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Numerical Counterexamples of Lorenz System in Implicit Time Scheme
Version 1 : Received: 29 January 2018 / Approved: 30 January 2018 / Online: 30 January 2018 (04:38:31 CET)
How to cite: Chen, X. Numerical Counterexamples of Lorenz System in Implicit Time Scheme. Preprints 2018, 2018010277. https://doi.org/10.20944/preprints201801.0277.v1. Chen, X. Numerical Counterexamples of Lorenz System in Implicit Time Scheme. Preprints 2018, 2018010277. https://doi.org/10.20944/preprints201801.0277.v1.
In nonlinear self-consistent system, Lorenz system (Lorenz equations) is a classic case with chaos solutions which are sensitively dependent on the initial conditions. As it is difficult to get the analytical solution, the numerical methods and qualitative analytical methods are widely used in many studies. In these papers, Runge-Kutta method is the one most often used to solve these differential equations. However, this method is still a method based on explicit time scheme, which would be the main reason for the chaotic solutions to Lorenz system. In this work, numerical experiments based on implicit time scheme and explicit scheme are setup for comparison, the results show that: in implicit time scheme, the numerical solutions (counterexamples) are without chaos; for an original volume, the volume shrinks exponentially fast to 0 in common.
Lorenz system; implicit time scheme; counterexamples
MATHEMATICS & COMPUTER SCIENCE, Numerical Analysis & Optimization
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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