Stability of quasi-simple heteroclinic cycles

The stability of heteroclinic cycles may be obtained from the value of the local stability index along each connection of the cycle. We establish a way of calculating the local stability index for quasi-simple cycles: cycles whose connections are one-dimensional and contained in flow-invariant spaces of equal dimension. These heteroclinic cycles exist both in symmetric and non-symmetric contexts. We make one assumption on the dynamics along the connections to ensure that the transition matrices have a convenient form. Our method applies to all simple heteroclinic cycles of type Z and to various heteroclinic cycles arising in population dynamics, namely non-simple heteroclinic cycles, as well as to cycles that are part of a heteroclinic network. We illustrate our results with a non-simple cycle present in a heteroclinic network of the Rock–Scissors–Paper game.     

Dynamics near the product of planar heteroclinic attractors

Motivated by problems in equivariant dynamics and connection selection in heteroclinic networks, Ashwin and Field investigated the product of planar dynamics where at least one of the factors was a planar homoclinic attractor. However, they were only able to obtain partial results in the case of a product of two planar homoclinic attractors. We give general results for the product of planar homoclinic and heteroclinic attractors. We show that the likely limit set of the basin of attraction of the product of two planar heteroclinic attractors is always the unique one-dimensional heteroclinic network which covers the heteroclinic attractors in the factors. The method we use is general and likely to apply to products of higher dimensional heteroclinic attractors as well as to situations where the product structure is broken but the cycles are preserved.     

Partial classification of heteroclinic behaviour associated with the perturbation of hexagonal planforms

Physical systems often exhibit pattern-forming instabilities. Equivariant bifurcation theory is often used to investigate the existence and stability of spatially doubly periodic solutions with respect to the hexagonal lattice. Previous studies have focused on the six- and twelve-dimensional representation of the hexagonal lattice where the symmetry of the model is perfect. Here, perturbation of group orbits of translation-free axial planforms in the six- and twelve-dimensional representations is considered. This problem is studied via the abstract action of the symmetry group of the perturbation on the group orbit of the planform. A partial classification for the behaviour of the group orbits is obtained, showing the existence of homoclinic and heteroclinic cycles between equilibria.     

Multi-cluster dynamics in coupled phase oscillator networks

In this paper, we examine robust clustering behaviour with multiple nontrivial clusters for identically and globally coupled phase oscillators. These systems are such that the dynamics is completely determined by the number of oscillators N and a single scalar function g(?) (the coupling function). Previous work has shown that (a) any clustering can stably appear via choice of a suitable coupling function and (b) open sets of coupling functions can generate heteroclinic network attractors between cluster states of saddle type, though there seem to be no examples where saddles with more than two nontrivial clusters are involved. In this work, we clarify the relationship between the coupling function and the dynamics. We focus on cases where the clusters are inequivalent in the sense of not being related by a temporal symmetry, and demonstrate that there are coupling functions that give robust heteroclinic networks between periodic states involving three or more nontrivial clusters. We consider an example for N = 6 oscillators where the clustering is into three inequivalent clusters. We also discuss some aspects of the bifurcation structure for periodic multi-cluster states and show that the transverse stability of inequivalent clusters can, to a large extent, be varied independently of the tangential stability.     

Stability analysis of autonomously controlled production networks

In this paper we present a stability analysis of autonomously controlled production networks from mathematical and engineering points of view. Roughly speaking stability of a system means that the defined state of the system remains bounded over time. The dynamics of a production network are modelled by differential equations (macroscopic approach) and discrete event simulation (microscopic approach), respectively. Both approaches are used to perform a stability analysis. As a result of the stability analysis of the macroscopic approach we calculate parameters, which guarantee stability of the network for arbitrary inputs. These results are refined for a certain (varying) input using the microscopic approach, where we derive the smallest maximal production rates of the plants for which stability of the overall system can be guaranteed. Furthermore, the microscopic approach includes two different autonomous control methods: the queue length estimator (QLE) and the pheromone based (PHE) method. These methods allow additional autonomous decision making on the shop floor level. The approach presented in this paper is to calculate stability conditions by mathematical systems theory to guarantee stability for production networks, to identify a stability region and to refine this region by simulations.     

Stability of multiple access communication networks

The problem of stability of different multiple access schemes in computer networks is discussed. The results for the main multiple access schemes — slottedaloha, controlledaloha, csma, csma/cd, polling systems and collision resolution algorithms are surveyed. The techniques used for the stability of these systems have been emphasized. An error in the first draft of this paper was pointed out by the reviewer.     

Nonreversible homoclinic snaking

Homoclinic snaking refers to the sinusoidal ‘snaking’ continuation curve of homoclinic orbits near a heteroclinic cycle connecting an equilibrium E and a periodic orbit P. Along this curve the homoclinic orbit performs more windings about the periodic orbit. Typically, this behaviour appears in reversible Hamiltonian systems. Here we discuss this phenomenon in systems without any particular structure. We give a rigorous analytical verification of homoclinic snaking under certain assumptions on the behaviour of the stable and unstable manifolds of E and P. We show how the snaking behaviour depends on the signs of the Floquet multipliers of P. Further we present a nonsnaking scenario. Finally, we show numerically that these assumptions are fulfilled in a model equation.     

一类Kelly-型排队网络的稳定性

排队网络可以用来模拟诸如通信网络这样的复杂系统。对排队网络的研究中的一个主要议题是建立其在某些特殊的规则下稳定的充要条件。本文的研究对象是一类具有两类顾客输入的Kelly-型排队网络。利用流体模型以及Lyapunov函数等工具,建立了该排队网络在所有非闲置的规则下稳定的充分条件。最后,对条件的充分性作了说明。     

一类一般化的排队网络族的全稳定性

本文对一类有两个工作站的一般化的排队网络族给出了其全稳定的一个充分条件。     

动力学系统的稳定性与非线性行为的一种分析计算方法

在非线性动力学系统中,分析系统参数的变化对系统稳定性影响和在系统不稳定的情况下系统参数对振动幅值的影响是十分重要的两个方面。针对常见的非线性动力学模型,提出了一个利用特征根判别、规范形计算等方法对系统非线性特性和稳定性进行分析计算的方法。应用该方法进行计算分析的实例说明该方法十分有效。     

优先权下大型交互对抗服务网络的稳定性研究

近年来,物联网、大数据与云计算为各种大型服务系统及其资源管理与任务调度提供了新的信息技术,也导致了这些服务系统中出现了大范围、松耦合、跨组织与异构性等多种关键特征.为了研究这些大型服务系统,交互对抗网络(Adversarial Networks)以其简便的确定性假设与直观的稳定性评估等优点而成为国际上一个热点且重要的研究方向.本文首先分析了一个具有优先权的环形交互对抗服务网络,并通过数学建模提出了一种确定性上界估计方法,由此讨论了这种网络的系统稳定性问题.然后,研究了系统性能指标并且给出了任意一个顾客在网络中运行时间的一个上界.最后,通过数值算例研究了网络中两类顾客总数的上界以及它们在网络中运行时间的上界是如何依赖网络中的一些关键参数.     

噪声扰动下的广义时滞细胞神经网络稳定性分析

本文考虑到不可避免的噪声干扰,针对一类广义时滞细胞神经网络,引入随机扰动项,利用随机微分方程理论,It(o)积分性质及分析技巧,研究了该系统在噪声扰动下的稳定性问题,并分别给出了判定平衡点全局稳定及全局指数稳定的充分条件.其结果以不等式形式给出,便于结论验证.最后,本文给出了主要定理的一个实例,表明结论的有效性.     

Switching near a network of rotating nodes

We study the dynamics of a Z ₂ ⊕ Z ₂-equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, in a network with rotating nodes, the rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network; close to the network, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.     

Hamiltonian stability in open surfaces with simple singularities

We extend to open surfaces with simple singularities the theorem proposed by Xavier Jarque and Zbigniew Nitecki for Hamiltonian flows in the plane that are structurally stable among Hamiltonian flows. We describe the Hamiltonian dynamics on {x² + y² = z²} and {xy = 0} by presenting characterization theorems for Hamiltonian stability and some natural consequences. The stability of planar Hamiltonian flows with an invariant line is also studied.     

Chaotic double cycling

We study the dynamics of a generic vector field in the neighbourhood of a heteroclinic cycle of non-trivial periodic solutions whose invariant manifolds meet transversely. The main result is the existence of chaotic double cycling: there are trajectories that follow the cycle making any prescribed number of turns near the periodic solutions, for any given bi-infinite sequence of turns. Using symbolic dynamics, arbitrarily close to the cycle, we find a robust and transitive set of initial conditions whose trajectories follow the cycle for all time and that is conjugate to a Markov shift over a finite alphabet. This conjugacy allows us to prove the existence of infinitely many heteroclinic and homoclinic subsidiary connections, which give rise to a heteroclinic network with infinitely many cycles and chaotic dynamics near them, exhibiting themselves switching and cycling. We construct an example of a vector field with Z ₃ symmetry in a five-dimensional sphere with a heteroclinic cycle having this property.     

Essentially asymptotically stable homoclinic networks

Melbourne [An example of a nonasymptotically stable attractor, Nonlinearity 4(3) (1991), pp. 835–844] discusses an example of a robust heteroclinic network that is not asymptotically stable but which has the strong attracting property called essential asymptotic stability. We establish that this phenomenon is possible for homoclinic networks, where all heteroclinic trajectories are symmetry related. Moreover, we study a transverse bifurcation from an asymptotically stable to an essentially asymptotically stable homoclinic network. The essentially asymptotically stable homoclinic network turns out to attract all nearby points except those on codimension-one stable manifolds of equilibria outside the homoclinic network.     

去合金化制备具有高循环稳定性的纳米多孔Sb/MCNT储钠负极材料

为提升Sb基负极材料的储钠循环性能, 通过简单的两步法(机械辅助化学合金化和酸溶解去合金化)制备纳米化和多孔化的去合金锑/多壁碳纳米管(De-Sb/MCNT)复合物, 采用不同方法表征材料的物理化学性质和储钠电化学性能。结果显示, De-Sb/MCNT材料的可逆比容量达到408.6 mAh·g^-1 (200 mA·g^-1), 首周库仑效率为69.2%; 在800 mA·g^-1循环330周后, 容量保持率仍可达88%, 展现出优异的储钠循环性能。这得益于机械辅助化学合金化/酸溶解去合金化对商品化Sb的“预粉化”作用, 促进了材料的纳米化和多孔化, 缓解了充放电过程中的体积膨胀, 实现了高的循环稳定性。这种常温合金化/去合金化的方法为制备循环稳定的储钠合金负极材料提供了新的途径。     

Stochastic analysis of biochemical reaction networks with absolute concentration robustness

It has recently been shown that structural conditions on the reaction network, rather than a ‘fine-tuning’ of system parameters, often suffice to impart ‘absolute concentration robustness’ (ACR) on a wide class of biologically relevant, deterministically modelled mass-action systems. We show here that fundamentally different conclusions about the long-term behaviour of such systems are reached if the systems are instead modelled with stochastic dynamics and a discrete state space. Specifically, we characterize a large class of models that exhibit convergence to a positive robust equilibrium in the deterministic setting, whereas trajectories of the corresponding stochastic models are necessarily absorbed by a set of states that reside on the boundary of the state space, i.e. the system undergoes an extinction event. If the time to extinction is large relative to the relevant timescales of the system, the process will appear to settle down to a stationary distribution long before the inevitable extinction will occur. This quasi-stationary distribution is considered for two systems taken from the literature, and results consistent with ACR are recovered by showing that the quasi-stationary distribution of the robust species approaches a Poisson distribution.