Periodic dynamics of coupled cell networks II: cyclic symmetry

A coupled cell network is a directed graph whose nodes represent dynamical systems and whose directed edges specify how those systems are coupled to each other. The typical dynamic behaviour of a network is strongly constrained by its topology. Especially important constraints arise from global (group) symmetries and local (groupoid) symmetries. The H/K theorem of Buono and Golubitsky characterises the possible spatio-temporal symmetries of time-periodic states of group-equivariant dynamical systems. A version of this theorem for group-symmetric networks has been proved by Josi? and Török. In networks, spatial symmetries correspond to synchrony of cells, and spatio-temporal symmetries correspond to phase relations between cells. Associated with any coupled cell network is a canonical class of admissible ODEs that respect the network topology. A pattern of synchrony or phase relations in a hyperbolic time-periodic state of such an ODE is rigid if the pattern persists under small admissible perturbations. We characterise rigid patterns of synchrony and rigid phase patterns in coupled cell networks, on the assumption that the periodic state is fully oscillatory (no cell is in equilibrium) and the network has a basic property, the rigid phase property. We conjecture that all networks have the rigid phase property, and that in any path-connected network an admissible ODE with a hyperbolic periodic state can always be perturbed to make the perturbed periodic state fully oscillatory. Our main result states that in any path-connected network with the rigid phase property, every rigid pattern of phase relations can be characterised in two stages. First, sets of cells form synchronous clumps according to a balanced equivalence relation. Second, the corresponding quotient network has a cyclic group of automorphisms, and the phase relations are induced by associating a fixed phase shift with a generator of this group. Thus the clumps of synchronous cells form a discrete rotating wave. As a corollary, we prove an analogue of the H/K theorem for any path-connected network. We also discuss the non-path-connected case.     

Probability symmetry broken upon rapid period-doubling bifurcation

A quadratic one-dimensional mapping with noise is used to model the phenomenon of violation of the equal probability of variants in the postbifurcation system evolution upon a fast change of the control parameter. The case of a period-doubling bifurcation, after which only one of the two types of oscillations with different phases may occur in the system, is analyzed. The laws of the phenomenon are considered and its possible mechanism is discussed.     

Pitchfork bifurcation and flip-flop operation with a coupled nonlinear system

A ring cavity system consisting of two hybrid Michelson interferometers coupled together with feedback is constructed. This system shows spatial bifurcation and can be utilized as an all-optical flip-flop device. We demonstrate experimentally a flip-flop operation by use of only a positive pulse in this system.     

Chaos in the Takens–Bogdanov bifurcation with O(2) symmetry

The Takens–Bogdanov bifurcation is a codimension-two bifurcation that provides a key to the presence of complex dynamics in many systems of physical interest. When the system is translation-invariant in one spatial dimension with no left-right preference the imposition of periodic boundary conditions leads to the Takens–Bogdanov bifurcation with O(2) symmetry. This bifurcation, analyzed by G. Dangelmayr and E. Knobloch, Phil. Trans. R. Soc. London A 322, 243 (1987), describes the interaction between steady states and travelling and standing waves in the nonlinear regime and predicts the presence of modulated travelling waves as well. The analysis reveals the presence of several global bifurcations near which the averaging method (used in the original analysis) fails. We show here, using a combination of numerical continuation and the construction of appropriate return maps, that near the global bifurcation that terminates the branch of modulated travelling waves, the normal form for the Takens–Bogdanov bifurcation admits cascades of period-doubling bifurcations as well as chaotic dynamics of Shil'nikov type. Thus chaos is present arbitrarily close to the codimension-two point.     

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.     

Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations

It has recently been proved by Golubitsky and coworkers that in any network of coupled dynamical systems, the possible 'rigid' patterns of synchrony of hyperbolic equilibria are determined by purely combinatorial properties of the network, known as 'balanced equivalence relations'. A pattern is 'rigid' if it persists under small 'admissible' perturbations of the differential equation — ones that respect the network structure. We discuss a natural generalisation of these ideas to time-periodic states, and motivate two basic conjectures, the Rigid Synchrony Conjecture and the Rigid Phase Conjecture. These conjectures state that for rigid hyperbolic time-periodic patterns, cells with synchronous dynamics must have synchronous input cells, and cells with phase-related dynamics must have input cells that have the same phase relations. We provide evidence supporting the two conjectures, by proving them for a special class of periodic orbits, which we call 'tame', under strong assumptions on the network architecture and the symmetries of the periodic state. The discussion takes place in the formal setting of coupled cell networks. We prove that rigid patterns of synchrony are balanced, together with the analogous result for rigid patterns of phase relations. The assumption on the network architecture simplifies the geometry of admissible vector fields, while tameness rules out patterns with non-trivial local or multilocal symmetry. The main idea is to perturb an admissible vector field in a way that retains sufficient control over the associated perturbed periodic orbit. We present two techniques for constructing these perturbations, both using a general theorem on groupoid-symmetrisation of vector fields, which has independent interest. In particular we introduce a method of 'patching' that makes local changes to an admissible vector field. Having established these results for all-to-all coupled networks and tame periodic orbits we prove more general versions that require these assumptions only on a suitable quotient network. These conditions are weaker and encompass a larger class of networks and periodic orbits. We give an example to show that rigidity cannot be relaxed to hyperbolicity. We also prove, without any technical assumptions, that rigidly synchronous or phase-related cells must be input-isomorphic, a necessary precondition for the two conjectures to hold.     

Rotational symmetry extraction based on local feature match

提出了一种基于局部特征匹配的旋转对称提取算法,为基于特征的逆向工程提供了一种新的对称约束的提取方法.该算法首先利用特征线将模型分割成对应各个特征类型的体素的集合；然后计算各个体素的形状分布曲线,通过形状分布曲线匹配的方式获取几何形状相似的体素,并利用最小包围盒从中提取大小相似的体素以构成对称体素的集合；最后利用迭代最近点算法计算对称体素的旋转对称信息并利用均值漂移算法优化提取的对称.实验结果表明,体素分割结果对应构成模型的各个特征,具有明显的工程语义,对称提取结果的误差较小并具有较强的鲁棒性.     

Stabilization of periodic orbits near a subcritical Hopf bifurcation in delay-coupled networks

We study networks of delay-coupled oscillators with the aim to extend time-delayed feedback control to networks. We show that unstable periodic orbits of a network can be stabilized by a noninvasive, delayed coupling. We state criteria for stabilizing the orbits by delay-coupling in networks and apply these to the case where the local dynamics is close to a subcritical Hopf bifurcation, which is representative of systems with torsion-free unstable periodic orbits. Using the multiple scale method and the master stability function approach, the network system is reduced to the normal form, and the characteristic equations for Floquet exponents are derived in an analytical form, which reveals the coupling parameters for successful stabilization. Finally, we illustrate the results by numerical simulations of the Lorenz system close to a subcritical Hopf bifurcation. The unstable periodic orbits in this system have no torsion, and hence cannot be stabilized by the conventional time delayed-feedback technique.     

Tensor methods for parameter estimation and bifurcation analysis of stochastic reaction networks

Stochastic modelling of gene regulatory networks provides an indispensable tool for understanding how random events at the molecular level influence cellular functions. A common challenge of stochastic models is to calibrate a large number of model parameters against the experimental data. Another difficulty is to study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation). In this paper, tensor-structured parametric analysis (TPA) is developed to address these computational challenges. It is based on recently proposed low-parametric tensor-structured representations of classical matrices and vectors. This approach enables simultaneous computation of the model properties for all parameter values within a parameter space. The TPA is illustrated by studying the parameter estimation, robustness, sensitivity and bifurcation structure in stochastic models of biochemical networks. A Matlab implementation of the TPA is available at http://www.stobifan.org.     

Small planar travelling waves in two-dimensional networks of coupled oscillators

The existence of several small planar travelling waves with arbitrary direction of propagation is shown for two-dimensional cubic networks of oscillators with linear nearest-neighbour coupling. The analysis is based on a spatial dynamics reformulation of the relevant advance–delay equation. Whereas important aspects of the analysis are discontinuous in the direction of travel, and continuity of the travelling wave is shown with respect to the direction.     

Bogdanov–Takens bifurcation in a neutral BAM neural networks model with delays

In this study, the authors first discuss the existence of Bogdanov–Takens and triple zero singularity of a five neurons neutral bidirectional associative memory neural networks model with two delays. Then, by utilising the centre manifold reduction and choosing suitable bifurcation parameters, the second‐order and the third‐order normal forms of the Bogdanov–Takens bifurcation for the system are obtained. Finally, the obtained normal form and numerical simulations show some interesting phenomena such as the existence of a stable fixed point, a pair of stable non‐trivial equilibria, a stable limit cycles, heteroclinic orbits, homoclinic orbits, coexistence of two stable non‐trivial equilibria and a stable limit cycles in the neighbourhood of the Bogdanov–Takens bifurcation critical point.Inspec keywords: neurophysiology, neural nets, bifurcation, delays, critical pointsOther keywords: Bogdanov‐Takens bifurcation critical point, neutral BAM neural networks, bidirectional associative memory, delays, triple zero singularity, neurons, centre manifold reduction, bifurcation parameters, second‐order normal forms, third‐order normal forms, numerical simulations, stable fixed point, stable nontrivial equilibria, stable limit cycles, heteroclinic orbits, homoclinic orbits     

Modified nodal analysis and the incorporation of multiphase, coupled networks

As a note to an original presentation ("Modified nodal analysis: an essential addition to electrical circuit theory and analysis", see ibid., vol.11, no.3, p.84-92, 2002), this paper adds a further, final comment on the efficacy of modified nodal analysis (MNA) in handling multi-phase, coupled impedance networks typical of which are multi-conductor transmission lines and cables in which the electrical length is short so that the network elements are discrete lumped impedances.     

基于局部位置更新的无线传感器网络路由协议

提出了在具有移动基站的无线传感器网络中的一种新的路由协议,该协议在基站移动时只需要在一个小的区域内更新基站的位置信息,因此既节省了传感器节点的能量,又使基站在移动过程中仍可保持与传感器节点的持续通信.理论分析和模拟研究表明,与全局更新基站位置信息的路由协议相比,该协议降低了基站位置信息更新的代价,减少了无线信道的冲突概率,减少了延迟,可适用于对延迟要求较高的大规模无线传感器网络.     

Least squares timestamp synchronization for local broadcast networks

The analysis of computer network experiments strongly relies on event log files recorded by the participating network nodes during the experiment. Timing related issues play an important role here for a number of central parameters like, e.g., end-to-end delay. As each node uses its local clock to timestamp the events in the log files, the large deviation of standard crystal oscillator based clocks imposes some big problems. We look at this issue in the case of networks with local broadcast media, where occurring transmissions can often be observed by multiple network nodes. We have developed a method to correct the timestamps in such an environment in a previous paper. Here, we present a second solution. For this new solution, we give bounds on the synchronization quality and compare the two approaches by means of simulation.     

Terabit optical local area networks for multiprocessing systems

The design of a scalable optical local area network formultiprocessing systems is described. Each workstation has aparallel-fiber-ribbon optical link to a centralized complementarymetal-oxide silicon (CMOS) switch core, implemented on a singlecompact printed circuit board (PCB). When the Motorola Optobusfiber technology is used, each workstation has a data bandwidth of 6.4Gbits/s to the core. A centralized switch core interconnecting 32workstations supports a 204-Gbit/s aggregate data bandwidth. Theswitch core is based on a conventional broadcast-and-selectarchitecture, implemented with parallel CMOS integrated circuits(IC's). The switch core scales well; by incorporation of theCMOS optoelectronic IC's with optical input-output, the electricalcore can be reduced to a single-chip optoelectronic IC with terabitcapacities. A prototype of an optoelectronic switch core has been fabricated and is described. The appeal of the architectureincludes its reliance on commercially available parallel-fibertechnology, its reliance on the well-developed markets of local areanetworks and networks of workstations, and its smooth scalability from the electrical to optical domains as technology matures.