Collective Dynamics of Deterministic and Noisy Oscillator Populations: Beyond Ott-Antonsen Theory

• Applicant: Professor Dr. Arkady Pikovsky
• Subject Area: Statistical Physics, Nonlinear Dynamics, Complex Systems, Soft and Fluid Matter, Biological Physics
• Term: from 2019 to 2025
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 405856192

• Final Report Year: 2023

Final Report Abstract

Populations of coupled oscillators are widely studied in different areas of physics and chemistry (arrays of lasers and of Josephson junctions, coupled electrochemical, spintronic, and nanomechanical oscillators), but they are also relevant for living systems (coupled yeast cells, gene-manipulated clocks in bacteria) and even for social phenomena (pedestrian synchrony on footbridges, synchronous hand clapping in opera houses). One of the promising approaches is to reduce the dynamics of a large number of coupled units to a few equations for relevant global variables (order parameters). Several such reductions have been described in the literature, and the project’s goal was to extend these methods to novel areas, for example, including the possibility of accounting for the effects of external noise on the oscillators. The main achievements are twofold. First, we demonstrated that Cauchy-distributed noise acting on the units allows for a significant simplification of the dynamical equations compared to the Gaussian noise. For such systems, we developed an exact description of possible stationary solutions for arbitrary multi-harmonic coupling between the oscillators (previously, only single-harmonic coupling was treated). Second, we developed an exact dynamical reduction for populations of oscillators with a distribution of natural frequencies and a singleharmonic coupling, valid for arbitrary initial conditions. These results open novel perspectives in analytical and numerical studies of oscillator ensembles.

Publications

• Low-dimensional description for ensembles of identical phase oscillators subject to Cauchy noise. Physical Review E, 102(5).
  Tönjes, Ralf & Pikovsky, Arkady
  
• Synchronization of oscillators with hyperbolic chaotic phases. Izvestiya VUZ. Applied Nonlinear Dynamics, 29(1).
  Arkady Pikovsky
  
• Exact finite-dimensional reduction for a population of noisy oscillators and its link to Ott–Antonsen and Watanabe–Strogatz theories. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(11).
  Cestnik, Rok & Pikovsky, Arkady
  
• Hierarchy of Exact Low-Dimensional Reductions for Populations of Coupled Oscillators. Physical Review Letters, 128(5).
  Cestnik, Rok & Pikovsky, Arkady
  
• Exact finite-dimensional description for networks of globally coupled spiking neurons. Physical Review E, 107(2).
  Pietras, Bastian; Cestnik, Rok & Pikovsky, Arkady
  
• Integrability of a Globally Coupled Complex Riccati Array: Quadratic Integrate-and-Fire Neurons, Phase Oscillators, and All in Between. Physical Review Letters, 132(5).
  Cestnik, Rok & Martens, Erik A.