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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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In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
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Quantum State Engineering of Light with Continuous-wave Optical Parametric Oscillators
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Phase description of stochastic oscillations.

Justus T C Schwabedal1, Arkady Pikovsky1

  • 1Department of Physics and Astronomy, Potsdam University, 14476 Potsdam, Germany.

Physical Review Letters
|August 29, 2014
PubMed
Summary

We introduce invariant isophases to describe noisy oscillations, providing a global phase variable even when deterministic phases are undefined. This method aids analysis of irregular oscillations and systems with stochastic switching or excitable behavior.

Area of Science:

  • Nonlinear dynamics
  • Statistical physics
  • Complex systems analysis

Background:

  • Stochastic oscillations are prevalent in various scientific fields.
  • Traditional phase descriptions often fail for noisy or irregular systems.
  • Generalizing phase concepts is crucial for understanding complex dynamics.

Purpose of the Study:

  • To develop a robust phase description for stochastic oscillations.
  • To introduce invariant isophases as a generalized phase concept.
  • To provide a method for analyzing systems where phase is ill-defined.

Main Methods:

  • Generalizing standard isophases to define average isophases.
  • Constructing isophases as state-space sections with constant mean first return time.

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  • Applying numerical methods to systems with noise-induced phenomena.

    • Main Results:

    • A global phase variable for noisy oscillations is obtained.
    • The method successfully describes noise-induced switching between limit cycles.
    • It also characterizes noise-induced oscillations in excitable systems.
    • A procedure for determining isophases in observed irregular oscillations is presented.

    Conclusions:

    • Invariant isophases offer a powerful tool for analyzing stochastic oscillations.
    • This approach refines phase description in data analysis of complex systems.
    • The method is applicable to diverse systems exhibiting irregular or noise-driven dynamics.