Dynamics of two van der Pol oscillators coupled via a bath

Erika Camacho, Richard Rand, Howard Howland

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this work we study a system of two van der Pol oscillators, x and y, coupled via a "bath" z: ẍ - ε(1 - x 2)ẋ + x = k(z - x) ÿ - ε(1 - y 2)ẏ + y = k(z - y) ż = k(x - z) + k(y - z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

Original languageEnglish (US)
Title of host publicationProceedings of the ASME Design Engineering Technical Conference
Pages2407-2414
Number of pages8
Volume5 C
StatePublished - 2003
Externally publishedYes
Event2003 ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference - Chicago, IL, United States
Duration: Sep 2 2003Sep 6 2003

Other

Other2003 ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference
CountryUnited States
CityChicago, IL
Period9/2/039/6/03

Fingerprint

Software packages
Melatonin

ASJC Scopus subject areas

  • Engineering(all)

Cite this

Camacho, E., Rand, R., & Howland, H. (2003). Dynamics of two van der Pol oscillators coupled via a bath. In Proceedings of the ASME Design Engineering Technical Conference (Vol. 5 C, pp. 2407-2414)

Dynamics of two van der Pol oscillators coupled via a bath. / Camacho, Erika; Rand, Richard; Howland, Howard.

Proceedings of the ASME Design Engineering Technical Conference. Vol. 5 C 2003. p. 2407-2414.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Camacho, E, Rand, R & Howland, H 2003, Dynamics of two van der Pol oscillators coupled via a bath. in Proceedings of the ASME Design Engineering Technical Conference. vol. 5 C, pp. 2407-2414, 2003 ASME Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Chicago, IL, United States, 9/2/03.
Camacho E, Rand R, Howland H. Dynamics of two van der Pol oscillators coupled via a bath. In Proceedings of the ASME Design Engineering Technical Conference. Vol. 5 C. 2003. p. 2407-2414
Camacho, Erika ; Rand, Richard ; Howland, Howard. / Dynamics of two van der Pol oscillators coupled via a bath. Proceedings of the ASME Design Engineering Technical Conference. Vol. 5 C 2003. pp. 2407-2414
@inproceedings{271672f9bea444b2b5d6a32ca3de1b81,
title = "Dynamics of two van der Pol oscillators coupled via a bath",
abstract = "In this work we study a system of two van der Pol oscillators, x and y, coupled via a {"}bath{"} z: ẍ - ε(1 - x 2)ẋ + x = k(z - x) {\"y} - ε(1 - y 2)ẏ + y = k(z - y) ż = k(x - z) + k(y - z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.",
author = "Erika Camacho and Richard Rand and Howard Howland",
year = "2003",
language = "English (US)",
volume = "5 C",
pages = "2407--2414",
booktitle = "Proceedings of the ASME Design Engineering Technical Conference",

}

TY - GEN

T1 - Dynamics of two van der Pol oscillators coupled via a bath

AU - Camacho, Erika

AU - Rand, Richard

AU - Howland, Howard

PY - 2003

Y1 - 2003

N2 - In this work we study a system of two van der Pol oscillators, x and y, coupled via a "bath" z: ẍ - ε(1 - x 2)ẋ + x = k(z - x) ÿ - ε(1 - y 2)ẏ + y = k(z - y) ż = k(x - z) + k(y - z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

AB - In this work we study a system of two van der Pol oscillators, x and y, coupled via a "bath" z: ẍ - ε(1 - x 2)ẋ + x = k(z - x) ÿ - ε(1 - y 2)ẏ + y = k(z - y) ż = k(x - z) + k(y - z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

UR - http://www.scopus.com/inward/record.url?scp=1842683655&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1842683655&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:1842683655

VL - 5 C

SP - 2407

EP - 2414

BT - Proceedings of the ASME Design Engineering Technical Conference

ER -