Course: Dynamical Systems 1

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Course title Dynamical Systems 1
Course code KMA/DS1
Organizational form of instruction Lecture + Exercise
Level of course Master
Year of study not specified
Semester Summer
Number of ECTS credits 3
Language of instruction Czech
Status of course Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
  • Rachůnková Irena, prof. RNDr. DrSc.
  • Fišer Jiří, RNDr. Ph.D.
Course content
1.Dynamical systems generated by a system of autonomous ordinary differential equations of the first order. (Basic definitions, deriving models.) 2.Dynamical systems generated by one autonomous differential equation of the first order. (Critical points and their stability, phase portraits.) 3.Elementary bifurcations of scalar dynamical systems. (Local bifurcations, bifurcation diagrams, saddle, pitchfork, transcritical bifurcation, hysteresis.) 4.System of two linear homogeneous equations with constant coefficients. (Global existence and uniqueness, Jordan canonical forms, types of solutions.) 5.Planar linear dynamical systems with canonical matrices. (Canonical phase portraits.) 6.Planar linear dynamical systems with general constant matrices. (Construction of phase portraits, eigenvectors and isoclines.) 7.Hyperbolic and nonhyperbolic matrices, classification of phase portraits. (Classification of phase portraits of all linear systems with constant coefficients by means of eigenvalues. Topological classification.) 8.Planar nonlinear dynamical systems. (Hyperbolic and nonhyperbolic critical points, linear variational equations, local topological equivalence, Grobman-Hartman Theorem, Flow-Box Theorem near regular points.) 9.Stability of hyperbolic critical points. (Asymptotic stability and instability of hyperbolic critical points.) 10.Local phase portraits near hyperbolic critical points. (Linearization. Node-source, focus-source, node-sink, focus-sink, saddle of planar nonlinear dynamical systems.) 11.Planar Hamiltonian systems. (Hamiltonian and its level sets. Conditions for center or saddle. Predator-pray population model.) 12.Conservative systems. (Potential function, symmetry of phase portrait. Conditions for center or saddle. Model of the planar pedulum.) 13.Investigation of particular models.

Learning activities and teaching methods
Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Work with Text (with Book, Textbook)
  • Attendace - 39 hours per semester
  • Preparation for the Course Credit - 35 hours per semester
  • Homework for Teaching - 15 hours per semester
Learning outcomes
Understand basic principles of the theory of dynamical systems, construction of dynamical models and their investigation.
Comprehension Explain main principles in the theory of dynamical systems and classify basic phase portraits. Interpret phase portraits of physical and population models.
Knowledge of Differential and Integral Calculus.

Assessment methods and criteria
Student performance

Credit: active participation in seminars.
Recommended literature
  • F. Verhulst. (1990). Nonlinear Differential Equations and Dynamical Systems. Springer-Verlag.
  • J. Hale, M. Kocak. (1991). Dynamics and Bifurcations. Springer-Verlag.

Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Science Discrete Mathematics (2015) Mathematics courses 1 Summer
Faculty of Science General Physics and Mathematical Physics (2014) Physics courses 1 Summer
Faculty of Science Applications of Mathematics in Economy (2015) Mathematics courses 1 Summer