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Related Concept Videos

Oscillations In An LC Circuit01:30

Oscillations In An LC Circuit

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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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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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Consider designing an oscillator circuit, a crucial component in various electronic devices and systems. The objective is to create an oscillator circuit with specific characteristics: a damped natural frequency of 4 kHz and a damping factor of 4 radians per second. To accomplish this, a parallel RLC circuit is employed, known for its ability to sustain oscillations at a resonant frequency. In this case, the damping factor is pivotal in achieving the desired performance.
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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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Fabrication and Testing of Microfluidic Optomechanical Oscillators
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Recent advances in coupled oscillator theory.

Bard Ermentrout1, Youngmin Park2, Dan Wilson3

  • 1Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|October 29, 2019
PubMed
Summary

This study extends weak coupling theory for oscillators to non-smooth systems, like the Izhikevich neuron. It introduces isostable reduction to explain complex behaviors beyond standard weak coupling paradigms.

Keywords:
isostable coordinatesphase reductionweakly coupled oscillators

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Area of Science:

  • Dynamical systems theory
  • Computational neuroscience
  • Nonlinear dynamics

Background:

  • The standard theory of weakly coupled oscillators effectively models smooth systems.
  • However, its application is limited in non-smooth systems and complex scenarios.
  • Existing models struggle to explain certain emergent behaviors in coupled dynamical systems.

Purpose of the Study:

  • To review and extend the theory of weakly coupled oscillators.
  • To investigate its limitations and applications in non-smooth systems.
  • To introduce and apply isostable reduction for explaining complex dynamical behaviors.

Main Methods:

  • Review of existing theory for weakly coupled oscillators.
  • Extension of the theory to accommodate non-smooth systems.
  • Application of isostable reduction techniques.
  • Analysis of the Izhikevich neuron model and coupled neuron systems.

Main Results:

  • The study demonstrates how to extend weak coupling theory for non-smooth systems.
  • Isostable reduction is introduced as a method to analyze behaviors not captured by standard theory.
  • The framework successfully explains bifurcations in coupled neuron stability.

Conclusions:

  • Weak coupling theory can be extended to non-smooth systems, particularly for models like the Izhikevich neuron.
  • Isostable reduction provides a powerful tool for understanding complex dynamics in coupled oscillators.
  • This approach offers new insights into phenomena like bifurcations in neural networks.