scholarly journals Synchrony in networks of coupled non-smooth dynamical systems: Extending the master stability function

2016 ◽  
Vol 27 (6) ◽  
pp. 904-922 ◽  
Author(s):  
STEPHEN COOMBES ◽  
RÜDIGER THUL
Keyword(s):  
Dynamical Systems ◽  
Variational Equation ◽  
Period Doubling ◽  
Star Graph ◽  
Stability Function ◽  
Bifurcation Of Limit Cycles ◽  
Master Stability Function ◽  
Coupled Networks ◽  
Low Dimensional ◽  
Insight Into

The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.

MASTER STABILITY FUNCTIONS FOR SYNCHRONIZED COUPLED SYSTEMS

1999 ◽  
Vol 09 (12) ◽  
pp. 2315-2320 ◽  
Author(s):  
LOUIS M. PECORA ◽  
THOMAS L. CARROLL
Keyword(s):  
Simple Form ◽  
Coupled Systems ◽  
Stability Function ◽  
Coupled Oscillator ◽  
Synchronous State ◽  
Master Stability Function ◽  
Stability Functions ◽  
The Stability

We show that many coupled oscillator array configurations considered in the literature can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function which solves, once and for all, the problem of synchronous stability for many couplings of that oscillator.

scholarly journals Ab Initio Study of Ferroelectric Critical Size of SnTe Low-Dimensional Nanostructures

Nanomaterials ◽  
2020 ◽  
Vol 10 (4) ◽  
pp. 732 ◽  
Author(s):  
Takahiro Shimada ◽  
Koichiro Minaguro ◽  
Tao Xu ◽  
Jie Wang ◽  
Takayuki Kitamura
Keyword(s):  
Density Functional ◽  
Critical Size ◽  
Density Functional Theory Calculations ◽  
Cell Width ◽  
Functional Theory ◽  
Unit Cells ◽  
Principle Density Functional Theory ◽  
Low Dimensional ◽  
Insight Into ◽  
Ferroelectric Perovskite

Beyond a ferroelectric critical thickness of several nanometers existed in conventional ferroelectric perovskite oxides, ferroelectricity in ultimately thin dimensions was recently discovered in SnTe monolayers. This discovery suggests the possibility that SnTe can sustain ferroelectricity during further low-dimensional miniaturization. Here, we investigate a ferroelectric critical size of low-dimensional SnTe nanostructures such as nanoribbons (1D) and nanoflakes (0D) using first-principle density-functional theory calculations. We demonstrate that the smallest (one-unit-cell width) SnTe nanoribbon can sustain ferroelectricity and there is no ferroelectric critical size in the SnTe nanoribbons. On the other hand, the SnTe nanoflakes form a vortex of polarization and lose their toroidal ferroelectricity below the surface area of 4 × 4 unit cells (about 25 Å on one side). We also reveal the atomic and electronic mechanism of the absence or presence of critical size in SnTe low-dimensional nanostructures. Our result provides an insight into intrinsic ferroelectric critical size for low-dimensional chalcogenide layered materials.

scholarly journals Synchrony in networks of Franklin bells

2019 ◽  
Vol 84 (5) ◽  
pp. 1001-1021 ◽  
Author(s):  
Mustafa Şayli ◽  
Yi Ming Lai ◽  
Rüdiger Thul ◽  
Stephen Coombes
Keyword(s):  
Linear Stability ◽  
Benjamin Franklin ◽  
Stability Function ◽  
Impact Oscillators ◽  
Network Topologies ◽  
Network States ◽  
Low Dimensional ◽  
Oscillatory Network ◽  
The Impact ◽  
Theoretical Results

Abstract The Franklin bell is an electro-mechanical oscillator that can generate a repeating chime in the presence of an electric field. Benjamin Franklin famously used it as a lightning detector. The chime arises from the impact of a metal ball on a metal bell. Thus, a network of Franklin bells can be regarded as a network of impact oscillators. Although the number of techniques for analysing impacting systems has grown in recent years, this has typically focused on low-dimensional systems and relatively little attention has been paid to networks. Here we redress this balance with a focus on synchronous oscillatory network states. We first study a single Franklin bell, showing how to construct periodic orbits and how to determine their linear stability and bifurcation. To cope with the non-smooth nature of the impacts we use saltation operators to develop the correct Floquet theory. We further introduce a new smoothing technique that circumvents the need for saltation and that recovers the saltation operators in some appropriate limit. We then consider the dynamics of a network of Franklin bells, showing how the master stability function approach can be adapted to treat the linear stability of the synchronous state for arbitrary network topologies. We use this to determine conditions for network induced instabilities. Direct numerical simulations are shown to be in excellent agreement with theoretical results.

scholarly journals A master stability function approach to cardiac alternans

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Yi Ming Lai ◽  
Joshua Veasy ◽  
Stephen Coombes ◽  
Rüdiger Thul
Keyword(s):  
Activity Patterns ◽  
Muscle Cells ◽  
Stability Function ◽  
Cardiac Muscle Cells ◽  
Network Nodes ◽  
Master Stability Function ◽  
Synchronous Network ◽  
Cardiac Alternans ◽  
Regular Rhythm ◽  
Network Of Networks

Abstract During a single heartbeat, muscle cells in the heart contract and relax. Under healthy conditions, the behaviour of these muscle cells is almost identical from one beat to the next. However, this regular rhythm can be disturbed giving rise to a variety of cardiac arrhythmias including cardiac alternans. Here, we focus on so-called microscopic calcium alternans and show how their complex spatial patterns can be understood with the help of the master stability function. Our work makes use of the fact that cardiac muscle cells can be conceptualised as a network of networks, and that calcium alternans correspond to an instability of the synchronous network state. In particular, we demonstrate how small changes in the coupling strength between network nodes can give rise to drastically different activity patterns in the network.

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