Course: Qualitative methods of analysis of non-linear systems

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Course title Qualitative methods of analysis of non-linear systems
Course code UMB/570
Organizational form of instruction Lecture + Lesson
Level of course Bachelor
Year of study not specified
Frequency of the course In each academic year, in the summer semester.
Semester Summer
Number of ECTS credits 5
Language of instruction Czech
Status of course Optional
Form of instruction unspecified
Work placements unspecified
Recommended optional programme components None
Lecturer(s)
  • Eisner Jan, Mgr. Dr.
Course content
Content of lectures: Local methods for equilibrial points to differential equations, Hartman-Grobman theorem, stable, unstable and central variet, Central variet theorem , Normal forms, Poincaré mapping, transversality, structural stability; Local bifurcation, Hopf bifurcation, Fractals, construction of attractors, Takens theorem, Mandelbrot and Julian sets, Sharkhov theorem, Smale horseshoe, symbolic dynamics, homoclinic orbits, Ljapunov exponents, strange attractors. Content of practicals: Solution of concrete models from biology with help of the computer software Mathematica, Matlab, GRIND, Locbif, Fractint.

Learning activities and teaching methods
Monologic (reading, lecture, briefing), Work with multi-media resources (texts, internet, IT technologies)
  • Class attendance - 42 hours per semester
  • Preparation for classes - 28 hours per semester
  • Preparation for exam - 42 hours per semester
Learning outcomes
Qualitative analysis of solutions to systems of ODEs , in particular classification of bifurcation points and the sensitivity to parameters.
Students learn methods of qualitative and sensitivity analysis of differential and difference equations, they learn how to find basic types of bifurcation points analytically an numerically with help of computer programs (Auto and xppaut).
Prerequisites
We assume knowledge of ordinary differential equations in the range of UMB/587
UMB/CV587
----- or -----
UMB/587

Assessment methods and criteria
Oral examination, Interim evaluation

Active attendance of practicals (excused absence permitted). Students should find basic types of bifurcation points either numerically or numerically for a given task (a system od ODE of a boundary value problem) and at oral exam they should show how they understand to the theoretical background.
Recommended literature
  • V.I. Arnold: Geometrical Methods in the Theory of ODE. A Series of Comprehensive Studies in Mathematics 250, Springer-Verlag, New York, 1983.
  • Y.A. Kuznetsov: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences 112, Springer-Verlag, New York, 1995;.


Study plans that include the course
Faculty Study plan (Version) Category of Branch/Specialization Recommended year of study Recommended semester
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Applied Mathematics (2010) Category: Mathematics courses 3 Recommended year of study:3, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (1) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 3 Recommended year of study:3, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 3 Recommended year of study:3, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 3 Recommended year of study:3, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Mathematics for future teachers (1) Category: Mathematics courses 3 Recommended year of study:3, Recommended semester: Summer
Faculty: Faculty of Science Study plan (Version): Secondary Schools Teacher Training in Mathematics (2012) Category: Pedagogy, teacher training and social care - Recommended year of study:-, Recommended semester: Summer