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Lecturer(s)
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Andres Jan, prof. RNDr. dr hab. DSc.
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Rachůnková Irena, prof. RNDr. DrSc.
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Course content
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1.Stability od hyperbolic and nonhyperbolic critical points. (Criteria about stability, asymptotic stability and instability based on linearization. Exponential stability.) 2.Liapunov stability of critical points. (Liapunov Theorem, Četaev Theorem, geometrical meaning of Liapunov functions.) 3.Investigation of stability of critical points in particular models. 4.Stable and unstable manifolds. (Existence of stable and unstable manifolds and their computing by means of power series.) 5.Local phase portraits near nonhyperbolic critical points. (In the presence of one zero eigenvalue and one nonzero eigenvalue, in the presence of purely imaginary eigenvalues.) 6.Center manifolds. (Existence of center manifolds and their computing by means of power series. Computing flow on center manifolds.) 7.Local bifurcations of planar dynamical systems. (Existence of bifurcations, bifurcation function, bifurcation equation, scalar equations on center manifolds, saddle-node bifurcation.) 8.Construction of local phase portraits of particular models. 9.Global bifurcations. (Examples of global bifurcations breaking a saddle connection, breaking a homoclinic loop.). 10.Periodic orbits. (Existence of periodic orbits, Poincaré-Bendixson Theorem, positively invariant sets, Bendixson's Criterion and Dulac's Criterion for the nonexistence of periodic orbits). 11.Stability of periodic orbits. (Poincaré map, orbital stability, instability and asymptotic stability, model of Van der Pol's oscillator.) 12.Investigation of stability of periodic orbits in particular models. 13.Hopf bifurcation. (Poincaré-Andronov-Hopf Theorem, bifurcation diagrams.)
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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
- 20 hours per semester
- Preparation for the Exam
- 60 hours per semester
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Learning outcomes
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Understand main principles of the theory of dynamical systems, construction of dynamical models and their investigation.
Application Apply the theory of dynamical systems to the study of various models in mathematics, physics, economics and biology.
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Prerequisites
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Knowledge of Scalar Dynamical Systems and Planar Linear Dynamical Systems.
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Assessment methods and criteria
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Oral exam, Student performance
Credit: active participation in seminars. Exam: to know and to understand the subject and to be able to apply it on standard models.
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Recommended literature
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J. B. Hubard, B. M. West. (1995). Differential Equations: A Dynamical Systems Approach. Springer.
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J. Guckenheimer, P. Holmes. (1993). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer.
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J. Hale, M. Kocak. (1996). Dynamics and Bifurcations. 2. edition. Springer-Verlag.
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Morris W. Hirsch, Stephen Smale, Robert L. Devaney. (2013). Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, Oxford.
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Steven H. Strogatz. (2014). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Boca Raton.
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Teschl, Gerald. (2012). Ordinary differential equations and dynamical systems. American Mathematical Society.
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