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Related Concept Videos

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Multimachine Stability01:25

Multimachine Stability

Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
In analyzing the system, the nodal equations represent the relationship between bus voltages, machine voltages, and machine currents. The nodal equation is given by:
Transfer Function to State Space01:23

Transfer Function to State Space

State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...

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Related Experiment Video

Updated: May 29, 2026

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline
10:44

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline

Published on: December 7, 2021

High-accuracy approximation of binary-state dynamics on networks.

James P Gleeson1

  • 1MACSI, Department of Mathematics & Statistics, University of Limerick, Ireland.

Physical Review Letters
|September 10, 2011
PubMed
Summary
This summary is machine-generated.

Master equations accurately approximate binary-state dynamics on random networks. Standard theories emerge as approximations, with applications in disease spread and spin models.

Related Experiment Videos

Last Updated: May 29, 2026

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline
10:44

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline

Published on: December 7, 2021

Area of Science:

  • Statistical Physics
  • Network Science
  • Epidemiology

Background:

  • Binary-state dynamics, such as the susceptible-infected-susceptible (SIS) model and Glauber spin dynamics, are fundamental in understanding complex systems.
  • These dynamics often occur on random networks, posing challenges for analytical treatment.
  • Master equations provide a powerful framework for describing the time evolution of probabilities in such systems.

Purpose of the Study:

  • To demonstrate that master equations accurately approximate binary-state dynamics on random networks.
  • To show that standard mean-field and pairwise theories arise as approximations to master equations.
  • To apply this framework to calculate critical phenomena in epidemic and spin models.

Main Methods:

  • Formulating master equations for binary-state dynamics on random networks.
  • Deriving standard mean-field and pairwise theories as approximate solutions to the master equations.
  • Applying the master equation approach to calculate SIS epidemic thresholds and critical points of nonequilibrium spin models.

Main Results:

  • Master equations provide accurate approximations for binary-state dynamics on random networks.
  • Standard mean-field and pairwise theories are shown to be specific approximations of the master equation solutions.
  • The master equation approach successfully calculates key quantities like SIS epidemic thresholds and critical points.

Conclusions:

  • Master equations offer a robust and accurate method for analyzing binary-state dynamics on complex networks.
  • This unified approach reconciles standard theories with a more fundamental description, providing deeper insights.
  • The demonstrated applications highlight the utility of master equations in diverse fields, from epidemiology to statistical mechanics.