### Introduction to Mathematical Chaos Theory for Engineers

#### Course Description

#### General Competencies

General knowledge on the history of the Mathematical theory of chaos. Mathematical calculation of basic elements of Mathematical theory of chaos: fixed points, stability, periodic points, bifurcation, sensitive dependence on intial conditions. Basic knowledge on fractal geometry: mathematical generation of fractals, self similarities, fractal dimensions. Basic criteria for chaotic behaviour of continuous iterators.

#### Learning Outcomes

- describe fundamental properties which characterize chaotic iterators
- distinguish linear, nonlinear, nonchaotic and chaotic iteration processes
- apply the Ljapunov method for estimating of sensitivity of processes
- explain the importance of fractal sets and geometry in describing of chaotic iterators
- manipulate with calculus of the theory of stability: stability of fixed points
- manipulate with calculus of the theory of bifurcation and chaos: periods 2 and 3
- analyze all four states on the path to chaos for continuous iterators
- create an own chaotic circuit from basic elements purchased in Chipoteka

#### Forms of Teaching

**Lectures**Based on mathematical skills and knowledge from first year of study, some new knowledge are adopted from mathematical theory on chatocal interators; computer simulation and visualization of fractal sets; basic knowledge on the fractal geometry; calculation of chatical properties; visualization of chaos on computer.

**Exams**Oral exam: students continuously can solve and present their homework; there exists a final exam for advanced students. Written exams: control test and fianl exam; seminar work.

**Consultations**By appointment.

**Seminars**Seminar work is free choice, but it is appropriate for higher estimates. By seminar, a student shows: creativity, independence and the level of adopted knowledge.

**Acquisition of Skills**Recognition of complex systems and process in nature, measurement of chatoc properties, work with fractal sets, introducing of new aspects of thinking.

#### Grading Method

Continuous Assessment | Exam | |||||
---|---|---|---|---|---|---|

Type | Threshold | Percent of Grade | Threshold | Percent of Grade | ||

Class participation | 0 % | 10 % | 0 % | 0 % | ||

Seminar/Project | 0 % | 30 % | 0 % | 30 % | ||

Mid Term Exam: Written | 0 % | 30 % | 0 % | |||

Final Exam: Written | 0 % | 30 % | ||||

Exam: Written | 0 % | 70 % |

#### Week by Week Schedule

- Reading of the first part of famous Gleick book on Chaos, an introduction for creativ way of thinkining, a computer demonstration on the existence of Chaos in the nature: 'vodenično kolo', mathematical and magnet pendulum, virbration, Lorentzov system
- Continous iterator, fixed points, stability of fixed points, iterators as continous mapping from [0,1] into [0,1], domain of stability for one parametric iterations
- Fractal sets which are appearing in the mathematical theory of chaos: Feigenbaum diagram and Cantor set
- A continuation of the reading of Gleick book; the background on the one-hump iterator; the basis on the 'shift' iterator
- Bifurcation of order 2, the cascade of bifucration, Feigenbaum number, the presentation of propositions for the working of a seminar from mathematical theory cahos
- Mathematical generation of fractals: iteration functional system 'IFS' and the mathematical basis for Julija sets and Mandelbrot set
- Sensitive dependence on initial conditions by definition, sensitive dependence on initial conditions for 'one-hump iterator' by using the Schwarzian derivative
- Sensitive dependence on initial conditions by Ljapunov coefficient, universality of Chaos, the set 'Sigma 2', 'Sigma 2' as a metric space, sensitive dependence on initial conditions for 'Shift' iterator
- Mathematical definition of Chaos: sensitive dependence on initial conditions, density of periodic points, transitivity of the set of periodic points
- Composition of higher order, periodic points, Sarkovskii theorem, period 3 implies chaos
- Computation of period 3 for continous iterators: critera for the existence and nonexistence of periodic points of order 3, methods for efective proving of period 3
- Chaotic properties of 'one-hump iterators': theory and examples
- Chaos in computer: chaotic behaviour of 'shift iterator'
- Presentation and estimation of student seminar on Chaos
- Presentation and estimation of student seminar on Chaos

#### Study Programmes

##### University graduate

#### Literature

*Chaos. A mathematical introduction*, Cambridge University Press

*Chaos and Fractals. New Frontiers of Science*, Springer

*Chaos, Fractals, and Dynamics*, Addison-Wesley

*Chaos for Engineers. Theory, Applications and Control*, Springer

*Chaos. An introduction to dynamical systems*, Springer, New York

#### Lecturers

#### General

**ID**34561

**4**ECTS

**L3**English Level

**L1**e-Learning

**45**Lectures

**0**Exercises

**0**Laboratory exercises

**0**Project laboratory

#### Grading System

**85**Excellent

**75**Very Good

**60**Good

**45**Acceptable