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Related Concept Videos

Damped Oscillations01:07

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In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
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Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so...
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An idealized LC circuit of zero resistance can oscillate without any source of emf by shifting the energy stored in the circuit between the electric and magnetic fields. In such an LC circuit, if the capacitor contains a charge q before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is given by
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An RLC circuit combines a resistor, inductor, and capacitor, connected in a series or parallel combination.
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A household microwave and lasers are examples of standing electromagnetic waves in a cavity. When two conducting metal plates are placed parallel at the nodal planes, it creates a cavity where standing waves are formed. The cavity between the two planes is analogous to a stretched string held at the points x = 0 and x = L. Here, the distance 'L' between the two planes must be an integer multiple of half of the wavelength. The wavelengths that satisfy this condition are given by:
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Generation and Coherent Control of Pulsed Quantum Frequency Combs
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Emergent Spaces for Coupled Oscillators.

Thomas N Thiem1, Mahdi Kooshkbaghi2, Tom Bertalan3

  • 1Department of Chemical and Biological Engineering, Princeton University, Princeton, NJ, United States.

Frontiers in Computational Neuroscience
|June 13, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a data-driven method using manifold learning to discover simplified variables and equations for complex dynamical systems. This approach successfully models coupled oscillators and can be extended to effective parameters and neuronal networks.

Keywords:
Kuramoto oscillatorscomplex networkscoupled systemsdiffusion mapsgeometric harmonicsmanifold learningneural networks

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Area of Science:

  • Complex Systems Science
  • Dynamical Systems Theory
  • Network Science

Background:

  • Coupled dynamical systems exhibit emergent behaviors often simplified via coarse-graining.
  • Traditional coarse-graining requires deep system knowledge and experience.
  • Discovering appropriate coarse variables and their dynamics is challenging.

Purpose of the Study:

  • Develop a systematic, data-driven approach for discovering coarse variables.
  • Learn evolution equations for these coarse variables.
  • Identify effective parameters for multi-parameter models.

Main Methods:

  • Manifold learning algorithms for coarse variable discovery.
  • Artificial neural networks templated on numerical integrators for learning dynamics.
  • Application to the Kuramoto phase oscillator model.
  • Analysis of Hodgkin-Huxley type neurons with Chung-Lu network.

Main Results:

  • Successfully identified a coarse variable equivalent to the Kuramoto order parameter.
  • Learned accurate ordinary differential equations (ODEs) for coarse variable dynamics.
  • Demonstrated discovery of effective parameters for complex models.
  • Validated the approach on neuronal network synchronization.

Conclusions:

  • The proposed methodology provides an integrated suite of tools for data-driven coarse-graining.
  • Enables discovery of coarse variables, effective parameters, and coarse-grained equations.
  • Offers a powerful framework for analyzing complex dynamical systems from observational data.