Related Experiment Video
Updated: May 26, 2026
06:44
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
Published on: September 23, 2025
Lyapunov exponents for multi-parameter tent and logistic maps
1School of Computing and Mathematics, University of Ulster, Newtownabbey, Northern Ireland.
Chaos (Woodbury, N.Y.)
|January 10, 2012
Summary
This study analyzes the behavior of logistic and tent maps with iteration-dependent parameters. We present analytic and numerical results for the global Lyapunov exponent, revealing insights into chaotic dynamics.
Area of Science:
- Dynamical Systems and Chaos Theory
- Nonlinear Dynamics
- Computational Physics
Background:
- The logistic and tent maps are fundamental models in chaos theory, often studied with constant control parameters.
- Understanding the influence of varying parameters on map behavior is crucial for exploring complex dynamical systems.
- Lyapunov exponents are key indicators of chaotic behavior, quantifying the rate of separation of nearby trajectories.
Purpose of the Study:
- To investigate the dynamics of logistic and tent maps when the control parameter varies with the iteration number.
- To derive analytic and numerical results for the global Lyapunov exponent under these non-autonomous conditions.
- To analyze the parameter space for chaotic behavior in multi-parameter tent maps.
Main Methods:
- Analytic derivation of the global Lyapunov exponent for the tent map with iteration-dependent parameters.
- Numerical simulations to compute the global Lyapunov exponent for the logistic map with iteration-dependent parameters.
- Fractional calculation of parameter space exhibiting positive global Lyapunov exponents for N-parameter tent maps.
Main Results:
- Analytic results for the global Lyapunov exponent were obtained for the tent map.
- Numerical results for the global Lyapunov exponent were obtained for the logistic map.
- The fraction of parameter space yielding positive global Lyapunov exponents was calculated for N-parameter tent maps.
Conclusions:
- Iteration-dependent parameters significantly alter the behavior of logistic and tent maps.
- The study provides a quantitative understanding of chaotic dynamics in non-autonomous systems.
- The findings contribute to the analysis of complex systems with time-varying control elements.
Related Concept Videos
01:14Parametric Survival Analysis: Weibull and Exponential Methods
Parametric survival analysis models survival data by assuming a specific probability distribution for the time until an event occurs. The Weibull and exponential distributions are two of the most commonly used methods in this context, due to their versatility and relatively straightforward application.
Weibull Distribution
The Weibull distribution is a flexible model used in parametric survival analysis. It can handle both increasing and decreasing hazard rates, depending on its shape parameter...
Weibull Distribution
The Weibull distribution is a flexible model used in parametric survival analysis. It can handle both increasing and decreasing hazard rates, depending on its shape parameter...
01:26Exponential Equations for Modeling Growth
Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is the relative...
01:29Exponential Equations with Logarithms: Problem Solving
In ecological studies, exponential models are often used to predict how populations grow over time under favorable conditions. These models assume that the growth rate is proportional to the current population, leading to continuous and compounding increases.The model expresses the population as a function of time, combining the initial population with a growth factor raised to an exponent involving the growth rate and time. To estimate how long it takes for a population to reach a specific...
01:29Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
01:20Region of Convergence of Laplace Tarnsform
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 substitution...
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 substitution...
01:28Types of Functions III
Logarithmic and piecewise functions play central roles in mathematical modeling, particularly when capturing nonlinear or segmented behaviors in real-world phenomena. Although these functions differ fundamentally in structure and application, both serve to represent complex relationships in simplified mathematical terms.A logarithmic function is defined as the inverse of an exponential function, expressed as These functions grow quickly for small values of x but slow down as x increases,...