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Ott-Antonsen attractiveness for parameter-dependent oscillatory systems.

Bastian Pietras1, Andreas Daffertshofer1

  • 1Faculty of Behavioural and Movement Sciences, MOVE Research Institute Amsterdam and Institute for Brain and Behavior Amsterdam, Vrije Universiteit Amsterdam, van der Boechorststraat 9, Amsterdam 1081 BT, The Netherlands.

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The Ott-Antonsen ansatz is extended to show that systems with parameter-dependent dynamics converge to the Ott-Antonsen manifold. This provides mathematical support for realistic oscillatory systems and neural networks.

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Area of Science:

  • Complex systems
  • Nonlinear dynamics
  • Mathematical physics

Background:

  • The Ott-Antonsen (OA) ansatz simplifies analysis of large coupled oscillator systems.
  • It assumes phase dynamics are independent of specific oscillators.
  • Its applicability to systems with oscillator-specific parameters is an open question.

Purpose of the Study:

  • To investigate the attractiveness of the OA manifold for systems with intrinsic parameter dependence.
  • To provide a rigorous mathematical framework for extended OA ansatz applications.
  • To confirm and mathematically underpin recent numerical observations.

Main Methods:

  • Extension of the Ott-Antonsen ansatz.
  • Mathematical proof of convergence for parameter-dependent oscillatory systems.
  • Analysis of systems with time-dependent and multi-dimensional parameters.

Main Results:

  • Demonstrated that parameter-dependent oscillatory systems converge to the OA manifold under specific conditions.
  • Provided mathematical underpinning for the OA ansatz in theta neuron networks.
  • Extended the validity of the OA ansatz to more complex scenarios.

Conclusions:

  • The extended OA ansatz is robust and applicable to a wider range of realistic systems.
  • Confirms convergence to the OA manifold even with oscillator-specific parameters.
  • Establishes the OA ansatz as a valuable tool for analyzing complex oscillatory networks.