Lecturer(s)


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, GrobmanHartman Theorem, FlowBox 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. Nodesource, focussource, nodesink, focussink, saddle of planar nonlinear dynamical systems.) 11.Planar Hamiltonian systems. (Hamiltonian and its level sets. Conditions for center or saddle. Predatorpray 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
 52 hours per semester
 Homework for Teaching
 20 hours per semester
 Preparation for the Course Credit
 50 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.

Prerequisites

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. SpringerVerlag.

J. Hale, M. Kocak. (1991). Dynamics and Bifurcations. SpringerVerlag.
