Related Experiment Video
Updated: Jun 6, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
Covariant Lyapunov vectors for rigid disk systems.
Hadrien Bosetti1, Harald A Posch
1Computational Physics Group, Faculty of Physics, University of Vienna, Boltzmanngasse 5, A-1090 Wien, Austria.
Chemical Physics
|December 15, 2010
Summary
Computer simulations reveal Lyapunov instability in a 2D hard-disk system. Covariant perturbation vectors are transversal, indicating a hyperbolic system with transverse stable and unstable manifolds.
Area of Science:
- Statistical Mechanics
- Dynamical Systems Theory
- Computational Physics
Background:
- Lyapunov instability is a key indicator of chaos in dynamical systems.
- Understanding the behavior of hard-disk systems is fundamental in statistical mechanics.
- Previous studies often focused on simpler models or smaller system sizes.
Purpose of the Study:
- To investigate Lyapunov instability in a 2D hard-disk system using extensive computer simulations.
- To analyze the Oseledec splitting and the properties of covariant perturbation vectors.
- To determine the nature of stable and unstable manifolds in this system.
Main Methods:
- Extensive computer simulations of a 2D hard-disk system with periodic boundary conditions.
- Computation of the full set of covariant perturbation vectors in tangent space.
- Analysis of the Oseledec splitting and Lyapunov spectrum.
Main Results:
- Lyapunov modes parallel to the x-axis were observed in the large system.
- Covariant perturbation vectors were found to be transversal but not generally orthogonal.
- The probability of vanishing angles between adjacent Lyapunov exponents approaches zero.
Conclusions:
- The 2D hard-disk system exhibits Lyapunov instability.
- The system is hyperbolic, characterized by transverse stable and unstable manifolds.
- The computed covariant vectors provide insights into the system's chaotic dynamics.
More Related Videos
Related Concept Videos
01:08Equation of Rotational Dynamics
Angular variables are introduced in rotational dynamics. Comparing the definitions of angular variables with the definitions of linear kinematic variables, it is seen that there is a mapping of the linear variables to the rotational ones. Linear displacement, velocity, and acceleration have their equivalents in rotational motion, which are angular displacement, angular velocity, and angular acceleration. Similar to the rotational variables, a mapping exists from Newton's second law of motion...
01:23Curvilinear Motion: Rectangular Components
Curvilinear motion characterizes the movement of a particle or object along a curved path, notably evident when envisioning a car navigating a winding road. If the car starts at point A, its position vector is established within a fixed frame of reference, where the ratio of the position vector to its magnitude signifies the unit vector pointing in the position vector's direction.
As the car advances, its position evolves over time. Quantifying the car's velocity involves computing the time...
As the car advances, its position evolves over time. Quantifying the car's velocity involves computing the time...
01:17Dynamics Of Circular Motion: Applications
Suppose a car moves on flat ground and turns to the left. The centripetal force causing the car to turn in a circular path is due to friction between the tires and the road. For this, a minimum coefficient of friction is needed, or the car will move in a larger-radius curve and leave the roadway. Let's now consider banked curves, where the slope of the road helps in negotiating the curve. The greater the angle of the curve, the faster one can take the curve. It is common for race tracks for...
01:21Equations of Motion: Rectangular Coordinates and Cylindrical Coordinates
Understanding the motion of particles is a fundamental aspect of classical mechanics, and the choice of the coordinate system plays a pivotal role in unraveling the complexities of their dynamics.
When a particle moves relative to an inertial frame, the equations of motion can be expressed using rectangular components. If the motion is confined to the x-y plane, the equations having the x and y coordinates only can be used to simplify the mathematical representation.
However, when particles...
When a particle moves relative to an inertial frame, the equations of motion can be expressed using rectangular components. If the motion is confined to the x-y plane, the equations having the x and y coordinates only can be used to simplify the mathematical representation.
However, when particles...
01:19Euler Equations of Motion
Imagine a rigid body that is rotating at an angular velocity of ω within an inertial frame of reference. Along with this, picture a second rotating frame that is attached to the body itself. This frame moves along with the body and possesses an angular velocity of Ω. The total moment about the center of mass is calculated by adding the rate of change of angular momentum about the center of mass in relation to the rotating frame and the cross-product of the body's angular velocity and its...
01:30Dynamics of Circular Motion
An object undergoing circular motion, like a race car, is accelerating because it is changing the direction of its velocity. This centrally directed acceleration is called centripetal acceleration. This acceleration acts along the radius of the curved path (thus is also referred to as radial acceleration).
Any acceleration must be produced by some force. Therefore, any force or combination of forces can cause centripetal acceleration. A few examples include the tension in the rope on a...
Any acceleration must be produced by some force. Therefore, any force or combination of forces can cause centripetal acceleration. A few examples include the tension in the rope on a...