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Oscillations about an Equilibrium Position

Oscillations about an Equilibrium Position

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...

Damped Oscillations

Damped Oscillations

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...

Types of Damping

Types of Damping

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...

BIBO stability of continuous and discrete -time systems

BIBO stability of continuous and discrete -time systems

System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.

Cyclic Processes And Isolated Systems

Cyclic Processes And Isolated Systems

A thermodynamic system with zero heat exchange and work is an isolated system. For these systems, the internal energy remains constant.
In the case of a non-isolated system, the change in the internal energy is zero only if the process is cyclic. A thermodynamic process is considered cyclic if the system undergoes a series of changes and returns to its initial state.
Consider a cyclic process that returns to its initial state, undergoing a four-step process. The heat transfer along each path...

Forced Oscillations

Forced Oscillations

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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Related Experiment Video

Updated: Jun 6, 2026

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

An Analog Macroscopic Technique for Studying Molecular Hydrodynamic Processes in Dense Gases and Liquids

Published on: December 4, 2017

Sustained oscillations for density dependent Markov processes.

Peter H Baxendale¹, Priscilla E Greenwood

¹Department of Mathematics, University of Southern California, Los Angeles, CA, USA. baxendal@usc.edu

Journal of Mathematical Biology

|November 16, 2010

Summary

Stochastic simulations reveal sustained oscillations in biological systems, unlike deterministic models. These oscillations follow a circular path modulated by a specific random process, explaining observed phenomena beyond expected noise levels.

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Area of Science:

* Mathematical Biology
* Computational Biology
* Systems Biology

Background:

* Deterministic models of bio-systems can exhibit damped oscillations.
* Stochastic counterparts of these models often display sustained oscillations.
* Identifying the underlying stochastic process in simulations is challenging.

Purpose of the Study:

* To characterize the stochastic process underlying sustained oscillations in bio-systems.
* To explain why stochastic models show persistent oscillations beyond noise levels.
* To provide a general limiting description of stochastic paths in such systems.

Main Methods:

* Analysis of local linear structure of deterministic dynamics.
* Derivation of a general limiting behavior for stochastic paths.
* Numerical simulations of established bio-system models.

Main Results:

* Stochastic paths exhibit circular motion modulated by a slowly varying Ornstein-Uhlenbeck process.
* This characterization explains sustained oscillations above expected noise.
* Validated using Volterra predator-prey, Sel'kov glycolysis, and damped linear oscillator models.

Conclusions:

* A unified theoretical framework for understanding stochastic oscillations in bio-systems.
* Provides insight into the dynamics of epidemics and biochemical pathways.
* Highlights the importance of stochasticity in biological modeling.