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Symmetry
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Symmetry Breaking
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Symmetry Breaking

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 March 2015) | Viewed by 37290

Special Issue Editor

Prof. Dr. Antonio Palacios

E-Mail Website
Guest Editor
Department of Mathematics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, USA
Interests: applied mathematics; bifurcations; symmetries; pattern formation and coupled nonlinear oscillators

Special Issue Information

Dear Colleagues,

The past two decades have seen an explosion of ideas and methods to study nonlinear dynamical systems with symmetry. These systems arise naturally at various length scales: in molecular dynamics, in animal gaits, in pattern-forming systems, in neural networks, underwater vehicle dynamics, in magnetic- and electric-field sensors, particle physics, in gyroscopic and navigational systems, hydroelastic rotating systems and boats/ships, and in complex systems such as telecommunication infrastructures and power grids. Motivated by the diversity of these and many other applications, we intend to dedicate a special issue of Symmetry to publish original research articles and/or comprehensive review articles on the phenomenon of Symmetry-Breaking. It is indeed well known that symmetry can restrict the type of solutions of systems of ordinary and partial differential equations, which often serve as models of systems that change in space and time, i.e., nonlinear dynamical systems. So it is reasonable to expect that certain aspects of the collective behavior of a system can be inferred from the presence of symmetry and from the symmetries that are broken when parameters change. Such scenario, which is commonly known as symmetry-breaking bifurcation, underlies the behavior of many natural and artificial systems and will be the central theme of this special issue of Symmetry. Research articles will include theoretical and experimental works addressing methods and results in the study on symmetry-breaking bifurcations and their applications. Interdisciplinary studies across various disciplines are also welcome.

Professor Antonio Palacios
Guest Editor

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

  • Symmetry-breaking
  • bifurcation theory
  • nonlinear dynamical systems
  • equivariant bifurcations
  • spatio-temporal dynamics
  • complex systems

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Published Papers (5 papers)

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Research

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234 KiB  
Article
Integrable (2 + 1)-Dimensional Spin Models with Self-Consistent Potentials
by Ratbay Myrzakulov, Galya Mamyrbekova, Gulgassyl Nugmanova and Muthusamy Lakshmanan
Symmetry 2015, 7(3), 1352-1375; https://doi.org/10.3390/sym7031352 - 3 Aug 2015
Cited by 58 | Viewed by 5446
Abstract
Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 [...] Read more.
Integrable spin systems possess interesting geometrical and gauge invariance properties and have important applications in applied magnetism and nanophysics. They are also intimately connected to the nonlinear Schrödinger family of equations. In this paper, we identify three different integrable spin systems in (2 + 1) dimensions by introducing the interaction of the spin field with more than one scalar potential, or vector potential, or both. We also obtain the associated Lax pairs. We discuss various interesting reductions in (2 + 1) and (1 + 1) dimensions. We also deduce the equivalent nonlinear Schrödinger family of equations, including the (2 + 1)-dimensional version of nonlinear Schrödinger–Hirota–Maxwell–Bloch equations, along with their Lax pairs. Full article
(This article belongs to the Special Issue Symmetry Breaking)
4654 KiB  
Article
Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator
by Ian Stewart
Symmetry 2014, 6(1), 23-66; https://doi.org/10.3390/sym6010023 - 3 Jan 2014
Cited by 15 | Viewed by 7504
Abstract
The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor [...] Read more.
The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Ζ2 x Ζ2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks. Full article
(This article belongs to the Special Issue Symmetry Breaking)
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837 KiB  
Article
Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points
by Guillaume Vasseur, Yannick Fagot-Revurat, Bertrand Kierren, Muriel Sicot and Daniel Malterre
Symmetry 2013, 5(4), 344-354; https://doi.org/10.3390/sym5040344 - 9 Dec 2013
Cited by 8 | Viewed by 12074
Abstract
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the [...] Read more.
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group. Full article
(This article belongs to the Special Issue Symmetry Breaking)
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Graphical abstract

821 KiB  
Article
Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations (ODEs): Equivariant Degree Approach
by Zalman Balanov, Wieslaw Krawcewicz, Zhichao Li and Mylinh Nguyen
Symmetry 2013, 5(4), 287-312; https://doi.org/10.3390/sym5040287 - 7 Nov 2013
Cited by 12 | Viewed by 5157
Abstract
In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree [...] Read more.
In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries. Full article
(This article belongs to the Special Issue Symmetry Breaking)
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Review

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1681 KiB  
Review
Symmetry-Breaking as a Paradigm to Design Highly-Sensitive Sensor Systems
by Antonio Palacios, Visarath In and Patrick Longhini
Symmetry 2015, 7(2), 1122-1150; https://doi.org/10.3390/sym7021122 - 19 Jun 2015
Cited by 5 | Viewed by 6307
Abstract
A large class of dynamic sensors have nonlinear input-output characteristics, often corresponding to a bistable potential energy function that controls the evolution of the sensor dynamics. These sensors include magnetic field sensors, e.g., the simple fluxgate magnetometer and the superconducting quantum interference device [...] Read more.
A large class of dynamic sensors have nonlinear input-output characteristics, often corresponding to a bistable potential energy function that controls the evolution of the sensor dynamics. These sensors include magnetic field sensors, e.g., the simple fluxgate magnetometer and the superconducting quantum interference device (SQUID), ferroelectric sensors and mechanical sensors, e.g., acoustic transducers, made with piezoelectric materials. Recently, the possibilities offered by new technologies and materials in realizing miniaturized devices with improved performance have led to renewed interest in a new generation of inexpensive, compact and low-power fluxgate magnetometers and electric-field sensors. In this article, we review the analysis of an alternative approach: a symmetry-based design for highly-sensitive sensor systems. The design incorporates a network architecture that produces collective oscillations induced by the coupling topology, i.e., which sensors are coupled to each other. Under certain symmetry groups, the oscillations in the network emerge via an infinite-period bifurcation, so that at birth, they exhibit a very large period of oscillation. This characteristic renders the oscillatory wave highly sensitive to symmetry-breaking effects, thus leading to a new detection mechanism. Model equations and bifurcation analysis are discussed in great detail. Results from experimental works on networks of fluxgate magnetometers are also included. Full article
(This article belongs to the Special Issue Symmetry Breaking)
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