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Related Concept Videos

Forced Oscillations01:06

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When an oscillator is forced with a periodic driving force, the motion may seem chaotic. The motions of such oscillators are known as transients. After the transients die out, the oscillator reaches a steady state, where the motion is periodic, and the displacement is determined.
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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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Types of Damping01:20

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If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...
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If a driven oscillator needs to resonate at a specific frequency, then very light damping is required. An example of light damping includes playing piano strings and many other musical instruments. Conversely, to achieve small-amplitude oscillations as in a car's suspension system, heavy damping is required. Heavy damping reduces the amplitude, but the tradeoff is that the system responds at more frequencies. Speed bumps and gravel roads prove that even a car's suspension system is not...
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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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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
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Dynamics of weakly coupled parametrically forced oscillators.

P Salgado Sánchez1, J Porter1, I Tinao1

  • 1Escuela Técnica Superior de Ingeniería Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Plaza de Cardenal Cisneros 3, 28040 Madrid, Spain.

Physical Review. E
|September 15, 2016
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Summary
This summary is machine-generated.

This study explores coupled parametric oscillators, revealing how symmetries and forcing phases dictate instabilities. Simulations and experiments confirm Hopf and saddle-node heteroclinic bifurcations in modulated cross waves.

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Area of Science:

  • Nonlinear Dynamics
  • Oscillations and Waves
  • Fluid Dynamics

Background:

  • Parametric oscillators exhibit complex dynamics near instability thresholds.
  • Coupling between oscillators introduces rich phenomena dependent on symmetries and forcing.
  • Understanding these instabilities is crucial for predicting system behavior.

Purpose of the Study:

  • To investigate the dynamics of two weakly coupled parametric oscillators near primary subharmonic instability.
  • To analyze the influence of permutation symmetries and forcing phases on instability nature.
  • To map detailed bifurcation sets and compare model predictions with experimental data.

Main Methods:

  • Analysis of coupled parametric oscillators in the vicinity of subharmonic instability.
  • Calculation of detailed bifurcation sets, including Bogdanov-Takens points.
  • Comparison of theoretical predictions with direct numerical simulations and experimental results on modulated cross waves.

Main Results:

  • Instability nature critically depends on remaining permutation symmetries and relative phases of forcing terms.
  • Complex transition series organized by Bogdanov-Takens points were identified.
  • Hopf bifurcation and subsequent saddle-node heteroclinic bifurcation were confirmed for out-of-phase forcing.

Conclusions:

  • The study elucidates the critical role of symmetry and phase in coupled parametric oscillator dynamics.
  • The findings provide a detailed understanding of bifurcations in such systems.
  • Experimental validation confirms the theoretical model's accuracy for modulated cross waves.