CSSS 2008 Argentina-Readings

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CSSS Argentina 2008

Week 1 (tentative)

Definition of chaos; examples in various fields

An extended example: the logistic map. Introduce: bifurcations; bifurcation diagram and its structure, incl. Feigenbaum number; fractals and their connection to chaos

Continuous-time dynamics: definition

Introduce concepts: state variables, state space, trajectory, initial condition, transient, attractor, basin of attraction, fixed point, stability, bifurcation, parameter

An extended example: the Lorenz system: history, physical meaning, trajectories, attractors, bifurcations (examples & definitions), types of attractors, stability: definition & mathematics, eigen. , un/stable manifolds, Lyapunov exponent and the connection to chaos

Numerical solvers: roles and issues


Projection vs section

Poincare sections in space & time

Delay-coordinate embedding

Examples: roulette, the SFI competition

Applications: filtering, control of chaos, synchronization & communication, spacecraft orbits, chaos in the solar system, harnessing the butterfly effect in fluids

Supporting Material:

Time Series Analysis - E. Bradley
Error in Numerical Methods - E. Bradley
Numerical Solution for Differential Equations - E. Bradley

Week 2


-G. Nicolis and C. Nicolis, Foundations of Complex Systems, World Scientific, Singapore (2007).
-W. Ebeling and G. Nicolis, Word frequency and entropy of symbolic sequences: a dynamical perspective, Chaos Solitons and Fractals 2, 635 (1992).
-G. Nicolis and P. Gaspard, Toward a probabilistic approach to complex systems, Chaos Solitons and Fractals 4, 41 (1994).
-G. Nicolis, Thermodynamics today, Physica A213, 1 (1995).
-G. Nicolis and D. Daems, Probabilistic and thermodynamic aspects of dynamical systems, Chaos 8, 311 (1998).
-G. Nicolis, Nonequilibrium Statistical Mechanics, in Encyclopedia of Nonlinear Science, A. Scott ed., Routledge, New York (2005).
-P. Gaspard, Chaos, Scattering and Statistical Mechanics, Cambridge University Press, Cambridge (1998).
-P. Gaspard, Time-reversed dynamical entropy and irreversibility in Markovian random processes, J. Stat. Phys. 117, 599 (2004).

Supporting Material:

Complex Systems - Scholarpedia
Interdisciplinary Center for Nonlinear Phenomena and Complex Systems - G. Nicolis
Foundations of Complex Systems - G. Nicolis, C. Nicolis
Figure 1
Figure 2
Figure 3

Supporting Material:

Simple rules yield complex food webs - R. Williams & N. Martinez
Network Structure and biodiversity loss in food webs: robustness increases with connectance - J. Dunne, R. Williams & N. Martinez

Supporting Material:

Classical thermodynamics and economic general equilibrium theory - E. Smith & D. Foley
Thermodynamics of natural selection I: Energy flow and the limits of organization - E. Smith
Thermodynamics of natural selection II: Chemical Carnot cycles - E. Smith
Thermodynamics of natural selection III: Landauer's principle in computation and chemistry - E. Smith