Summary
It is proposed to carry out an interdisciplinary research
project to analyse the effect of noise on the dynamics of multistable
chaotic systems, and to tackle the problem of control in such systems.
The results of the fundamental research will be applied to the analysis
of the dynamics of nonlinear optical devices and of cardiovascular flow.
The research programme is motivated by the fact that almost
all systems, whether in nature or in technology, are noisy and nonlinear.
Frequently, they are characterised by two or more coexisting attractors
that can often be chaotic. Analysis of the stochastic dynamics, and
the development of methods for controlling such systems, are long-standing
and challenging problems. They are of broad interdisciplinary interest
from the point of view of fundamental research, and of great importance
in view of a host of practical applications.
The remarkable progress achieved recently in the fields
of chaos control, synchronisation and fluctuational non-equilibrium
dynamics - in which the co-applicants have played not insignificant
roles - provides the necessary basis for the proposed research. In the
course of earlier collaborative investigations, the complementary expertise
of some of the participants has already been shown to provide a potent
combination for the solution of problems in chaotic and stochastic dynamics.
It is now envisaged that joint efforts, involving specialists
in fundamental and applied physics, will lead to a substantial breakthrough
in the understanding and control stochastic dynamics in multistable
chaotic systems. It is anticipated in particular that the investigation
of the statistical properties of fluctuational trajectories in chaotic
systems and maps, and in the analysis of synchronisation, will reveal
optimal paths underlying the processes of noise induced escape and inter-well
transitions. Their existence will allow one to consider stochastic dynamics
in such systems in general terms using the methods of nonlinear dynamics.
A knowledge of the optimal paths and corresponding optimal forces will
provide a solution of the energy-optimal switching problem in chaotic
multistable systems, and thus pave the way to the design of new methods
of chaos control.
The understanding of stochastic dynamics in multistable
chaotic systems thereby gained will then be applied immediately to analyse
stochastic dynamics in semiconductor lasers and in cardiovascular flow.
The fundamental part of the proposed research will be focused initially
on the detailed investigation of stochastic dynamics in continuous chaotic
systems with non-hyperbolic (CR4, CO, CR5, CR6) and quasi-hyperbolic
(CR5, CO, CR4, CR6) attractors, and weakly dissipative systems with
homoclinic tangles (CR8, CR1, CR6, CR3). The research will be then extended
to the analysis of the stochastic dynamics in chaotic maps (CR3, CR4,
CR5, CO). Another important extension is related to an analysis of the
effect of noise on synchronisation in a system of coupled oscillators
(CR7, CR5, CO, CR4). Attempts will also be made to extend the technique
to analyse the spatio-temporal dynamics of an ensemble of coupled FitzHugh
- Nagumo systems (CR7, CR6, CR4). The results of the fundamental research
will be applied to the analysis of the stochastic dynamics in semiconductor
laser with modulated current or delayed feedback (CR2, CR4, CR5). The
latter will open the way to analysis of the whole new class of systems
described by the delayed differential equations. The key ingredient
here is a new technique (CR2) of constructing a series of finite dimensional
analytical maps.
Finally, the topical and challenging problem of modelling
cardiovascular dynamics will be considered (CR9, CR5, CO, CR4, CR7).
The research will be based on the recent suggestion that the circulatory
system can usefully be modelled (CR9) as a system of five coupled, autonomous,
nonlinear oscillators. The results of the modelling will be compared
with the results of physiological experiments (CR9, CO).
