On phase-locking of oscillators with delay coupling

H.G.E. Meijer; T. Kalmár-Nag; S.A. van Gils

On phase-locking of oscillators with delay coupling

H.G.E. Meijer
, T. Kalmár-Nag
, S.A. van Gils

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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Abstract

We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.

Original language	English
Title of host publication	Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)
Editors	D. Bernardini, G. Rega, F. Romeo
Publisher	EUROMECH
Pages	MS-11
Number of pages	2
ISBN (Print)	978-88-906234-2-4
Publication status	Published - Jul 2011
Event	7th European Nonlinear Dynamics Conference, ENOC 2011 - Rome, Italy Duration: 24 Jul 2011 → 29 Jul 2011 Conference number: 7

Conference

Conference	7th European Nonlinear Dynamics Conference, ENOC 2011
Abbreviated title	ENOC
Country/Territory	Italy
City	Rome
Period	24/07/11 → 29/07/11

Keywords

Delay equations
Stability
Numerical bifurcation analysis
Synchronization

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Cite this

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title = "On phase-locking of oscillators with delay coupling",

abstract = "We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.",

keywords = "Delay equations, Stability, Numerical bifurcation analysis, Synchronization",

author = "H.G.E. Meijer and T. Kalm{\'a}r-Nag and \{van Gils\}, S.A.",

year = "2011",

month = jul,

language = "English",

isbn = "978-88-906234-2-4",

pages = "MS--11",

editor = "D. Bernardini and G. Rega and F. Romeo",

booktitle = "Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)",

publisher = "EUROMECH",

note = "7th European Nonlinear Dynamics Conference, ENOC 2011, ENOC ; Conference date: 24-07-2011 Through 29-07-2011",

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TY - GEN

T1 - On phase-locking of oscillators with delay coupling

AU - Meijer, H.G.E.

AU - Kalmár-Nag, T.

AU - van Gils, S.A.

N1 - Conference code: 7

PY - 2011/7

Y1 - 2011/7

N2 - We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.

AB - We consider two oscillators with delayed direct and velocity coupling. The oscillators have frequencies close or equal to 1:1 resonance. Due to the coupling the oscillations of the subsystems are in or out of phase. For these synchronized and anti-phase solutions, we use averaging for analytical stability results for small parameters. We also determine bifurcation curves of the delay system numerically. We identify regions in the parameter space (two coupling constants and the delay) where both solutions are stable or only one. For small parameters the averaging and numerical results are in good agreement. For larger values of the delay, we find multiple synchronized and anti-phase solutions. For small detuning we show that a minimal coupling value is needed to have almost synchronous or anti-phase behaviour.

KW - Delay equations

KW - Stability

KW - Numerical bifurcation analysis

KW - Synchronization

M3 - Conference contribution

SN - 978-88-906234-2-4

SP - MS-11

BT - Proceedings of the 7th European Nonlinear Dynamics Conference (ENOC 2011)

A2 - Bernardini, D.

A2 - Rega, G.

A2 - Romeo, F.

PB - EUROMECH

T2 - 7th European Nonlinear Dynamics Conference, ENOC 2011

Y2 - 24 July 2011 through 29 July 2011

ER -

On phase-locking of oscillators with delay coupling

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