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Discontinuity, Nonlinearity, and Complexity 2(4) (2013) 345--355 | DOI:10.5890/DNC.2013.11.004

Peng Yang$^{2}$, Zhiliang Lin$^{2}$, Shijun Liao$^{1}$,$^{2}$,$^{3}$,$^{4}$

$^{1}$ State Key Laboratory of Ocean Engineering

$^{2}$ School of Naval Architecture, Ocean and Civil Engineering

$^{3}$ Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China

$^{4}$ Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia

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Abstract

The method of the so-called “Clean Numerical Simulation” (CNS) is applied to gain reliable chaotic solutions of Saltzman’s dynamic system, a simplified model for convection flows of fluid. Based on the high-order Taylor series method with data in multiple precision library and a validation check of global reliability of result, the CNS provides us a practical way to gain reliable, accurate enough solutions of chaotic dynamic systems in a finite but long enough time interval. The parallel computation is used to greatly increase the computational efficiency. The numerical noises of the CNS can be controlled to be so small that even the influence of the micro-level inherent uncertainty of initial conditions can be investigated in details. It is found that the micro-level inherent physical uncertainty (i.e. the unavoidable statistical fluctuation of temperature and velocity of fluid) of initial conditions of chaotic Saltzman’s system transfers into macroscopic randomness. This suggests that chaos might be a bridge between micro-level inherent physical uncertainty and macroscopic randomness. The current work illustrates that the above conclusion holds not only for Lorenz equation with three ODEs but also for Saltzman’s equation with up-to nine ODEs, and thus has general meanings.

Acknowledgments

Thanks to Dr. P.F. Wang for his kindly assistance on multiple precision library and parallel computation. This work is partly supported by the State Key Lab of Ocean Engineering (Approval No. GKZD010056-6) and the National Natural Science Foundation of China (Approval No. 11272209).

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