1,529 research outputs found
Synchronisation in networks of delay-coupled type-I excitable systems
We use a generic model for type-I excitability (known as the SNIPER or SNIC
model) to describe the local dynamics of nodes within a network in the presence
of non-zero coupling delays. Utilising the method of the Master Stability
Function, we investigate the stability of the zero-lag synchronised dynamics of
the network nodes and its dependence on the two coupling parameters, namely the
coupling strength and delay time. Unlike in the FitzHugh-Nagumo model (a model
for type-II excitability), there are parameter ranges where the stability of
synchronisation depends on the coupling strength and delay time. One important
implication of these results is that there exist complex networks for which the
adding of inhibitory links in a small-world fashion may not only lead to a loss
of stable synchronisation, but may also restabilise synchronisation or
introduce multiple transitions between synchronisation and desynchronisation.
To underline the scope of our results, we show using the Stuart-Landau model
that such multiple transitions do not only occur in excitable systems, but also
in oscillatory ones.Comment: 10 pages, 9 figure
Inferring Network Topology from Complex Dynamics
Inferring network topology from dynamical observations is a fundamental
problem pervading research on complex systems. Here, we present a simple,
direct method to infer the structural connection topology of a network, given
an observation of one collective dynamical trajectory. The general theoretical
framework is applicable to arbitrary network dynamical systems described by
ordinary differential equations. No interference (external driving) is required
and the type of dynamics is not restricted in any way. In particular, the
observed dynamics may be arbitrarily complex; stationary, invariant or
transient; synchronous or asynchronous and chaotic or periodic. Presupposing a
knowledge of the functional form of the dynamical units and of the coupling
functions between them, we present an analytical solution to the inverse
problem of finding the network topology. Robust reconstruction is achieved in
any sufficiently long generic observation of the system. We extend our method
to simultaneously reconstruct both the entire network topology and all
parameters appearing linear in the system's equations of motion. Reconstruction
of network topology and system parameters is viable even in the presence of
substantial external noise.Comment: 11 pages, 4 figure
Node dynamics on graphs: dynamical heterogeneity in consensus, synchronisation and final value approximation for complex networks
Here we consider a range of Laplacian-based dynamics on graphs such as dynamical invariance and coarse-graining, and node-specific properties such as convergence, observability and
consensus-value prediction. Firstly, using the intrinsic relationship between the external equitable partition (EEP) and the spectral properties of the graph Laplacian, we characterise convergence
and observability properties of consensus dynamics on networks. In particular, we
establish the relationship between the original consensus dynamics and the associated consensus
of the quotient graph under varied initial conditions. We show that the EEP with respect
to a node can reveal nodes in the graph with increased rate of asymptotic convergence to the consensus value as characterised by the second smallest eigenvalue of the quotient Laplacian.
Secondly, we extend this characterisation of the relationship between the EEP and Laplacian based dynamics to study the synchronisation of coupled oscillator dynamics on networks. We
show that the existence of a non-trivial EEP describes partial synchronisation dynamics for nodes within cells of the partition. Considering linearised stability analysis, the existence of a nontrivial EEP with respect to an individual node can imply an increased rate of asymptotic convergence
to the synchronisation manifold, or a decreased rate of de-synchronisation, analogous to the linear consensus case. We show that high degree 'hub' nodes in large complex networks such as Erdős-Rényi, scale free and entangled graphs are more likely to exhibit such dynamical
heterogeneity under both linear consensus and non-linear coupled oscillator dynamics.
Finally, we consider a separate but related problem concerning the ability of a node to compute the final value for discrete consensus dynamics given only a finite number of its own state values.
We develop an algorithm to compute an approximation to the consensus value by individual nodes that is ϵ close to the true consensus value, and show that in most cases this is possible for substantially less steps than required for true convergence of the system dynamics. Again considering a variety of complex networks we show that, on average, high degree nodes, and
nodes belonging to graphs with fast asymptotic convergence, approximate the consensus value employing fewer steps.Open Acces
Synchronisation in dynamically coupled maps
The central aim of this thesis is to better understand the dynamics of a set of dynamically
coupled map systems previously introduced by Ito & Kaneko in a series of papers (Phys.
Rev. Lett. 88 (2002), no. 2, 028701 and Phys. Rev. E 67 (2003), no. 4, 046226). The current
work extends Ito & Kaneko’s studies to clarify the changes in macrodynamics induced
by the differences in microdynamics between the two systems. A third system is also
introduced that has a minor change to the microdynamics from nonlinear to linear output
function in the externally coupled system.
The dynamics of these three dynamically-coupled maps is also compared with their
simplified systems with static coupling. The previous studies of these dynamically-coupled
maps showed a partitioning of the parameter space into regions of different macrodynamics.
Here, an in-depth study is presented of the behaviour of the systems as they cross the
boundary between one region and another. The behaviour across this boundary is shown to
be much more complicated than suggested in the previous studies.
These three systems of dynamically-coupled maps all differ in the form of their microscopic
couplings, yet two of the systems are shown to produce similar macrodynamics,
whereas the third differs dramatically by almost any measure of the macrodynamics.
The time it takes for the systems to synchronise, both the dynamically-coupled and
static-coupled systems, is investigated. It is shown that the introduction of dynamicalcouplings
stops the systems from synchronising quasi-instantaneously. Details of potential
consequences of this in the field of neuroscience are discussed.
A brief study of the effect of driving the systems with external stimuli is presented.
The different microscopic coupling forms cause different responses to the external stimuli.
Some of the responses are similar to that observed by the visual cortex area of the brain
Structural engineering of evolving complex dynamical networks
Networks are ubiquitous in nature and many natural and man-made systems can be modelled as networked systems. Complex networks, systems comprising a number of nodes that are connected through edges, have been frequently used to model large-scale systems from various disciplines such as biology, ecology, and engineering. Dynamical systems interacting through a network may exhibit collective behaviours such as synchronisation, consensus, opinion formation, flocking and unusual phase transitions. Evolution of such collective behaviours is highly dependent on the structure of the interaction network. Optimisation of network topology to improve collective behaviours and network robustness can be achieved by intelligently modifying the network structure. Here, it is referred to as "Engineering of the Network". Although coupled dynamical systems can develop spontaneous synchronous patterns if their coupling strength lies in an appropriate range, in some applications one needs to control a fraction of nodes, known as driver nodes, in order to facilitate the synchrony. This thesis addresses the problem of identifying the set of best drivers, leading to the best pinning control performance. The eigen-ratio of the augmented Laplacian matrix, that is the largest eigenvalue divided by the second smallest one, is chosen as the controllability metric. The approach introduced in this thesis is to obtain the set of optimal drivers based on sensitivity analysis of the eigen-ratio, which requires only a single computation of the eigenvector associated with the largest eigenvalue, and thus is applicable for large-scale networks. This leads to a new "controllability centrality" metric for each subset of nodes. Simulation results reveal the effectiveness of the proposed metric in predicting the most important driver(s) correctly.     Interactions in complex networks might also facilitate the propagation of undesired effects, such as node/edge failure, which may crucially affect the performance of collective behaviours. In order to study the effect of node failure on network synchronisation, an analytical metric is proposed that measures the effect of a node removal on any desired eigenvalue of the Laplacian matrix. Using this metric, which is based on the local multiplicity of each eigenvalue at each node, one can approximate the impact of any node removal on the spectrum of a graph. The metric is computationally efficient as it only needs a single eigen-decomposition of the Laplacian matrix. It also provides a reliable approximation for the "Laplacian energy" of a network. Simulation results verify the accuracy of this metric in networks with different topologies. This thesis also considers formation control as an application of network synchronisation and studies the "rigidity maintenance" problem, which is one of the major challenges in this field. This problem is to preserve the rigidity of the sensing graph in a formation during motion, taking into consideration constraints such as line-of-sight requirements, sensing ranges and power limitations. By introducing a "Lattice of Configurations" for each node, a distributed rigidity maintenance algorithm is proposed to preserve the rigidity of the sensing network when failure in a sensing link would result in loss of rigidity. The proposed algorithm recovers rigidity by activating, almost always, the minimum number of new sensing links and considers real-time constraints of practical formations. A sufficient condition for this problem is proved and tested via numerical simulations. Based on the above results, a number of other areas and applications of network dynamics are studied and expounded upon in this thesis
Pinning dynamic systems of networks with Markovian switching couplings and controller-node set
In this paper, we study pinning control problem of coupled dynamical systems
with stochastically switching couplings and stochastically selected
controller-node set. Here, the coupling matrices and the controller-node sets
change with time, induced by a continuous-time Markovian chain. By constructing
Lyapunov functions, we establish tractable sufficient conditions for
exponentially stability of the coupled system. Two scenarios are considered
here. First, we prove that if each subsystem in the switching system, i.e. with
the fixed coupling, can be stabilized by the fixed pinning controller-node set,
and in addition, the Markovian switching is sufficiently slow, then the
time-varying dynamical system is stabilized. Second, in particular, for the
problem of spatial pinning control of network with mobile agents, we conclude
that if the system with the average coupling and pinning gains can be
stabilized and the switching is sufficiently fast, the time-varying system is
stabilized. Two numerical examples are provided to demonstrate the validity of
these theoretical results, including a switching dynamical system between
several stable sub-systems, and a dynamical system with mobile nodes and
spatial pinning control towards the nodes when these nodes are being in a
pre-designed region.Comment: 9 pages; 3 figure
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