February 1 --- May 31, 2004
Duan Jinqiao (duan@iit.edu) Illinois
Institute of Technology, Chicago, USA
Jing Zhujun (jingzj@math.ac.cn) The
Chinese Academy of Sciences
Lu Kening (klu@math.msu.edu) Michigan
State University, USA
Shang Zaijiu (zaijiu@math.ac.cn) The Chinese Academy of Sciences
Ye Xiangdong (yexd@ustc.edu.cn) The University
of Science and Technology of China
This programme seeks to inspire new research on current issues in nonlinear and
stochastic dynamical systems. Lecturers and participants (including young
researchers and graduate students) will discuss research topics on:
Nonautonomous dynamical systems, random dynamical systems, ergodic
theory,
stochastic partial differential equations, infinite dimensional dynamical
systems, invariant manifolds, bifurcation, and applications to
engineering and science.
Session 2: Ergodic and Topological Dynamical Systems (April 15---May 31, 2004)
Session 3: Applied Dynamical Systems (February 1---March 15, 2004)
It has long been clear that it is indispensable to take nonlinearity into
account,
in order to better understand complex systems. There is a growing recognition
of a role for the inclusion of nonautonomous terms and stochastic terms in the
modeling of complex, multi-scale phenomena in scientific and engineering
systems.
Taking nonautonomous and stochastic effects into account is of central
importance for the
development of mathematical models of these systems.
Macroscopic models in the form of ordinary/partial differential
equations contain
such nonautonomous terms or randomness as time-dependent forcing, stochastic
forcing, uncertain parameters, random sources or inputs, and time-dependent or
random initial and boundary conditions. Nonlinear stochastic or nonautonomous
ordinary/partial differential
equations are appropriate models for nonautonomously or randomly influenced
systems.
The addition of such terms has led to interesting new mathematical problems
at the interface between two fields among probability theory, dynamical
systems, bifurcation, topology, and numerical analysis.
The central issues in nonlinear and stochastic dynamical systems include
irregular behavior,
invariant measures, ergodic theory, invariant manifolds and attractors,
averaging principle and effective reduction, efficient numerical simulation,
and bifurcation. The core questions in the modeling, analysis, simulation
and prediction of complex engineering and scientific systems under uncertainty
include: exploring appropriate ways to take stochastic effects
into account; understanding the impact of randomness on the evolution of
complex systems;
and designing efficient numerical algorithms to simulate random phenomena.
During the last decade significant progress has been made towards building a
comprehensive
theory of nonautonomous and random dynamical systems. However, some
fundamental
issues remain unsolved, such as, a random dynamics approach for general
stochastic partial
differential equations; interactions among noise, nonlinearity, multiple scales
and instability;
random pattern formation; and efficient numerical methods for stochastic
partial differential
equations.
There have been some promising applications of nonlinear and stochastic
dynamics to many
fields of engineering and science, such as environmental fluid dynamics,
geophysical flows, climate dynamics, electronical and telecommunication
systems, and finance,
to name just a few. On the other hand, the applications of these
random dynamics ideas to
engineering and scientific problems have not yet been fully explored. In fact
such an endeavor
is still in its infancy. It appears that it would be a timely effort to foster
such promising
interaction of the theory of nonautonomous and stochastic dynamical systems and
these applied fields. It is expected that such interaction may lead to new
advances in the theory of nonlinear and stochastic dynamical systems. For
example, problems arising in the context of climate change and geophysical
flows have inspired interesting research topics about the interaction between
noise and multiple scales, and statistics and dynamics (ergodicity).
This programme seeks to build bridges between the theoretical and applied
communities by
promoting themes of common interest in emerging areas of nonlinear and
stochastic
dynamics, and to educate graduate students, postdoctoral fellows and young
researchers for working in these emerging areas.