SYNCHRONIZATION STABILITY IN COUPLED OSCILLATOR ARRAYS: SOLUTION FOR ARBITRARY CONFIGURATIONS

2000 ◽  
Vol 10 (02) ◽  
pp. 273-290 ◽  
Author(s):  
LOU PECORA ◽  
TOM CARROLL ◽  
GREGG JOHNSON ◽  
DOUG MAR ◽  
KENNETH S. FINK
Keyword(s):  
Coupled Oscillators ◽  
Stability Function ◽  
Crucial Issue ◽  
Asymptotic Results ◽  
Coupled Oscillator ◽  
Master Stability Function ◽  
Synchronization Problem ◽  
Oscillator Arrays ◽  
The Stability ◽  
General Guidance

The stability of the state of motion in which a collection of coupled oscillators are in identical synchrony is often a primary and crucial issue. When synchronization stability is needed for many different configurations of the oscillators the problem can become computationally intense. In addition, there is often no general guidance on how to change a configuration to enhance or diminsh stability, depending on the requirements. We have recently introduced a concept called the Master Stability Function that is designed to accomplish two goals: (1) decrease the numerical load in calculating synchronization stability and (2) provide guidance in designing coupling configurations that conform to the stability required. In doing this, we develop a very general formulation of the identical synchronization problem, show that asymptotic results can be derived for very general cases, and demonstrate that simple oscillator configurations can probe the Master Stability Function.

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.

Three coupled oscillators as a universal probe of synchronization stability in coupled oscillator arrays

2000 ◽  
Vol 61 (5) ◽  
pp. 5080-5090 ◽  
Author(s):  
Kenneth S. Fink ◽  
Gregg Johnson ◽  
Tom Carroll ◽  
Doug Mar ◽  
Lou Pecora
Keyword(s):  
Coupled Oscillators ◽  
Coupled Oscillator ◽  
Oscillator Arrays

scholarly journals Recent advances in coupled oscillator theory

2019 ◽  
Vol 377 (2160) ◽  
pp. 20190092 ◽  
Author(s):  
Bard Ermentrout ◽  
Youngmin Park ◽  
Dan Wilson
Keyword(s):  
Social Sciences ◽  
Coupled Oscillators ◽  
Weak Coupling ◽  
Standard Theory ◽  
Theme Issue ◽  
Coupled Oscillator ◽  
Dynamical Interaction ◽  
Weakly Coupled ◽  
The Stability ◽  
Oscillator Theory

We review the theory of weakly coupled oscillators for smooth systems. We then examine situations where application of the standard theory falls short and illustrate how it can be extended. Specific examples are given to non-smooth systems with applications to the Izhikevich neuron. We then introduce the idea of isostable reduction to explore behaviours that the weak coupling paradigm cannot explain. In an additional example, we show how bifurcations that change the stability of phase-locked solutions in a pair of identical coupled neurons can be understood using the notion of isostable reduction. This article is part of the theme issue ‘Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences’.

A fast technique for calculating master stability function

2020 ◽  
Vol 34 (05) ◽  
pp. 2050024
Author(s):  
Shirin Panahi ◽  
Sajad Jafari
Keyword(s):  
Numerical Simulation ◽  
Complex Dynamical Networks ◽  
Computational Time ◽  
Stability Function ◽  
Efficient Tool ◽  
New Approach ◽  
Master Stability Function ◽  
Floquet Exponent ◽  
Extensive Numerical Simulation ◽  
The Stability

Investigating the stability of the synchronization manifold is a critical topic in the field of complex dynamical networks. Master stability function (MSF) is known as a powerful and efficient tool for the study of synchronization in complex identical networks. The network can be synchronized whenever the MSF is negative. MSF uses the Lyapunov or Floquet exponent theory to determine the stability of the synchronization state. Both of these methods need extensive numerical simulation and a long computational time. In this paper, a new approach to calculate MSF is proposed. The accuracy of the results and time of simulations are tested on seven different known oscillators and also compared with the conventional methods of MSF. The results show that the proposed technique is faster and more efficient than the existing methods.

The linearity of the master stability function

2021 ◽  
Author(s):  
Janarthanan Ramadoss ◽  
Karthikeyan Rajagopal ◽  
Hayder Natiq ◽  
Iqtadar Hussain
Keyword(s):  
Root Mean Square Error ◽  
Network Topology ◽  
Coupled Oscillators ◽  
Local Stability ◽  
Mean Square ◽  
Single Variable ◽  
Stability Function ◽  
Master Stability Function ◽  
Straight Line ◽  
Variable Coupling

Abstract The master stability function (MSF) is an approach to evaluate the local stability of the synchronization in coupled oscillators. Computing the MSF of a network according to its parameters results in a curve whose shape is dependent on the nodes’ dynamics, network topology, coupling function, and coupling strength. This paper calculates the MSF of networks of two diffusively coupled oscillators by considering different single variable and multi-variable couplings. Then, the linearity of the MSF is investigated by fitting a straight line to the MSF curve, and the root mean square error is obtained. It is observed that the multi-variable coupling with equal coefficients on all variables results in a linear MSF regardless of the dynamics of the nodes.

Delay stabilization of periodic orbits in coupled oscillator systems

2010 ◽  
Vol 368 (1911) ◽  
pp. 319-341 ◽  
Author(s):  
B. Fiedler ◽  
V. Flunkert ◽  
P. Hövel ◽  
E. Schöll
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Orbits ◽  
Coupled Oscillators ◽  
Coupled Oscillator ◽  
Non Invasive ◽  
Coupling Parameters ◽  
Necessary And Sufficient ◽  
Time Delayed Feedback ◽  
Coupled Oscillator Systems

We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator’s nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.

Revisiting an old concept: the coupled oscillator model for VCD. Part 2: implications of the generalised coupled oscillator mechanism for the VCD robustness concept

2016 ◽  
Vol 18 (31) ◽  
pp. 21213-21225 ◽  
Author(s):  
Valentin Paul Nicu
Keyword(s):  
Relative Orientation ◽  
Oscillator Model ◽  
Coupled Oscillator ◽  
Displacement Vectors ◽  
The Stability ◽  
Coupled Oscillator Model

The generalised coupled oscillator (GCO) mechanism implies that the stability of the computed VCD sign should be assigned by monitoring the uncertainties in the relative orientation of the GCO fragments and in the nuclear displacement vectors, i.e. not the magnitude of the dissymmetry factor.

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