Cardinal Wyszyński University in Warsaw
F a c u l t y o f M a t h e m a t i c s a n d N a t u r a l S c i e n c e s
S Y L L A B U S
Course title: XXXX. MODELLING OF COMPLEX SYSTEMS Major: Physics
Graduate Program: XXXX Level of course: BSc Semester: First
Type of course: Compulsory
Academic hours: lectures – 30 hours; practical exercises – 30 hours ECTS: XXXX
Lecturer: Assoc. Prof. Vesselin Tonchev, PhD
COURSE SUMMARY
This course is aimed at the better understanding of the complexity as an emergent property of simple models. The proper instrument to study models’ behavior is a programming language of high level and as such, the GNU Octave is chosen - being part of the GNU Project it is a free software.
During the course a maximum efficiency is sought towards accumulation of practical skills. The exercises are aimed at creating working codes in which two sides of the model are studied – the qualitative and the quantitative (thus the principle “a plot and a number”).
In view of the orientation of the bachelor’s degree program basic models are introduced and studied, all they organized around two main themes in the natural sciences – Order and.
Disorder. The main programming tools aimed at the former are the Cellular Automata while for the latter as an archetypical example is chosen the Diffusion. It is the Logistic Map that yet in the beginning of the course provides the ground to distinguish between Disorder and Deterministic Chaos while at the end the Ising model gives a deeper perspective on the “Order-Disorder”
transition. Additional view on the determinism is provided by introducing the Burton-Cabrera- Frank model for the evolution of vicinal crystal surfaces.
On a technical level, the programming language tools and some elementary applications are introduced during the lecture classes, while the practical assignments end with working procedures. Students are expected to write programs alone neglecting the usage of ready products. Skills for working with files, finding extrema, working with vectors, matrices, finding eigenvalues spectra, numerical differentiation and integration, numerical solution of ordinary differential equations, and dealing with data from computer experiments are pursued.
The latest updated syllabus is approved by the Council of the XXXX (minutes from XXX) and by the Council of the Faculty of Mathematics and Natural Sciences (minutes from XXX).
L E C T U R E T H E M E S
І. Programming with GNU Octave – 6+6 hours
The concept of algorithm. Languages for algorithm description, flowcharts. Basic symbols, basic structures: chain, branching, loop. Languages for scientific modelling. GNU Octave - semantic structure and program structure. Data types and structures. Expressions and assignments. Input and output, control, loops. Common programming rules, testing and realization. Arrays, string arrays. Representation of numbers, types of errors connected with the representation and calculation. Significant digits. Errors due to rounding. Arithmetics with fixed and floating point, with single and double precision. Rounding. Speed of computation. Main mechanisms for error cumulation, control and estimation.
ІІ. Numerical methods with Gnu Octave – 6+6 hours
Finding of function extrema, Golden Mean in 1D. Numerical differentiation and numerical integration – simple rules. Operations with vectors and matrices, finding the largest eigenvalue of a matrix by interpolation (von Mises method). Integration of ordinary differential equations (ODE), Runge-Kutta method(s).
III. Models – 18+18 hours
Deterministic chaos, logistic map and other maps, bifurcations and Feigenbaum number(s).
Random number generation, testing of a self-made random number generator. Diffusion, studying the trajectory of a single walker, Diffusion-Limited Agregation – fractal dimension(s).
Cellular Automata in 1D with 16 and 256 states, studying time evolution and finding fractal dimension(s). Cellular Automata in 2D, “Life” game, Diffusion-Limited Aggragation (again).
The model of Burton-Cabrera-Frank and related models, integration of the ODE’s, monitoring.
The Ising model, Scaling and Universality, the method of Transfer-Matrix, finding the critical temperature of the order-disorder transition and the critical indices.
R E F E R E N C E S
Only freely available on Internet sources are referenced during the course.
WRITTEN TESTS
1. Mid-term tests on section І and II.
REQUIREMENTS
Mandatory practical exercises. Preparation and defense of term project based on extensions of the models introduced and studied in section III. The final grade includes 70 % term project defense + 30 % mid-term tests.
