Stability of entrainment of a continuum of coupled oscillators

Snyder, Jordan; Zlotnik, Anatoly; Hagberg, Aric

doi:10.1063/1.4994567

Title: Stability of entrainment of a continuum of coupled oscillators

Abstract

Complex natural and engineered systems are ubiquitous, and their behavior is challenging to characterize and control. Here, we examine the design of the entrainment process for an uncountably infinite collection of coupled phase oscillators that are all subject to the same periodic driving signal. In the absence of coupling, an appropriately designed input can result in each oscillator attaining the frequency of the driving signal, with a phase offset determined by its natural frequency. We also consider a special case of interacting oscillators in which the coupling tends to destabilize the phase configuration to which the driving signal would send the collection in the absence of coupling. In this setting, we derive stability results that characterize the trade-off between the effects of driving and coupling, and compare these results to the well-known Kuramoto model of a collection of free-running coupled oscillators.

Authors:

^[1];

^[2];

^[2]

Univ. of California, Davis, CA (United States). Dept. of Mathematics
Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Applied Mathematics and Plasma Physics

Publication Date:: Thu Oct 05 00:00:00 EDT 2017

Research Org.:: Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1408843

Alternate Identifier(s):: OSTI ID: 1398105

Report Number(s):: LA-UR-17-26419
Journal ID: ISSN 1054-1500

Grant/Contract Number:: AC52-06NA25396; W911NF-13-1-0340; W911NF-09-2-0053

Resource Type:: Accepted Manuscript

Journal Name:: Chaos: An Interdisciplinary Journal of Nonlinear Science

Additional Journal Information:: Journal Volume: 27; Journal Issue: 10; Journal ID: ISSN 1054-1500

Publisher:: American Institute of Physics (AIP)

Country of Publication:: United States

Language:: English

Subject:: 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICS AND COMPUTING; Mathematics

Citation Formats


                    Snyder, Jordan, Zlotnik, Anatoly, and Hagberg, Aric. Stability of entrainment of a continuum of coupled oscillators.  United States: N. p., 2017. 
Web.  doi:10.1063/1.4994567.

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                    Snyder, Jordan, Zlotnik, Anatoly, & Hagberg, Aric. Stability of entrainment of a continuum of coupled oscillators.  United States.  https://doi.org/10.1063/1.4994567

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                    Snyder, Jordan, Zlotnik, Anatoly, and Hagberg, Aric. Thu .  
"Stability of entrainment of a continuum of coupled oscillators".  United States.  https://doi.org/10.1063/1.4994567.  https://www.osti.gov/servlets/purl/1408843.

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@article{osti_1408843,

  title        = {Stability of entrainment of a continuum of coupled oscillators},

  author       = {Snyder, Jordan and Zlotnik, Anatoly and Hagberg, Aric},

  abstractNote = {Complex natural and engineered systems are ubiquitous, and their behavior is challenging to characterize and control. Here, we examine the design of the entrainment process for an uncountably infinite collection of coupled phase oscillators that are all subject to the same periodic driving signal. In the absence of coupling, an appropriately designed input can result in each oscillator attaining the frequency of the driving signal, with a phase offset determined by its natural frequency. We also consider a special case of interacting oscillators in which the coupling tends to destabilize the phase configuration to which the driving signal would send the collection in the absence of coupling. In this setting, we derive stability results that characterize the trade-off between the effects of driving and coupling, and compare these results to the well-known Kuramoto model of a collection of free-running coupled oscillators.},

  doi          = {10.1063/1.4994567},

  journal      = {Chaos: An Interdisciplinary Journal of Nonlinear Science},

  number       = 10,

  volume       = 27,

  place        = {United States},

  year         = {Thu Oct 05 00:00:00 EDT 2017},

  month        = {Thu Oct 05 00:00:00 EDT 2017}

}

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Works referencing / citing this record:

Control of neural systems at multiple scales using model-free, deep reinforcement learning
journal, July 2018

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Scientific Reports, Vol. 8, Issue 1
DOI: 10.1038/s41598-018-29134-x

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Title: Stability of entrainment of a continuum of coupled oscillators

Abstract

Citation Formats

Low dimensional behavior of large systems of globally coupled oscillators journal, September 2008

Phase reduction approach to synchronisation of nonlinear oscillators journal, October 2015

Synchronization and desynchronization in epilepsy: controversies and hypotheses: Synchronization in epilepsy journal, January 2013

The Geometry of Biological Time. journal, June 1983

Distributed control in a mean-field cortical network model: Implications for seizure suppression journal, August 2012

Emerging Coherence in a Population of Chemical Oscillators journal, May 2002

Circadian Rhythm Generation and Entrainment in Astrocytes journal, January 2005

Optimal Subharmonic Entrainment of Weakly Forced Nonlinear Oscillators journal, January 2014

Neural Synchrony in Brain Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology journal, October 2006

Globally Coupled Oscillator Networks book, January 2003

Biological rhythms and the behavior of populations of coupled oscillators journal, July 1967

Neural Synchrony in Brain Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology journal, October 2006

Long time evolution of phase oscillator systems journal, June 2009

Jump Bifurcation and Hysteresis in an Infinite-Dimensional Dynamical System of Coupled Spins journal, February 1990

Perception's shadow: long-distance synchronization of human brain activity journal, February 1999

Optimal Entrainment of Neural Oscillator Ensembles preprint, January 2012

Kuramoto model with uniformly spaced frequencies: Finite- N asymptotics of the locking threshold journal, June 2016

Optimal entrainment of neural oscillator ensembles journal, July 2012

Long Time Evolution of Phase Oscillator Systems text, January 2009

Thermodynamic limit of the first-order phase transition in the Kuramoto model journal, October 2005

Thermodynamic limit of the first-order phase transition in the Kuramoto model text, January 2005

Amplitude and Frequency Dependence of Spike Timing: Implications for Dynamic Regulation journal, July 2003

From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators journal, September 2000

Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization journal, June 2007

Synchronization in Complex Oscillator Networks and Smart Grids text, January 2012

Synchronization in complex oscillator networks and smart grids journal, January 2013

Optimal Waveform for Fast Entrainment of Weakly Forced Nonlinear Oscillators journal, July 2013

Cooperative Phenomena in Coupled Oscillator Systems under External Fields journal, January 1988

The Kuramoto model: A simple paradigm for synchronization phenomena journal, April 2005

A Study of Locking Phenomena in Oscillators journal, June 1946

External periodic driving of large systems of globally coupled phase oscillators journal, September 2008

Perception's shadow: long-distance synchronization of human brain activity journal, February 1999

Comment on “Long time evolution of phase oscillator systems” [Chaos 19, 023117 (2009)] journal, June 2011

A study of locking phenomena in oscillators journal, January 1973

Synchronization and desynchronization in epilepsy: controversies and hypotheses text, January 2013

The Theory of Weakly Coupled Oscillators book, November 2011

Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators text, January 2008

Interplay of coupling and common noise at the transition to synchrony in oscillator populations journal, December 2016

Interplay of coupling and common noise at the transition to synchrony in oscillator populations text, January 2016

Phase-selective entrainment of nonlinear oscillator ensembles journal, March 2016

Kuramoto model with uniformly spaced frequencies:Finite-N asymptotics of the locking threshold text, January 2015

Globally Coupled Oscillator Networks book, January 2003

Type I Membranes, Phase Resetting Curves, and Synchrony journal, July 1996

Stability diagram for the forced Kuramoto model journal, December 2008

Silicon carbide (SiC) membrane nanomechanical resonators with multiple vibrational modes conference, June 2011

Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization journal, June 2007

Control of Synchronization of Brain Dynamics Leads to Control of Epileptic Seizures in Rodents journal, June 2009

Entrainment Limit of Weakly Forced Nonlinear Oscillators book, January 2015

Control of neural systems at multiple scales using model-free, deep reinforcement learning journal, July 2018

Low dimensional behavior of large systems of globally coupled oscillators
journal, September 2008

Phase reduction approach to synchronisation of nonlinear oscillators
journal, October 2015

Synchronization and desynchronization in epilepsy: controversies and hypotheses: Synchronization in epilepsy
journal, January 2013

The Geometry of Biological Time.
journal, June 1983

Distributed control in a mean-field cortical network model: Implications for seizure suppression
journal, August 2012

Emerging Coherence in a Population of Chemical Oscillators
journal, May 2002

Circadian Rhythm Generation and Entrainment in Astrocytes
journal, January 2005

Optimal Subharmonic Entrainment of Weakly Forced Nonlinear Oscillators
journal, January 2014

Neural Synchrony in Brain Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology
journal, October 2006

Globally Coupled Oscillator Networks
book, January 2003

Biological rhythms and the behavior of populations of coupled oscillators
journal, July 1967

Neural Synchrony in Brain Disorders: Relevance for Cognitive Dysfunctions and Pathophysiology
journal, October 2006

Long time evolution of phase oscillator systems
journal, June 2009

Jump Bifurcation and Hysteresis in an Infinite-Dimensional Dynamical System of Coupled Spins
journal, February 1990

Perception's shadow: long-distance synchronization of human brain activity
journal, February 1999

Optimal Entrainment of Neural Oscillator Ensembles
preprint, January 2012

Kuramoto model with uniformly spaced frequencies: Finite- $N$ asymptotics of the locking threshold
journal, June 2016

Optimal entrainment of neural oscillator ensembles
journal, July 2012

Long Time Evolution of Phase Oscillator Systems
text, January 2009

Thermodynamic limit of the first-order phase transition in the Kuramoto model
journal, October 2005

Thermodynamic limit of the first-order phase transition in the Kuramoto model
text, January 2005

Amplitude and Frequency Dependence of Spike Timing: Implications for Dynamic Regulation
journal, July 2003

From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators
journal, September 2000

Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization
journal, June 2007

Synchronization in Complex Oscillator Networks and Smart Grids
text, January 2012

Synchronization in complex oscillator networks and smart grids
journal, January 2013

Optimal Waveform for Fast Entrainment of Weakly Forced Nonlinear Oscillators
journal, July 2013

Cooperative Phenomena in Coupled Oscillator Systems under External Fields
journal, January 1988

The Kuramoto model: A simple paradigm for synchronization phenomena
journal, April 2005

A Study of Locking Phenomena in Oscillators
journal, June 1946

External periodic driving of large systems of globally coupled phase oscillators
journal, September 2008

Perception's shadow: long-distance synchronization of human brain activity
journal, February 1999

Comment on “Long time evolution of phase oscillator systems” [Chaos 19, 023117 (2009)]
journal, June 2011

A study of locking phenomena in oscillators
journal, January 1973

Synchronization and desynchronization in epilepsy: controversies and hypotheses
text, January 2013

The Theory of Weakly Coupled Oscillators
book, November 2011

Low Dimensional Behavior of Large Systems of Globally Coupled Oscillators
text, January 2008

Interplay of coupling and common noise at the transition to synchrony in oscillator populations
journal, December 2016

Interplay of coupling and common noise at the transition to synchrony in oscillator populations
text, January 2016

Phase-selective entrainment of nonlinear oscillator ensembles
journal, March 2016

Kuramoto model with uniformly spaced frequencies:Finite-N asymptotics of the locking threshold
text, January 2015

Globally Coupled Oscillator Networks
book, January 2003

Type I Membranes, Phase Resetting Curves, and Synchrony
journal, July 1996

Stability diagram for the forced Kuramoto model
journal, December 2008

Silicon carbide (SiC) membrane nanomechanical resonators with multiple vibrational modes
conference, June 2011

Engineering Complex Dynamical Structures: Sequential Patterns and Desynchronization
journal, June 2007

Control of Synchronization of Brain Dynamics Leads to Control of Epileptic Seizures in Rodents
journal, June 2009

Entrainment Limit of Weakly Forced Nonlinear Oscillators
book, January 2015

Control of neural systems at multiple scales using model-free, deep reinforcement learning
journal, July 2018