TY  - JOUR
A1  - Omel'chenko, Oleh
T1  - Mathematical framework for breathing chimera states
JF  - Journal of nonlinear science
N2  - About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such patterns, called chimera states, have been the subject of intensive study but mostly in the stationary case when the coarse-grained system dynamics remains unchanged over time. Nonstationary coherence-incoherence patterns, in particular periodically breathing chimera states, were also reported, however not investigated systematically because of their complexity. In this paper we suggest a semi-analytic solution to the above problem providing a mathematical framework for the analysis of breathing chimera states in a ring of nonlocally coupled phase oscillators. Our approach relies on the consideration of an integro-differential equation describing the long-term coarse-grained dynamics of the oscillator system. For this equation we specify a class of solutions relevant to breathing chimera states. We derive a self-consistency equation for these solutions and carry out their stability analysis. We show that our approach correctly predicts macroscopic features of breathing chimera states. Moreover, we point out its potential application to other models which can be studied using the Ott-Antonsen reduction technique.
KW  - Coupled oscillators
KW  - Breathing chimera states
KW  - Coherence-incoherence
KW  - patterns
KW  - Ott-Antonsen equation
KW  - Periodic solutions
KW  - Stability
Y1  - 2022
U6  - https://doi.org/10.1007/s00332-021-09779-1
SN  - 0938-8974
SN  - 1432-1467
VL  - 32
IS  - 2
PB  - Springer
CY  - New York
ER  -