Background & Justification for Undertaking the Project

The dynamics of real systems, whether in nature or in technology, are often characterised by a number of coexisting states (some of which may be chaotic) and by the presence of fluctuations. Examples are numerous and include lasers [1], the cardiovascular system [9.4, 9.5], and chemical reactions [2]. The long-time evolution of such systems is dominated by transitions between different states [3.5, 0.4].
An understanding of stochastic dynamics and of methods of control in multistable systems is a long-standing and challenging problem of broad interdisciplinary interest, both from the point of view of fundamental research and in view of practical applications.

The inherent difficulties of tackling these questions are manifest. The prediction ofbifurcations and phase diagrams are still largely open problems, even in such relatively simple systems such as the driven Duffing oscillator [3]. The analysis is further complicated by the presence of chaos and fractal boundaries of the basins of attraction. Fluctuations are inevitable in real systems and introduce qualitatively new features to their dynamics [4, 5.4]. The different levels of complexity make it especially difficult to model the dynamics of real systems that display simultaneously multistability, chaos and fluctuations. For example building a dynamical model of cardiovascular system represents an extremely complex and as yet unsolved problem [9.4, 9.5].

Substantial progress has been achieved in recent years in the investigation of fluctuations and chaos, and the co-applicants have played a not insignificant role in this. The idea of stabilisation of the unstable periodic orbits embedded within the chaotic attractor [9a] has led to a major breakthrough in the theory of the control of chaos [9b]. In multistable systems noise induces hopping between the coexisting (now metastable) chaotic and nonchaotic states. Thus, the task of controlling complexity in multistable systems goes far beyond the simple control-of-chaos idea. As well as stabilising the system on one of the attractors, one needs to control switching between them. Thus far, new switching strategies have been restricted to the possibility of linearising the flow in the neighbourhood of the stable state [3.5]. The efficiency of switching can be significantly increased by the use of targeting methods [12] or through the new energy-optimal switching technique [4.10].

The notion of synchronisation [13ab] provides a general approach to the understanding of complex behaviour in oscillatory systems. It has recently been extended to systems of interacting chaotic oscillators [13c]. The phenomenon of synchronisation underlies a variety of modern techniques for chaos control and secure communications. In many cases the effect of noise on synchronisation can be considered in terms of stochastic phase dynamics in a multistable potential. Such an approach can also be applied to the analysis of synchronisation in coupled non-identical oscillators modelling the cardiovascular system.

The Hamiltonian theory of fluctuations [6] offers [0.1, 0.4] deep insights into the stochastic dynamics of non-equilibrium systems. The central idea is that, in the course of a fluctuation, the system follows very closely one of the deterministic (noise-free) optimal paths [6b]. The latter are physically observable and have recently been shown [0.4] to underlie fluctuational escape from chaotic attractors [5.4] and to provide a solution to the energy-optimal control problem of switching from such attractors [4.10]. Methods for the analysis of fluctuational paths have recently been extended to encompass the analysis of noise-induced transitions in mutistable systems [1.2].

It was proven at an earlier stage of the collaboration that the complementary expertise of researchers in the analysis of chaos, synchronisation, and fluctuations can be very effectively combined for the solution of problems in chaotic and stochastic dynamics (see the numerous joint publications in the reference list). It is now envisaged that an extension and enlargement of the collaboration will lead to substantial breakthroughs in the understanding and control of complexity in stochastic and chaotic multistable systems. Such collaboration will also pave the way to the application of the results of the fundamental research to the analysis of the stochastic dynamics of nonlinear optical devices and of cardiovascular flow.

The fundamental research will be focused on detailed investigations of the fluctuational escape problem in continuous chaotic systems with non-hyperbolic and quasi-hyperbolic attractors, on potential systems with homoclinic tangles, and on noise-induced inter-well transitions in these systems. The research will be extended to the analysis of stochastic dynamics in chaotic maps and the influence of noise on synchronisation in systems of coupled oscillators. Attempts will also be made to investigate the effect of noise on spatio-temporal dynamics in extended systems.

Applied research will be focused on the development of new control strategies for chaotic multistable systems that can take account of a large number of coexisting attractors for a given set of parameters [3.5]. These strategies will combine stabilisation and targeting techniques with the energy-optimal control of switching between coexisting states. The new techniques will be applied to the analysis of the control problem in semiconductor lasers.

The latter problem is particularly topical in the context of new laser-based communication technologies. It has long been known that lasers display a rich variety of nonlinear phenomena including mutistability and chaos. Recently it has been shown that lasers operating in a chaotic regime can be used e.g. for secure communications [1d]. On the other hand it is well known that fluctuational transitions limit substantially the application of semiconductor lasers in communications [1b]. It is therefore proposed to investigate multistable chaotic dynamics, noise-induced transitions between different states, and new methods of control in semiconductor lasers. The key ingredients here will be a new technique [2.1, 2.4] for the construction of analytic maps for the analysis of laser dynamics, and new methods for the analysis of fluctuational transitions in semiconductor lasers [1bc].

Finally a challenging, important and highly topical problem of nonlinear dynamical modelling will be considered: cardiovascular flow. Numerous experiments show evidence of inherently nonlinear dynamics and determinism in the time series characterising cardiovascular flow [10]; it is evident that random fluctuations clearly play a role, but one that is not yet understood. The physiological origin of the nonlinearities, and the clinical significance of various nonlinear characteristic parameters, remains a subject of intensive research. The low frequency spectral components in blood-flow have recently been resolved, using improved data acquisition and time-series analysis techniques [9.4, 9.5]. These studies suggest that the circulatory system can usefully be modelled as a system of five coupled, autonomous, nonlinear oscillators corresponding respectively to cardiac rhythm, respiration, myogenic activity, neurogenic activity and endothelial signalling. It is therefore proposed to model the dynamical properties of cardiovascular flow with a system of five coupled oscillators and, in particular, to investigate the extent to which the model parameters correlate with abnormalities in the blood-flow in a cohort of patients with differing severities of heart failure.