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Once the fields have been calculated using Maxwell's four equations, the Lorentz force equation gives the force that the fields exert on a charged particle moving with a certain velocity. The Lorentz force equation combines the force of the electric field and of the magnetic field on the moving charge. Maxwell's equations and the Lorentz force law together encompass all the laws of electricity and magnetism. The symmetry that Maxwell introduced into his mathematical framework may not be...
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The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
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Large Scale Energy Efficient Sensor Network Routing Using a Quantum Processor Unit
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Recent advances in symmetric and network dynamics.

Martin Golubitsky1, Ian Stewart2

  • 1Mathematical Biosciences Institute, Ohio State University, Columbus, Ohio 43210, USA.

Chaos (Woodbury, N.Y.)
|October 3, 2015
PubMed
Summary
This summary is machine-generated.

This study reviews three decades of research on symmetric dynamical systems and pattern formation. Symmetry-breaking is key to understanding pattern creation in physical and biological systems.

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Area of Science:

  • Dynamical Systems Theory
  • Pattern Formation
  • Symmetry Analysis

Background:

  • Over 30 years of research on symmetric dynamical systems and networks.
  • Focus on pattern formation, where constraints alter generic phenomena.
  • Exploration of time-periodic states, mode interactions, and non-compact symmetry groups.

Purpose of the Study:

  • To summarize key findings in symmetric dynamical systems and networks over three decades.
  • To highlight the role of symmetry-breaking in pattern formation.
  • To review applications in physical and biological systems.

Main Methods:

  • Analysis of dynamics and bifurcations in symmetric systems.
  • Application of equivariant Hopf bifurcation and the H/K theorem.
  • Investigation of mode interactions and systems with non-compact symmetry groups.

Main Results:

  • Symmetry-breaking is crucial for pattern creation.
  • Equivariant dynamics and mode interactions organize bifurcations.
  • Networks of coupled systems exhibit patterns through synchrony and phase relations.

Conclusions:

  • Symmetry analysis provides a powerful framework for understanding pattern formation.
  • Methods developed for single systems are transferable to networks.
  • Applications span diverse fields from fluid dynamics to biological processes.