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Lecturer(s)
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Krajščáková Věra, Mgr.
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Rachůnková Irena, prof. RNDr. DrSc.
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Tomeček Jan, doc. RNDr. Ph.D.
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Course content
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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.
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Learning activities and teaching methods
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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
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Learning outcomes
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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.
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Prerequisites
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Knowledge of Differential and Integral Calculus.
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Assessment methods and criteria
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Student performance
Credit: active participation in seminars.
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Recommended literature
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C. Robinson. (1998). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. 2. edition. CRC Press, Boca Raton.
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F. Verhulst. (1996). Nonlinear Differential Equations and Dynamical Systems. 2.edition. Springer-Verlag.
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I. Rachůnková, J. Fišer. (2014). Dynamické systémy 1. Univerzita Palackého v Olomouci, Olomouc.
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J. Hale, M. Kocak. (1996). Dynamics and Bifurcations. 2. edition. Springer-Verlag.
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J. Kalas, M. Ráb. (2012). Obyčejné diferenciální rovnice. Vyd. 3.. Brno: Masarykova univerzita.
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L. Perko. (2001). Differential Equations and Dynamical Systems. 3. edition. Springer, New York.
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S. Wiggins. (2003). Introduction to Applied Nonlinear Dynamical Systems and Chaos. 2. edition. Springer, New York.
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