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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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Stochastic Noise Application for the Assessment of Medial Vestibular Nucleus Neuron Sensitivity In Vitro

Stochastic Noise Application for the Assessment of Medial Vestibular Nucleus Neuron Sensitivity In Vitro

Published on: August 28, 2019

Noise-sustained fluctuations in stochastic dynamics with a delay.

Paolo D'Odorico¹, Francesco Laio, Luca Ridolfi

¹Department of Environmental Sciences, University of Virginia, 291 McCormick Road, Charlottesville, Virginia 22904-4123, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|June 12, 2012

Summary

Noise can surprisingly enhance and sustain periodic oscillations in systems with delays. This study explores how random drivers interact with time delays, revealing conditions for these noise-induced dynamics in linear systems.

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Area of Science:

Mathematical modeling
Dynamical systems theory
Stochastic processes

Background:

Delayed responses are common in natural and engineered systems.
Time delays can cause oscillations and instabilities, even in linear systems.
The interplay between random drivers (noise) and time delays is not well understood.

Purpose of the Study:

To investigate if noise can induce or sustain periodic oscillations in systems with time delays.
To identify conditions governing the emergence and disappearance of these noise-induced dynamics.

Main Methods:

Analysis of a linear delayed stochastic differential equation.
Inclusion of both additive and multiplicative noise terms.
Examination of system dynamics under varying noise intensities and delay parameters.

Main Results:

Noise can significantly enhance and prolong transient periodic oscillations present in deterministic delayed systems.
Specific conditions related to noise type (additive/multiplicative) and system parameters influence the emergence and persistence of these oscillations.
Demonstration of noise-induced transient periodicity in a linear framework.

Conclusions:

The interaction of noise and time delays can lead to emergent oscillatory behaviors.
Understanding these dynamics is crucial for modeling systems with inherent delays and random fluctuations.
This research provides a foundation for further studies on noise-induced phenomena in complex delayed systems.