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

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.

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.

Frequency Modulated Coupled Oscillator Arrays

2001 ◽  
Author(s):  
Ragip Ispir ◽  
Akira Miyata ◽  
Kazunari Kawahata ◽  
Masao Ida ◽  
Yohei Ishikawa
Keyword(s):  
Coupled Oscillator ◽  
Oscillator Arrays

Wave propagation of phase difference in coupled oscillator arrays

2016 ◽  
pp. 133-162 ◽  
Author(s):  
Masayuki Yamauchi Yamauchi ◽  
Yoshihito Todani Todani ◽  
Syohei Fujimoto Fujimoto
Keyword(s):  
Wave Propagation ◽  
Phase Difference ◽  
Coupled Oscillator ◽  
Oscillator Arrays

STABILITY OF SYNCHRONIZATION IN NETWORKS OF DIGITAL PHASE-LOCKED LOOPS

1995 ◽  
Vol 05 (04) ◽  
pp. 983-990 ◽  
Author(s):  
GUILLERMO GOLDSZTEIN ◽  
STEVEN H. STROGATZ
Keyword(s):  
Linear Stability ◽  
Phase Locked Loops ◽  
Explicit Formulas ◽  
Coupled Oscillator ◽  
Transient Time ◽  
Digital Phase ◽  
Digital Phase Locked Loops ◽  
Oscillator Arrays ◽  
Specific Phase

We analyze the linear stability of the synchronized state in networks of N identical digital phase-locked loops. These are pulse-coupled oscillator arrays in which the frequency (rather than the phase) of each oscillator is updated discontinuously whenever that oscillator reaches a specific phase in its cycle. Three different coupling configurations are studied: one-way rings, two-way rings, and globally coupled arrays. In each case we obtain explicit formulas for the transient time to lock, the critical gain at which the synchronized state loses stability, and the period of the bifurcating solution at the onset of instability. Our results explain the numerical observations of de Sousa Vieira, Lichtenberg, and Lieberman.

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