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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 400^{th}-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.