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Symmetry
Special Issues
Slow-Fast Dynamical Systems
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Slow-Fast Dynamical Systems

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (31 January 2022)

Special Issue Information

Dear Colleagues,

Slow–fast systems, i.e., nonlinear dynamic systems in which two or more variables are governed by very different time scales, appear in many branches of natural science. Examples are found in chemical systems (the most familiar of which is the Belousov–Zhabotinsky reaction), in models of the electrical activity of neuron and cardiac cells, population dynamics, plasma physics, lasers, and nonlinear optical systems, to name just a few. Mathematically, they are typically characterized by the existence of critical manifolds on which the slow dynamics takes place.

The geometry, stability and symmetry properties of these sets are at the origin of a wealth of dynamical phenomena including canard orbits, special trajectories following a repelling slow manifold for a considerable amount of time, and complex oscillatory patterns that can either be periodic or chaotic. This Special Issue is intended to collect contributions on slow–fast dynamics at mathematical, numerical, and experimental levels, with models and experiments ranging from biology to chemistry to physics.

Topics of interest include, but are not limited to, canard explosions and excitability in higher dimensions, bursting and mixed-mode oscillations, symmetries in the context of singular perturbation problems, symmetry-breaking phenomena, model-reduction techniques, etc. Both research and review papers are welcomed.

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • singularly-perturbed systems and symmetry
  • canard explosions
  • invariant manifolds
  • mixed-mode oscillations and bursting
  • networks
  • slow-fast dynamics and symmetries of the limiting problems
  • slow-fast Hamiltonian systems and slow evolution to symmetry

Published Papers (2 papers)

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Research

9 pages, 849 KiB  
Article
Slow Invariant Manifold of Laser with Feedback
by Jean-Marc Ginoux and Riccardo Meucci
Symmetry 2021, 13(10), 1898; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13101898 - 08 Oct 2021
Cited by 2 | Viewed by 1442
Abstract
Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant [...] Read more.
Previous studies have demonstrated, experimentally and theoretically, the existence of slow–fast evolutions, i.e., slow chaotic spiking sequences in the dynamics of a semiconductor laser with AC-coupled optoelectronic feedback. In this work, the so-called Flow Curvature Method was used, which provides the slow invariant manifold analytical equation of such a laser model and also highlights its symmetries if any exist. This equation and its graphical representation in the phase space enable, on the one hand, discriminating the slow evolution of the trajectory curves from the fast one and, on the other hand, improving our understanding of this slow–fast regime. Full article
(This article belongs to the Special Issue Slow-Fast Dynamical Systems)
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19 pages, 711 KiB  
Article
Evolution to Mirror-Symmetry in Rotating Systems
by Ferdinand Verhulst
Symmetry 2021, 13(7), 1189; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13071189 - 01 Jul 2021
Cited by 1 | Viewed by 1447
Abstract
A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and ω>0. The time-dependence produces slow [...] Read more.
A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and ω>0. The time-dependence produces slow evolution to discrete (mirror) symmetry in one of the degrees-of-freedom. This changes the dynamics drastically depending on the frequency ratio ω and the timescale of evolution. We analyse the cases ω=1,2,3 where the ratio’s 1,2 turn out to be the most interesting. In an initial phase we find 2 adiabatic invariants with changes near the end of evolution. A remarkable feature is the vanishing and emergence of normal modes, stability changes and strong changes of the velocity distribution in phase-space. The problem is inspired by the dynamics of axisymmetric, rotating galaxies that evolve slowly to mirror symmetry with respect to the galactic plane, the model formulation is quite general. Full article
(This article belongs to the Special Issue Slow-Fast Dynamical Systems)
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