# 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
**

**COMPLEXITY - HIERARCHICAL STRUCTURES AND SCALING IN PHYSICS**

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.