Activities

Nonlinear Dynamics, Chaos and Integrable Systems

Nonlinear dynamics was born following the realization that the laws of nature can generate, under certain conditions, a variety of complex behaviors. The latter may be manifested in the form of a multiplicity of states, abrupt transitions, spatial patterns and temporal rhythms, propagating waves, or processes in which large scale regularities are intertwined with random looking episodes to which one refers as deterministic chaos. These phenomena are encountered in a wide variety of seemingly very different systems, from atomic and molecular physics to fluid dynamics, chemical kinetics, optics and materials science and are known to affect the functioning of processes of industrial interest. They also provide the natural archetype for understanding a large body of phenomena in branches which traditionally are outside the realm of physics in which emergence, cooperativity and evolution play a key role. Nonlinear science is thus a markedly interdisciplinary field, suggesting connections and encouraging cross-fertilization between different scientific branches.

The research in the broad field of nonlinear science is articulated around three major topics.

Fundamental aspects of nonlinear dynamics and chaos

The objective is to develop universal tools for classifying and characterizing the complex behavior generated by nonlinear systems. Local bifurcations and normal forms, global bifurcations particularly in connection with homoclinic orbits, periodic orbit theory, semi-classical quantization, and symbolic dynamics are being applied to abstract dynamical systems and models designed to interpret experimental data. Of special interest is the probabilistic approach to nonlinear systems, leading to interesting connections between the dynamical behavior observed and the spectral properties of the associated Liouville operator.

Dynamical systems and the foundations of statistical mechanics and thermodynamics

The ubiquity of dynamical chaos at the microscopic level has been at the origin of new insights on the microscopic basis of macroscopic level transport phenomena and other manifestations of irreversibility. Current subjects of interest include the connection between Lyapunov exponents and the viscosity and diffusion coefficients; the fractal structure of nonequilibrium states and of the associated hydrodynamic modes; and fluctuation theorems for the entropy production.

A second subtopic concerns nanoscale systems, in which the interaction between the thermodynamic fluctuations and/or the geometry of the surrounding medium interferes significantly with the evolution. Problems of current interest include the relaxation of a small quantum system in contact with one or several reservoirs; the analysis of Maxwell demon like devices such as ratchets or Brownian motors; the extension of thermodynamics to include fluctuations; and nonlinear dynamical processes on low-dimensional surfaces, fractal sets or dispersed media.

Integrable systems

The most familiar signature of integrability is the emergence, in a spatially extended system, of particle-like excitations referred to as solitons. Integrable systems are also essential as reference models about which perturbation approaches to study non integrable systems can be developed. Of special interest in this context are "nearly integrable" systems, which may exhibit a transition from integrability to chaos. On the other hand there is also the recent question of complex behaviour displayed by particular systems that are modeled by completely integrable equations (soliton instabilities and spider web solutions).

A comprehensive picture of an integrable system with infinitely many degrees of freedom is still lacking. The research program in this area includes the study of continuous integrable systems by a combination of analytic, geometric and algebraic methods; the extent to which this symmetry-based approach applies to discrete nonlinear lattices; and the development of a quantum theory of major soliton equations using the methods of quantum groups. In connection with this last issue further attention should be paid to the correspondence and contrast between classical and quantum many particle systems.

Associated Research teams

Center for Nonlinear Phenomena and Complex Systems (ULB)
TENA (VUB)