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Research Interests of Dr. Liao in Nonlinear Dynamics
(1) Clean Numerical Simulation (CNS)
This is a fine numerical technique to precisely solve chaotic dynamic systems: even the numerical noises due to truncation and round-off errors can be neglected and the chaotic solutions are reliable in a considerably long interval.
The CNS is based on the high-order Taylor expansion with high-accurate data. For example, by means of the CNS and using 400th-order Taylor expansion and data in accuracy of 800-digit placed to the right of a decimal point, Liao obtained, for the first time, a reliable chaotic solution of Lorenz equation in a rather long interval 0 < t < 1100. Such kind of high-precision data is possible by means of either computer algebra systems such as Mathematica and Maple, or Multiple Precision Arithmetic Library.
The CNS provides us a powerful tool to gain very accurate, reliable chaotic solutions in a considerably ling interval. For details, please refer to the article
Liao, S.J.: On the reliability of computed chaotic solutions of non-linear differential equations. Tellus, 61A:550 – 564 (2009) [ arXiv:0901.2986 ]
(2) Chaos: A bridge from micro-level uncertainty to macroscopic randomness
It is well-known that chaotic dynamic systems have the so-called sensitive dependence on initial condition (SDIC). Due to the SDIC and the impossibility of our measuring initial conditions in arbitrary-precision, the long-term prediction of chaotic dynamic systems is impossible - this is the famous Òbutterfly-effectÓ of chaos.
However, Lorenz equation is a simplified model from Navier-Stokes equations that is based on the so-called continuum-assumption: fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. From the viewpoint of the continuum-assumption, the physical variables such as velocity, temperature, mass density and so on, are statistical quantities and therefore have inherent statistical fluctuation: seriously speaking, the initial conditions of Lorenz equation contain inherent micro-level uncertainty, which has nothing to do with the measuring ability of our human being at all.
In general, such kind of micro-level uncertainty is much smaller than truncation and round-off errors of traditional numerical techniques. Thus, traditional numerical methods are too inaccurate to investigate the propagation of such kind of micro-level uncertainty of a chaotic dynamic system with the SDIC. Fortunately, the propagation of such kind of micro-level uncertainty can be precisely investigated by means of the CNS: using Lorenz equation as an example, it is found for the first time that, due to the SDIC of a chaotic dynamic system, the micro-level uncertainty of initial conditions transfers into the macroscopic randomness. Thus, chaos might be a bridge from the micro-level uncertainty to macroscopic randomness. For details, please refer to:
Liao, S.J.: Chaos – A bridge from micro-level uncertainty to macroscopic randomness. Commun. Nonlin. Sci. Numer. Simula. Accepted (online available) [ arXiv.1108.4472 ]
Note that chaos of Lorenz equation has close relationship to the turbulence. So, we suggest that the turbulence is such a kind of flows of fluid, which are so instable that micro-level uncertainty transfers into macroscopic randomness. For detailed discussion, please refer to the above-mentioned article.