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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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The arithmetic mean is usually skewed towards the larger values in the data set. Therefore, to avoid this inherent bias towards smaller values, the harmonic mean is used.
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Eddy currents can produce significant drag on motion, called magnetic damping. For instance, when a metallic pendulum bob swings between the poles of a strong magnet, significant drag acts on the bob as it enters and leaves the field, quickly damping the motion.
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Related Experiment Video

Updated: Apr 2, 2026

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Noise-enhanced stickiness in the Harper map.

J R Homan1, J D Meiss1

  • 1Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309-0526, USA.

Chaos (Woodbury, N.Y.)
|April 1, 2026
PubMed
Summary

Noise in dynamical systems alters the Poincaré recurrence statistic (PRS). Perturbations allow trajectories to enter previously inaccessible regions, leading to longer return times and modified PRS decay patterns.

Area of Science:

  • Dynamical systems
  • Statistical mechanics
  • Chaos theory

Background:

  • The Poincaré recurrence statistic (PRS) describes the probability of a system returning to its initial state.
  • In deterministic systems, PRS decay is often power-law due to 'stickiness' near invariant structures.
  • Understanding recurrence is crucial for predicting system behavior over time.

Purpose of the Study:

  • To investigate the effect of noise on the Poincaré recurrence statistic in dynamical systems.
  • To analyze how noise perturbs trajectories and influences recurrence times.
  • To elucidate the mechanisms behind altered PRS decay in the presence of noise.

Main Methods:

  • Analysis of the Poincaré recurrence statistic (PRS) under noise perturbations.

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  • Study of the Harper map as a representative dynamical system.
  • Comparison of trapping, visit, and recurrence times.
  • Modeling with a finite-state Markov chain.

    • Main Results:

    • Noise allows trajectories to access previously inaccessible regions, enhancing trapping.
    • Noisy PRS exhibits an extended intermediate-time tail before asymptotic exponential decay.
    • Recurrence times are significantly influenced by noise-induced entry into islands.
    • Markov models confirm noise's role in causing slower PRS decay.

    Conclusions:

    • Noise fundamentally alters the recurrence properties of dynamical systems.
    • The enhanced trapping effect due to noise leads to observable changes in the PRS.
    • Noise-induced transitions between regions are key to understanding modified recurrence statistics.