丰富了对离散系统的多时间尺度动力学机理的认识.     

Synthesizing attractors of Hindmarsh–Rose neuronal systems

In this paper a periodic parameter-switching scheme is applied to the Hindmarsh–Rose neuronal system to synthesize certain attractors. Results show numerically, via computer graphic simulations, that the obtained synthesized attractor belongs to the class of all admissible attractors for the Hindmarsh–Rose neuronal system and matches the averaged attractor obtained with the control parameter replaced with the averaged switched parameter values. This feature allows us to imagine that living beings are able to maintain vital behavior while the control parameter switches so that their dynamical behavior is suitable for the given environment.     

A novel strange attractor with a stretched loop

This short paper introduces a new 3D strange attractor topologically different from any other known chaotic attractors. The intentionally constructed model of three autonomous first-order differential equations derives from the coupling-induced complexity of the well-established 2D Lotka?CVolterra oscillator. Its chaotification process via an anti-equilibrium feedback allows the exploration of a new domain of dynamical behavior including chaotic patterns. To focus a rapid presentation, a fixed set of parameters is selected linked to the widest range of dynamics. Indeed, the new system leads to a chaotic attractor exhibiting a double scroll bridged by a loop. It mutates to a single scroll with a very stretched loop by the variation of one parameter. Indexes of stability of the equilibrium points corresponding to the two typical strange attractors are also investigated. To encompass the global behavior of the new low-dimensional dissipative dynamical model, diagrams of bifurcation displaying chaotic bubbles and windows of periodic oscillations are computed. Besides, the dominant exponent of the Lyapunov spectrum is positive reporting the chaotic nature of the system. Eventually, the novel chaotic model is suitable for digital signal encryption in the field of communication with a rich set of keys.     

Kawakami映射的超混沌行为研究

通过对一类平面二维映射系统非线性动力学行为的分析,发现该系统存在一个奇怪吸引子,该吸引子具有两个正Lyapunov指数和分数维。通过该系统不动点的分析揭示了该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。     

A new procedure for exploring chaotic attractors in nonlinear dynamical systems under random excitations

Due to uncertain push-pull action across boundaries between different attractive domains by random excitations,attractors of a dynamical system will drift in the phase space,which readily leads to colliding and mixing with each other,so it is very difficult to identify irregular signals evolving from arbitrary initial states.Here,periodic attractors from the simple cell mapping method are further iterated by a specific Poincare’ map in order to observe more elaborate structures and drifts as well as possible dynamical bifurcations.The panorama of a chaotic attractor can also be displayed to a great extent by this newly developed procedure.From the positions and the variations of attractors in the phase space,the action mechanism of bounded noise excitation is studied in detail.Several numerical examples are employed to illustrate the present procedure.It is seen that the dynamical identification and the bifurcation analysis can be effectively performed by this procedure.     

Development of Dominating Waves From Small Disturbances in Falling Viscous-Liquid Films

For wavy liquid films, the principle of selection of the periodic solutions realized experimentally as regular waves is justified. By means of numerical methods, the bifurcations of the families of steady periodic waves and the attractors of the corresponding nonstationary problem are systematically studied. A comparison of the bifurcations and the attractors shows that, when several periodic solutions exist for a given wave number, the solution with the maximum wave amplitude and the maximum phase velocity develops from small initial disturbances (the dominating wave regime). With wave number variation, near the bifurcation points the attractor passes discontinuously from one family to another. This passage is accompanied by the appearance of two-periodic solutions in small neighborhoods of these points. The relations between the calculated parameters of the dominating waves are in a good agreement with all the available experimental data.     

参数激励耦合系统的复杂动力学行为分析

分析了耦合van der Pol振子参数共振条件下的复杂动力学行为．基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解．随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3—D环面解，它们都会随参数的变化进一步导致混吨．发现在系统的混沌区域中，其混吨吸引子随参数的变化会突然发生变化，分解为两个对称的混吨吸引子．值得注意的是，系统首先是由于2—D环面解破裂产生混吨，该混吨吸引子破裂后演变为新的混吨吸引子，却由倒倍周期分岔走向3—D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混吨吸引子在一定参数条件下会相互转换．     

Continuity of Dynamical Structures for Nonautonomous Evolution Equations Under Singular Perturbations

In this paper we study the continuity of invariant sets for nonautonomous infinite-dimensional dynamical systems under singular perturbations. We extend the existing results on lower-semicontinuity of attractors of autonomous and nonautonomous dynamical systems. This is accomplished through a detailed analysis of the structure of the invariant sets and its behavior under perturbation. We prove that a bounded hyperbolic global solutions persists under singular perturbations and that their nonlinear unstable manifold behave continuously. To accomplish this, we need to establish results on roughness of exponential dichotomies under these singular perturbations. Our results imply that, if the limiting pullback attractor of a nonautonomous dynamical system is the closure of a countable union of unstable manifolds of global bounded hyperbolic solutions, then it behaves continuously (upper and lower) under singular perturbations.     

Dynamical analysis of two coupled parametrically excited van der Pol oscillators

The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to different types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.     

考虑时滞效应时耦合化学反应系统的动力学行为

在耦合自催化反应系统中,采用数值分析方法研究了考虑时滞效应和流速扰动时子系统的动力学行为.与原系统相比,该系统呈现出更加丰富的动力学现象.反应过程中出现了结构复杂的混沌吸引子和由在周期解邻域内振荡而产生的概周期运动,并且存在混沌由倍周期分岔演变为新的混沌吸引子的过程.这些结果对于解释耦合化学反应系统中的复杂现象、揭示其反应机理具有一定的指导意义.     

Global attractor for Klein-Gordon-Schroedinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schroedinger equation （KGS lattice system）. We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using ＂End Tails＂ of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

Global attractor for Klein-Gordon-Schrodinger lattice system

We considered the longtime behavior of solutions of a coupled lattice dynamical system of Klein-Gordon-Schrodinger equation (KGS lattice system). We first proved the existence of a global attractor for the system considered here by introducing an equivalent norm and using "End Tails" of solutions. Then we estimated the upper bound of the Kolmogorov delta-entropy of the global attractor by applying element decomposition and the covering property of a polyhedron by balls of radii delta in the finite dimensional space. Finally, we presented an approximation to the global attractor by the global attractors of finite-dimensional ordinary differential systems.     

双频1:2激励下修正蔡氏振子两尺度耦合行为

不同尺度耦合系统存在的复杂振荡及其分岔机理一直是当前国内外研究的热点课题之一. 目前相关工作大都是针对单频周期激励频域两尺度系统,而对于含有两个或两个以上周期激励系统尺度效应的研究则相对较少. 为深入揭示多频激励系统的不同尺度效应,本文以修正的四维蔡氏电路为例,通过引入两个频率不同的周期电流源,建立了双频1:2周期激励两尺度动力学模型. 当两激励频率之间存在严格共振关系,且周期激励频率远小于系统的固有频率时,可以将两周期激励项转换为单一周期激励项的函数形式. 将该单一周期激励项视为慢变参数,给出了不同激励幅值下快子系统随慢变参数变化的平衡曲线及其分岔行为的演化过程,重点考察了3种较为典型的不同外激励幅值下系统的簇发振荡行为. 结合转换相图,揭示了各种簇发振荡的产生机理. 系统的轨线会随慢变参数的变化,沿相应的稳定平衡曲线运动,而fold分岔会导致轨迹在不同稳定平衡曲线上的跳跃,产生相应的激发态. 激发态可以用从分岔点向相应稳定平衡曲线的暂态过程来近似,其振荡幅值的变化和振荡频率也可用相应平衡点特征值的实部和虚部来描述,并进一步指出随着外激励幅值的改变,导致系统参与簇发振荡的平衡曲线分岔点越多,其相应簇发振荡吸引子的结构也越复杂.      

Random attractors

In this paper, we generalize the notion of an attractor for the stochastic dynamical system introduced in [7]. We prove that the stochastic attractor satisfies most of the properties satisfied by the usual attractor in the theory of deterministic dynamical systems. We also show that our results apply to the stochastic Navier-Stokes equation, the white noise-driven Burgers equation, and a nonlinear stochastic wave equation.     

一类二维非自治离散系统中的复杂簇发振荡结构1）

簇发振荡是多时间尺度系统复杂动力学行为的典型代表,簇发振荡的动力学机制与分类问题是簇发研究的重要问题之一,但当前学者们所揭示的簇发振荡的结构大多较为简单.研究以非自治离散Duffing系统为例,探讨具有复杂分岔结构的新型簇发振荡模式,并将其分为两大类,一类经由Fold分岔所诱发的对称式簇发,另一类经由延迟倍周期分岔所诱发的非对称式簇发.快子系统的分岔表现为典型的含有两个Fold分岔点的S形不动点曲线,其上、下稳定支可经由倍周期（即Flip）分岔通向混沌.当非自治项（即慢变量）穿越Fold分岔点时,系统的轨线可以向上、下稳定支的各种吸引子（例如,周期轨道和混沌）进行转迁,因此得到了经由Fold分岔所诱发的各种对称式簇发;而当非自治项无法穿越Fold分岔点,但可以穿越Flip分岔点时,系统产生了延迟Flip分岔现象.基于此,得到了经由延迟Flip分岔所诱发的各种非对称簇发.特别地,文中所报道的簇发振荡模式展现出复杂的反向Flip分岔结构.研究结果表明,这与非自治项缓慢地反向穿越快子系统的Flip分岔点有关.研究结果丰富了离散系统簇发的动力学机理和分类.     

Determining Asymptotic Behavior from the Dynamics on Attracting Sets

Two tracking properties for trajectories on attracting sets are studied. We prove that trajectories on the full phase space can be followed arbitrarily closely by skipping from one solution on the global attractor to another. A sufficient condition for asymptotic completeness of invariant exponential attractors is found, obtaining similar results as in the theory of inertial manifolds. Furthermore, such sets are shown to be retracts of the phase space, which implies that they are simply connected.