The Numerical Connection between Map and its Differential Equation: Logistic and Other Systems
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Marcos Roberto de Lima
Abstract
The logistic map, in this article, is used to show the propagation of the numerical error for numerically solved logistic ODE's. When this equation is solved numerically (discretization procedure) the solution shows a chaotic behavior by varying the control parameter (integration step-size). However, the logistic ODE dx/dt = (k − 1)x − kx2 also has an analytical solution, i.e., not presenting the signature of chaos. The connection between the two limits, analytical solution and chaos, allows us to establish a region of acceptability for the numerical error. Therefore the limits of the numerical process can be found in between the limits of the analytical and chaotic solutions.
This study can be extended to any other ODE or map. For example, there is a similar numerical connection between the chaotic bi-dimensional Cat Map with its respective integrable ODE, which has an analytical solution.
Our discussion applied to those particular systems is generic, that is, the solutions obtained from the numerical discretization can be more instable than those obtained from the continuous procedure. The important point is to find a compromise between the step-size and the propagation length in order to obtain the best numerical solution since the instabilities can be unavoidable.
©2013 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Vapor Films under Influence of High Heat Fluxes: Nongravity Surface Waves and Film Explosive Disintegration
- Dynamic Constitutive Behavior and Fracture of Lanthanum Metal Subjected to Impact Compression at Different Temperatures and Impact Tension
- An Application of the Modified Reductive Perturbation Method to a Generalized Boussinesq Equation
- Switching Design for the Asymptotic Stability and Stabilization of Nonlinear Uncertain Stochastic Discrete-time Systems
- A Novel Two-fluid Model for the Identification of Possible Multiple Solutions in Slightly Inclined Pipelines
- Adaptive Synchronization of Filippov Systems with Unknown Parameters via Sliding Mode Control
- Chebyshev Split-step Scheme for the Sine-Gordon Equation in 2+1 Dimensions
- The Numerical Connection between Map and its Differential Equation: Logistic and Other Systems
- Exact Travelling Wave Solutions for the Modified Fornberg-Whitham Equation
Artikel in diesem Heft
- Frontmatter
- Vapor Films under Influence of High Heat Fluxes: Nongravity Surface Waves and Film Explosive Disintegration
- Dynamic Constitutive Behavior and Fracture of Lanthanum Metal Subjected to Impact Compression at Different Temperatures and Impact Tension
- An Application of the Modified Reductive Perturbation Method to a Generalized Boussinesq Equation
- Switching Design for the Asymptotic Stability and Stabilization of Nonlinear Uncertain Stochastic Discrete-time Systems
- A Novel Two-fluid Model for the Identification of Possible Multiple Solutions in Slightly Inclined Pipelines
- Adaptive Synchronization of Filippov Systems with Unknown Parameters via Sliding Mode Control
- Chebyshev Split-step Scheme for the Sine-Gordon Equation in 2+1 Dimensions
- The Numerical Connection between Map and its Differential Equation: Logistic and Other Systems
- Exact Travelling Wave Solutions for the Modified Fornberg-Whitham Equation
