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Stability analysis of mean field models described by Fokker‐Planck equations
Author(s) -
Frank T.D.
Publication year - 2002
Publication title -
annalen der physik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.009
H-Index - 68
eISSN - 1521-3889
pISSN - 0003-3804
DOI - 10.1002/1521-3889(200211)11:10/11<707::aid-andp707>3.0.co;2-a
Subject(s) - stability (learning theory) , statistical physics , physics , mean field theory , field (mathematics) , kuramoto model , lyapunov function , range (aeronautics) , fokker–planck equation , linear stability , basis (linear algebra) , mathematics , mathematical analysis , quantum mechanics , partial differential equation , instability , nonlinear system , computer science , geometry , topology (electrical circuits) , materials science , combinatorics , machine learning , synchronization (alternating current) , pure mathematics , composite material
A mean field model for weakly coupled oscillators is studied that can describe transitions from stationary uniform distributions to traveling wave solutions. The mean field model is reminiscent of a XY model with infinite‐range interactions. A linear stability analysis of the uniform distribution has been carried out by Kuramoto . We show that the stability analysis can also be carried out on the basis of Lyapunov's direct method. In doing so, we illustrate that the correspondence of Lyapunov's direct method and linear stability analysis that is well‐known for deterministic autonomous systems also holds for mean field models.