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research article

Slow-fast Dynamics of Strongly Coupled Adaptive Frequency Oscillators

Righetti, Ludovic  
•
Buchli, Jonas  
•
Ijspeert, Auke Jan  
January 1, 2021
Siam Journal On Applied Dynamical Systems

Oscillators have two main limitations: their synchronization properties are limited (i.e., they have a finite synchronization region) and they have no memory of past interactions (i.e., they return to their intrinsic frequency whenever the entraining signal disappears). We previously proposed a general mechanism to transform an oscillator into an adaptive frequency oscillator which adapts its parameters to learn the frequency of any input signal. The synchronization region then becomes infinite and the oscillator retains the entrainment frequency when the driving signal disappears. While this mechanism has been successfully used in various applications, such as robot control or observer design for active prosthesis, a formal understanding of its properties is still missing. In this paper, we study the adaptation mechanism in the case of strongly coupled phase oscillators and show that nontrivial slow-fast dynamics is at the origin of the adaptation. We show the existence of a layered structure of stable and unstable invariant slow manifolds and demonstrate how the input signal forces the dynamics to jump between these manifolds at regular intervals, leading to exponential convergence of the frequency adaptation. We extend the idea to a network of oscillators with amplitude adaptation and show that the slow invariant manifolds structure persists. Numerical simulations validate our analysis and extend the discussion to more complex cases.

  • Details
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Type
research article
DOI
10.1137/20M1357755
Web of Science ID

WOS:000736132900001

Author(s)
Righetti, Ludovic  
•
Buchli, Jonas  
•
Ijspeert, Auke Jan  
Date Issued

2021-01-01

Publisher

SIAM PUBLICATIONS

Published in
Siam Journal On Applied Dynamical Systems
Volume

20

Issue

4

Start page

1985

End page

2012

Subjects

Mathematics, Applied

•

Physics, Mathematical

•

Mathematics

•

Physics

•

key words

•

oscillators

•

synchronization

•

frequency adaptation

•

slow-fast dynamics

•

central pattern generators

•

model

•

locomotion

•

populations

Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
BIOROB  
Available on Infoscience
January 15, 2022
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/184502
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