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Synchronization in the quaternionic Kuramoto model
Authors
Ting-Yang Hsiao
Yun-Feng Lo
Winnie Wang
Publication date
4 September 2023
Publisher
View
on
arXiv
Abstract
In this paper, we propose an
N
N
N
oscillators Kuramoto model with quaternions
H
\mathbb{H}
H
. In case the coupling strength is strong, a sufficient condition of synchronization is established for general
N
⩾
2
N\geqslant 2
N
⩾
2
. On the other hand, we analyze the case when the coupling strength is weak. For
N
=
2
N=2
N
=
2
, when coupling strength is weak (below the critical coupling strength
λ
c
\lambda_c
λ
c
​
), we show that new periodic orbits emerge near each equilibrium point, and hence phase-locking state exists. This phenomenon is different from the real Kuramoto system since it is impossible to arrive at any synchronization when
λ
<
λ
c
\lambda<\lambda_c
λ
<
λ
c
​
. A theorem is proved which states that the closed contours form a set of "Baumkuchen" that is dense near each equilibrium point. In other words, the trajectory of phase difference lies on a
4
D
4D
4
D
-torus surface. Therefore, this implies that the phase-locking state is Lyapunov stable but not asymptotically stable. The proof uses a new infinite buffer method ("
δ
/
n
\delta/n
δ
/
n
criterion") and a Lyapunov function argument. This has been studied both analytically and numerically. For
N
=
3
N=3
N
=
3
, we consider Lion Dance flow, the analog of Cherry flow, to demonstrate that the quaternionic synchronization exists even when the coupling strength is "super weak" (when
λ
/
ω
3
\lambda/\omega 3
λ
/
ω
3
, the stable manifold of Lion Dance flow exists, and the number of these equilibria is
⌊
N
−
1
2
⌋
\lfloor \frac{N-1}{2}\rfloor
⌊
2
N
−
1
​
⌋
. Therefore, we conjecture that quaternionic synchronization always exists.Comment: 35 pages, 6 figure
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oai:arXiv.org:2309.01893
Last time updated on 12/09/2023