Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# 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 (doi: 10.20944/preprints201801.0277.v1). Chen, X. Numerical Counterexamples of Lorenz System in Implicit Time Scheme. Preprints 2018, 2018010277 (doi: 10.20944/preprints201801.0277.v1).

## Abstract

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.

## Keywords

Lorenz system; implicit time scheme; counterexamples

## Subject

MATHEMATICS & COMPUTER SCIENCE, Numerical Analysis & Optimization

Comment 1
Commenter:
The commenter has declared there is no conflict of interests.
Comment: This is a very interesting problem that making the initial conditions with all points except the ones on Z-axis and two attractors in X-Y-Z space, after a long simulation, as there is no points 'disappear'. Is this contrast to that in the three-dimensional space, the measure of any simple two-dimensional space (as the numerical solutions in explicit time scheme are converged to a curved surface (Lorenz 1963)) is 0, or in another word, can all points in X-Y-Z space except the ones on Z-axis and two attractors in X-Y-Z space be mapped to the special curved surface (the map is a surjection)?
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