The objective of the school is to give a comprehensive, self-contained, and methodically unified survey of nonlinear dynamics in non-equilibrium systems. The emphasis is on analytical methods, based on qualitative theory of dynamical systems, bifurcation expansions and multiscale analysis. This will allow the participants to gain understanding of the various non-equilibrium structures: fronts, waves, solitons, periodic arrays, etc. The analysis will be complemented by hands-on modeling experience. The ultimate aim is to give the participants analytical and computational tools required in various applications of nonlinear dynamics in physics, chemistry and biology.
A new shift in the understanding of stochastic processes has
emerged in the last decades. Since they were first analyzed about one hundred
years ago (the pioneering works of Einstein, Langevin, Perrin, Smoluchovsky,
etc.), the noise terms in physical, chemical, engineering and biological systems
had commonly been considered to be a nuisance, something to be avoided or, in
those cases where the noise is intrinsic to the system, to be minimize as much
as possible. This point of view has changed recently with the discovery that in
nolinear systems noise can actually have a constructive effect that induces new
ordering phenomena. This change was originated in the already classic works on
stochastic resonance, originally aiming to explain the observed periodicity of
the Earth's ice ages as a subtle entanglement between a nonlinear climatic
model, the periodic changes in solar radiation due to the variations in the
ellipticity of the earth's orbit, and noise in the form of random variations of
the total solar emission. This work instantly opened a door. Many other
constructive effects have since been found, such as noise-induced transitions
and noise-induced phase transitions, noise-induced transport (Brownian motors,
ratchets, etc), noise-sustained patterns, synchronization induced by noise, etc.
In all the examples, the nonlinearities play an essential role. Many
applications of those phenomena have been found in different fields, mainly of
physical and biological interest.
Stochastic processes are usually not covered, or only very superficially, in the academic programs and therefore there is need to start at a low level in the training. The proposed course will address a wide audience, including, physicists, chemists, biologists, mathematicians, electrical engineers, etc. Since one of the major tools in this field is that of numerical simulations, special attention will be devoted to the application of numerical algorithms for stochastic processes. A computer network will be available.