Graduate School in Nonlinear Science

Sponsored by the Danish Research Academy

MIDIT                               OFD                           CATS
Modelling, Nonlinear Dynamics       Optics and Fluid Dynamics     Chaos and Turbulence Studies
and Irreversible Thermodynamics     Risø National Laboratory      Niels Bohr Institute and 
Technical University of Denmark     Building 128                  Department of Chemistry
Building 321                        P.O. Box 49                   University of Copenhagen 
DK-2800 Lyngby                      DK-4000 Roskilde              DK-2100 Copenhagen Ø
Denmark                             Denmark                       Denmark


by Remo Badii
Paul Scherrer Institute
Villigen, Switzerland
and Antonio Politi
Istituto Nazionale di Ottica
Florence, Italy

at MIDIT, IMM, Bldg. 305, Room 027, DTU

Wednesday June 23 - 15.00 h

Lecture 1: Thermodynamic formalism for dynamical systems (Badii, 40 min)
Parallel between local dimensions, entropies, Lyapunov exponents and thermodynamic observables. Differences among thermodynamic ensembles in nonlinear dynamics. Scaling with a continuum of exponents, phase transitions.

5 min Break

Lecture 2: Formal Languages (Badii, 40 min)
Computational modelling of symbolic sequences. The Chomsky hierarchy. Other languages relevant to physics: substitutions. Physical characterization of formal languages (excluding singular continuous spectra: see A. Politi's lecture 3), computational characterization of physical systems (excluding generalized shifts and Turing machines: see A. Politi's lecture 2).

Wednesday June 30 - 15.00 h

Lecture 3: Complexity indicators (Badii, 40 min)
Compression of information, coding. Algorithmic information, uncomputability. Other algorithmic indicators. Physical measures, time- versus scaling- unpredictability (excluding topological complexity: see A. Politi's lecture 3), Feigenbaum's scaling function (possible shapes, convergence).

5 min Break

Lecture 4: Turbulence: two-scale approach and test of energy cascade models (Badii, 40 min)
Statistical self-affinity of turbulent time series, two-scale estimate of exponents. Models for the energy dissipation's scaling exponent, asymptotic behaviour, new scaling relations.

Thursday July 1 - 15.00 h

Lecture 1: Symbolic encoding (Politi, 40 min)
Advantages, meaning and procedures to encode dynamical systems. General introduction and discussion of some experimental systems; specific case of Hamiltonian dynamics.

5 min Break

Lecture 2: Model classes (Politi, 40 min)
Discussion about sources of ``complex'' behaviour; mutual connections between coupled-maps, automata, and continuous-time systems. Connections with Turing machines.

Wednesday July 7 - 15.00 h

Lecture 3: Tools (Politi, 40 min)
Correlation functions/power spectra and applications to models with singular continuous spectra. Topological complexity.

5 min Break

Lecture 4: Space-time chaos (Politi, 40 min)
Introduction of the general tools for the characterization of space-time chaos. Lyapunov spectra (``chronotopic approach''), entropies, fractal dimensions.