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Damped Oscillations

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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.
Although friction and other non-conservative...

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The starting point for expressing the modes of standing waves is understanding the boundary conditions that the waves must follow. The boundary conditions are derived from the physical understanding of how the standing waves are sustained, that is, how the vibrating particles of the medium behave at the boundaries imposed on them.
For a tube open at one end and closed at the other filled with air, the modes are such that there is always an antinode at the open end and a node at the closed end.

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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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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 because...

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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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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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Induction of Microstreaming by Nonspherical Bubble Oscillations in an Acoustic Levitation System

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Characterizing mixed mode oscillations shaped by noise and bifurcation structure.

Peter Borowski¹, Rachel Kuske, Yue-Xian Li

¹Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada. peterphysik@gmail.com

Chaos (Woodbury, N.Y.)

|January 5, 2011

Summary

This study introduces new measures to differentiate between neuronal models generating mixed mode oscillations (MMOs). These methods analyze subthreshold dynamics to classify MMO mechanisms, aiding in understanding complex neuronal behavior.

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Area of Science:

Neuroscience
Computational Biology
Dynamical Systems

Background:

Neuronal systems often exhibit mixed mode oscillations (MMOs), characterized by small amplitude oscillations interspersed with spikes.
Both deterministic and stochastic models exist for MMO generation, but distinguishing their underlying mechanisms is challenging.
Stochastic models can produce MMOs via noise-driven mechanisms distinct from deterministic routes.

Purpose of the Study:

To develop and present a suite of quantitative measures for distinguishing between different models and classifying routes to MMO generation.
To analyze the influence of model parameters, resets, and return mechanisms on MMO dynamics.
To provide a novel approach using noise levels to differentiate model types and MMO mechanisms.

Main Methods:

Focusing on subthreshold oscillations, analysis includes interspike interval density, amplitude trends, and a coherence measure.
Measures were developed and tested on a biophysical model for stellate cells and a FitzHugh-Nagumo-type model.
Application to related models and exploration of noise level as a distinguishing factor.

Main Results:

The developed measures effectively distinguish between different MMO-generating models and classify underlying mechanisms.
Analysis revealed the significant influence of model parameters and reset/return mechanisms on MMO characteristics.
Noise level was identified as a key factor in differentiating model types and MMO generation routes.

Conclusions:

The suite of measures provides a robust framework for analyzing and classifying MMOs in computational models.
These methods can be applied to experimental time series to elucidate underlying dynamical structures.
The approach allows for the exploitation of intrinsic or extrinsic noise to reveal system dynamics.