TY - GEN
T1 - Synchronisation of Weakly Coupled Oscillators
AU - Martins, Rogério
N1 - Funding Information:
This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UIDB/00297/2020 (Centro de Matemática e Aplicações).
Publisher Copyright:
© 2021, Springer Nature Switzerland AG.
PY - 2021
Y1 - 2021
N2 - The synchronization phenomenon was reported for the first time by Christiaan Huygens, when he noticed the strange tendency of a couple of clocks to synchronise their movements. More recently this phenomena was shown to be ubiquitous in nature and it is broadly studied by its applications, for example in biological cycles. We consider the problem of synchronization of a general network of linearly coupled oscillators, not necessarily identical. In this case the existence of a linear synchronization space is not expected, so we present an approach based on the proof of the existence of a synchronization manifold, the so-called generalised synchronization. Based on some results developed by R. Smith and on Wazewski’s principle, a general theory on the existence of invariant manifolds that attract the solutions of the system that are bounded in the future, is presented. Applications and estimates on parameters for the existence of synchronization are presented for several examples: systems of coupled pendulum type equations, coupled Lorenz systems of equations, and oscillators coupled through a medium, among many others.
AB - The synchronization phenomenon was reported for the first time by Christiaan Huygens, when he noticed the strange tendency of a couple of clocks to synchronise their movements. More recently this phenomena was shown to be ubiquitous in nature and it is broadly studied by its applications, for example in biological cycles. We consider the problem of synchronization of a general network of linearly coupled oscillators, not necessarily identical. In this case the existence of a linear synchronization space is not expected, so we present an approach based on the proof of the existence of a synchronization manifold, the so-called generalised synchronization. Based on some results developed by R. Smith and on Wazewski’s principle, a general theory on the existence of invariant manifolds that attract the solutions of the system that are bounded in the future, is presented. Applications and estimates on parameters for the existence of synchronization are presented for several examples: systems of coupled pendulum type equations, coupled Lorenz systems of equations, and oscillators coupled through a medium, among many others.
KW - Coupled oscillators
KW - Dissipative systems
KW - Invariant manifolds
KW - Synchronisation
UR - http://www.scopus.com/inward/record.url?scp=85116939485&partnerID=8YFLogxK
U2 - 10.1007/978-3-030-78163-7_14
DO - 10.1007/978-3-030-78163-7_14
M3 - Conference contribution
AN - SCOPUS:85116939485
SN - 978-3-030-78162-0
T3 - Springer Proceedings in Mathematics and Statistics
SP - 323
EP - 354
BT - Modeling, Dynamics, Optimization and Bioeconomics IV - DGS VI JOLATE and ICABR, Selected Contributions
A2 - Pinto, Alberto
A2 - Zilberman, David
PB - Springer
T2 - 6th International Conference on Dynamics Games and Science, DGS-VI-2018, 19th Latin American Conference on Economic Theory, JOLATE 2018 and 21st International Consortium on Applied Bioeconomy Research, ICABR 2017
Y2 - 30 May 2017 through 2 June 2017
ER -