Catherine Nicolis

Foundations of Complex Systems

An approach to Complexity from the perspective of fundamental science is outlined, drawing on the cross-fertilization of concepts and tools from nonlinear dynamics, statistical physics, probability and information theories, data analysis and numerical simulation. Emphasis is placed on the intertwining between different levels of description and on the probabilistic dimension of complex systems, in connection with the issue of prediction.

Bifurcation and predictability analysis of a low-order atmospheric circulation model

A comprehensive bifurcation analysis of a low-order atmospheric circulation model is carried out. It is shown that the model admits a codimension-2 saddle-node-Hopf bifurcation. The principal mechanisms leading to the appearance of complex dynamics around this bifurcation are described and various routes to chaotic behavior are identified, such as the transition through the period doubling cascade, the breakdown of an invariant torus and homoclinic bifurcations of a saddle-focus. Non-trivial limit sets in the form of a chaotic attractor or a chaotic repeller are found in some parameter ranges. Their presence implies an enhanced unpredictability of the system for parameter values corresponding to the winter season.

Foundations of Complex Systems: Emergence, Information and Prediction

This book provides a self-contained presentation of the physical and mathematical laws governing complex systems. Complex systems arising in natural, engineering, environmental, life and social sciences are approached from a unifying point of view using an array of methodologies such as microscopic and macroscopic level formulations, deterministic and probabilistic tools, modeling and simulation. This book can be used as a textbook by graduate students, researchers and teachers in science, as well as non-experts who wish to have an overview of one of the most open, markedly interdisciplinary and fast-growing branches of present-day science.

Chaotic dynamics, Markov partitions, and Zipf's law

A chaotic dynamics model creating Markovian strings of symbols as well as sequences of “words” is presented, and its possible relevance to Zipfs law is discussed.

Enhancement of the nucleation of protein crystals by the presence of an intermediate phase: a kinetic model

The role of an intermediate state in the kinetics of the passage between an initial and a final state is analyzed using a generic model involving one-order parameter and a sixth-order Landau-type potential. Applied to the problem of nucleation of protein crystals the model shows that the rate of nucleation can be enhanced in a large part of the state diagram by the presence of an intermediate fluid phase. Some conditions under which this enhancement can be optimized are brought out. These predictions are compared to the results of recent experimental and theoretical investigations on protein crystallization.

Extreme value distributions in chaotic dynamics

A theory of extremes is developed for chaotic dynamical systems and illustrated on representative models of fully developed chaos and intermitent chaos. The cumulative distribution and its associated density for the largest value occurring in a data set, for monotonically increasing (or decreasing) sequences, and for local maxima are evaluated both analytically and numerically. Substantial differences from the classical statistical theory of extremes are found, arising from the deterministic origin of the underlying dynamics.

Efficiency of encounter-controlled reaction between diffusing reactants in a finite lattice

Numerically-exact values of the mean walk length before first encounter for reactants diffusing on a lattice are calculated by generalizing a Markovian method introduced previously. The approach developed is illustrated by presenting results for two random walkers diffusing simultaneously on a square-planar lattice; the results obtained are compared with those calculated assuming one of the reactant partners is stationary. In almost all cases, it is found that the former process is the more efficient. Analytic arguments are presented and Monte Carlo studies are carried out to support the results obtained.

Pattern formation and fluctuation-induced transitions in protein crystallization

A kinetic model of protein crystallization accounting for the nucleation stage, the growth and competition of solid particles and the formation of macroscopic patterns is developed. Different versions are considered corresponding successively, to a continuous one-dimensional crystallization reactor, a coarse grained two-box model and a model describing the evolution of the space averaged values of fluid and solid material. The analysis brings out the high multiplicity of the patterns. It provides information on their stability as well as on the kinetics of transitions between different states under the influence of the fluctuations.

Nonlinear dynamics and the two-slit delayed experiment

Abstract The two-slit delayed experiment is re-examined from the standpoint of nonlinear dynamics in the presence of multiple attractors and fractal basin boundaries. It is suggested that the results may be interpreted as the response of the underlying system to a temporary switch of one control parameter, rather than as a retroaction between this system and the observer.

From Short-Scale Atmospheric Variability to Global Climate Dynamics: Toward a Systematic Theory of Averaging

Abstract Traditionally, climate is defined by the properties of the averages of the meteorological fields over an appropriate time interval. In this paper the properties of the time-averaged observables of a red noise atmosphere and of a simplified model of thermal convection are investigated both analytically and numerically and are compared to those of the original finescale variables. It is shown that averaging tends to reduce the domain of variability and the attractor dimension favors persistence of initial correlations and enhances predictability. The implications of these findings in the real-world atmosphere are briefly assessed.