SubjectsSubjects(version: 853)
Course, academic year 2019/2020
  
Dynamical Systems - P413007
Title: Dynamické systémy
Guaranteed by: Department of Mathematics (413)
Actual: from 2019
Semester: both
Points: 0
E-Credits: 0
Examination process:
Hours per week, examination: 3/0 other [hours/week]
Capacity: winter:unknown / unknown (unknown)
summer:unknown / unknown (unknown)
Min. number of students: unlimited
Language: Czech
Teaching methods: full-time
Level:  
For type: doctoral
Note: course is intended for doctoral students only
can be fulfilled in the future
you can enroll for the course in winter and in summer semester
Guarantor: Schreiber Igor prof. Ing. CSc.
Němcová Jana Mgr. Ph.D.
Interchangeability : D413011
Is interchangeable with: AP413007
Annotation -
Last update: Janovská Drahoslava prof. RNDr. CSc. (26.09.2018)
The course deals with solutions of differential equations occurring in chemical-engineering and chemical-technological fields. Emphasis is placed on the qualitative theory of dynamic systems described by differential equations.
Literature -
Last update: Janovská Drahoslava prof. RNDr. CSc. (26.09.2018)

A. Katok, B. Hasselblatt: Introduction to the Modern Theory of Dynamical Systems (Encyclopedia of Mathematics and its Applications). Cambridge University Press 1996.

Clark Robinson: Dynamical Systems: Stability, Symbolic Dynamics, and Chaos (Studies in Advanced Mathematics). CRC Press 1998.

Bernardo, M., Budd, C., Champneys, A.R., Kowalczyk, P. : Piecewise-smooth Dynamical Systems. Theory and Applications. Springer-Verlag London, 2008.

James D. Meiss, Differential Dynamical Systems (Monographs on Mathematical Modeling and Computation), SIAM, 434 pp, 2007.

Syllabus -
Last update: Schreiber Igor prof. Ing. CSc. (12.11.2018)

LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS:

1)Exponential of a linear operator, fundamental matrix. Stability for linear systems: stable, unstable and center subspaces. Sinks and sources.

2)Classification of the equilibrium point for a planar linear system: saddles, nodes, foci and centers. Phase portrait in dimension two. Non homogeneous linear systems.

LOCAL THEORY FOR NON LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS:

3)Invariant sets. Hyperbolic equilibrium points. Linearization. Stable manifold theorem.

4)Center manifold theorem. The Hartman-Grobman theorem.

5)Hyperbolic equilibrium points for planar systems: saddles, nodes, foci and centers. Stability and Liapunov function.

6)Non hyperbolic equilibrium points for a planar system. Center manifold theory. Normal form theory. Gradient and Hamiltonian systems.

7)Theory of local bifurcations

GLOBAL THEORY FOR NON LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS:

8) Dynamical systems. Limits sets and attractors. Periodic orbits, limit cycles and separatrix cycles. Homoclinic and heteroclinic orbits. Compound separatrix cycles. Dulac theorem.

9) Chaotic invariant sets.

10)Poincaré map for a cycle. Poincaré map for a focus. Planar case and Poincaré map.

11)The Stable manifold theorem for periodic orbits. Floquet theorem.

12)The Center manifold theorem for periodic orbits. Liouvile theorem for the fundamental matrix. Characteristic exponents and multipliers for a period orbit.

13)Theory of global bifurcations.

14) Applications in chemistry, physics and biology.

Registration requirements -
Last update: Borská Lucie RNDr. Ph.D. (16.09.2019)

none

 
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