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Nonlinear dynamics of rapidly driven systems

Afshin Besharat* and Alexander A. Penin†

  • *Contact author: abeshara@ualberta.ca
  • †Contact author: penin@ualberta.ca
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Phys. Rev. E 114, 054213 – Published 10 November, 2026

DOI: https://doi.org/10.1103/46tw-51sh

Abstract

We consider systems characterized by the presence of a rapidly oscillating force. A general method is presented for the construction of the effective action governing the large-scale nonlinear dynamics of such systems order by order in inverse powers of the oscillation frequency ω. The explicit expression for the effective Lagrangian is derived up to O(1/ω6) next-to-next-to-leading approximation in the strong-drive regime. The general structure of the high-frequency expansion reveals a broad class of nonlinear systems whose transition curves are identical to those of the linear Mathieu equation, which enables a fully nonperturbative stability analysis in the case of strong driving and nonlinearity. The method is generalized to velocity-dependent forces and configuration space with curvature, characteristic to systems with constraints. Several applications are discussed in detail, including the dynamical magnetic trapping of electric charges.

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Phys. Rev. E 114, 054213– Published 10 November, 2026
  • Received 30 July 2026
  • Accepted 21 October 2026
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