Related Experiment Video
Updated: Oct 2, 2025
04:35
Development of an Individual-Tree Basal Area Increment Model using a Linear Mixed-Effects Approach
Published on: July 3, 2020
3.4K
Lyapunov exponent in the Vicsek model.
L H Miranda-Filho1, T A Sobral2, A J F de Souza1
1Departamento de Física, Universidade Federal Rural de Pernambuco, Rua Manoel de Medeiros, s/n, Dois Irmãos, 52171-900, Recife, Brazil.
Physical Review. E
|February 23, 2022
Summary
This study quantifies chaotic behavior in self-propelled particle (SPP) systems using the largest Lyapunov exponent (LLE). A modified Vicsek model reveals a chaotic regime and noise-dependent LLE near phase transitions.
Area of Science:
- Physics
- Complex Systems
- Statistical Mechanics
Background:
- The Vicsek model is a standard for flocking dynamics in self-propelled particles (SPPs).
- Quantifying chaotic behavior in these systems has been a significant challenge.
- Existing models lack direct measures for chaos.
Purpose of the Study:
- To introduce a method for measuring chaotic behavior in Vicsek systems.
- To investigate the dynamical phase transition using the largest Lyapunov exponent (LLE).
- To propose a smooth Vicsek model for improved computational analysis.
Main Methods:
- Numerical computation of the largest Lyapunov exponent (LLE).
- Analysis of dynamical evolution in tangent space for up to two million SPPs.
- Development and application of a smooth Vicsek model to overcome computational discontinuities.
Main Results:
- Identification of a chaotic regime in the collective behavior of SPPs.
- The LLE was successfully computed, providing a direct measure of chaos.
- Observed significant changes in LLE dependence on noise near known Vicsek model transition points.
Conclusions:
- The largest Lyapunov exponent (LLE) provides a robust measure for chaotic dynamics in Vicsek systems.
- A smooth Vicsek model facilitates accurate computation of chaos.
- Noise acts as a critical control parameter influencing chaotic behavior and phase transitions.
Related Concept Videos
01:22Second Derivatives and Laplace Operator
1.8K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
Consider a scalar function. The curl of its...
1.8K
01:25Poisson's And Laplace's Equation
3.6K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
3.6K
01:20Region of Convergence of Laplace Tarnsform
750
The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This...
750
01:16Properties of Laplace Transform-II
311
Time differentiation, convolution, integration, and periodicity are fundamental concepts in analyzing functions and signals over time. Each concept provides a unique perspective on how functions evolve, interact, and repeat, offering essential tools for various scientific and engineering applications.
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...
311
01:22Definition of Laplace Transform
3.3K
The Laplace transform is an indispensable mathematical technique for simplifying the resolution of differential equations by converting them into more manageable algebraic expressions. The Laplace transform of a function is denoted by L[x(t)], where x(t) is the time-domain function. The laplace transform is mathematically expressed as
3.3K
01:26Linear Approximation in Frequency Domain
147
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....
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....
147