Theory and Applications in Nonlinear Oscillators: 2nd Edition

A special issue of Dynamics (ISSN 2673-8716).

Deadline for manuscript submissions: 30 September 2025 | Viewed by 744

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Laboratory of Nonlinear Systems, Circuits & Coplexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece
Interests: electrical and electronics engineering; mathematical modeling; control theory; engineering, applied and computational mathematics; numerical analysis; mathematical analysis; numerical modeling; modeling and simulation; robotics
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Special Issue Information

Dear Colleagues,

Oscillations are fundamental to numerous physical, chemical, biological, and mechanical systems and play a vital role in many applications. Recent research has extensively explored oscillations and vibrations, with a particular focus on the intriguing characteristics of nonlinear oscillations. These oscillations can elucidate complex phenomena and offer solutions to mechanical, electrical, and other challenges. Emerging scientific fields, such as nonlinear targeted energy transfer and hidden oscillations, highlight the dynamic nature of this research area.

This Special Issue aims to create a platform for scientists to share their latest developments, discoveries, and advancements in both the theoretical and practical aspects of nonlinear oscillators. Topics that will be covered include nonlinear oscillations, hidden attractors, energy transfer, bifurcation theory, mathematical modeling of nonlinear oscillators, synchronization and chaos control, nonlinear circuits, mechanical applications in oscillations, and more.

Dr. Christos Volos
Guest Editor

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Keywords

  • bifurcation theory
  • chaos
  • control
  • dynamical systems
  • energy transfer
  • hidden attractors
  • mathematical modeling
  • mechanical applications
  • nonlinear circuits
  • nonlinear oscillations
  • synchronization

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Published Papers (1 paper)

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Research

16 pages, 367 KB  
Article
Generalized Miller Formulae for Quantum Anharmonic Oscillators
by Maximilian T. Meyer and Arno Schindlmayr
Dynamics 2025, 5(3), 34; https://doi.org/10.3390/dynamics5030034 - 28 Aug 2025
Viewed by 168
Abstract
Miller’s rule originated as an empirical relation between the nonlinear and linear optical coefficients of materials. It is now accepted as a useful tool for guiding experiments and computational materials discovery, but its theoretical foundation had long been limited to a derivation for [...] Read more.
Miller’s rule originated as an empirical relation between the nonlinear and linear optical coefficients of materials. It is now accepted as a useful tool for guiding experiments and computational materials discovery, but its theoretical foundation had long been limited to a derivation for the classical Lorentz model with a weak anharmonic perturbation. Recently, we developed a mathematical framework which enabled us to prove that Miller’s rule is equally valid for quantum anharmonic oscillators, despite different dynamics due to zero-point fluctuations and further quantum-mechanical effects. However, our previous derivation applied only to one-dimensional oscillators and to the special case of second- and third-harmonic generation in a monochromatic electric field. Here we extend the proof to three-dimensional quantum anharmonic oscillators and also treat all orders of the nonlinear response to an arbitrary multi-frequency field. This makes the results applicable to a much larger range of physical systems and nonlinear optical processes. The obtained generalized Miller formulae rigorously express all tensor elements of the frequency-dependent nonlinear susceptibilities in terms of the linear susceptibility and thus allow a computationally inexpensive quantitative prediction of arbitrary parametric frequency-mixing processes from a small initial dataset. Full article
(This article belongs to the Special Issue Theory and Applications in Nonlinear Oscillators: 2nd Edition)
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