Synchronized States Observed in Coupled Four Oscillators

Hiroyuki KITAJIMA; Hiroshi KAWAKAMI; Tetsuo HATTORI

doi:10.1093/ietfec/e88-a.3.712

Synchronized States Observed in Coupled Four Oscillators

Hiroyuki KITAJIMA, Hiroshi KAWAKAMI, Tetsuo HATTORI

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In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E88-A No.3 pp.712-717

Publication Date: 2005/03/01

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DOI: 10.1093/ietfec/e88-a.3.712

Type of Manuscript: PAPER

Category: Nonlinear Problems

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Hiroyuki KITAJIMA, Hiroshi KAWAKAMI, Tetsuo HATTORI, "Synchronized States Observed in Coupled Four Oscillators" in IEICE TRANSACTIONS on Fundamentals, vol. E88-A, no. 3, pp. 712-717, March 2005, doi: 10.1093/ietfec/e88-a.3.712.
Abstract: In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.
URL: https://globals.ieice.org/en_transactions/fundamentals/10.1093/ietfec/e88-a.3.712/_p

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@ARTICLE{e88-a_3_712,
author={Hiroyuki KITAJIMA, Hiroshi KAWAKAMI, Tetsuo HATTORI, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Synchronized States Observed in Coupled Four Oscillators},
year={2005},
volume={E88-A},
number={3},
pages={712-717},
abstract={In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.},
keywords={},
doi={10.1093/ietfec/e88-a.3.712},
ISSN={},
month={March},}

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TY - JOUR
TI - Synchronized States Observed in Coupled Four Oscillators
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 712
EP - 717
AU - Hiroyuki KITAJIMA
AU - Hiroshi KAWAKAMI
AU - Tetsuo HATTORI
PY - 2005
DO - 10.1093/ietfec/e88-a.3.712
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E88-A
IS - 3
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - March 2005
AB - In this paper, we examine oscillatory modes generated by the Hopf bifurcations of equilibrium points except for the origin in a system of coupled four oscillators. (The bifurcation analyses of the origin for many coupled oscillators were already done.) The Hopf bifurcations of the equilibrium points with strong symmetrical property and the generated oscillatory modes are classified. We observe four-phase, in-phase and a pair of anti-phase synchronized states. Even in a system of four coupled oscillators, we discover the existence of a stable three-phase oscillation. By the numerical bifurcation analysis of generated periodic oscillations we find out successive period-doubling bifurcations as the route to chaos and show some of them are symmetry-breaking bifurcations. As a result of the symmetry-breaking period-doubling bifurcations, a periodic solution with complete synchronization becomes a chaotic solution with phase synchronization.
ER -