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Lecturer(s)
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Course content
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A. The language of dynamical systems 1. A summary of linear systems 2. De-motivation: Chaos and the Lorenz attractor 3. Case-study: Driven damped oscillations 4. Relation of non-linear to linear systems: The Hartman-Grobman Theorem 5. Chaos forbidden: The Poincare-Bendixon Theorem B. Fixed-point methods 1. Fixed-point theorems 2. Standard applications 3. BVP for nonlinear ODEs 4. nonlinear PDEs: Classical approach 5. nonlinear PDEs: Modern approach 6. modern approach to nonlinear evolution PDEs. C. Monotonicity methods 1. Monotonicity and the Browder-Minty Theorem 2. Application: The method of lower and upper solutions 3. Pseudo-monotonicity and the Brezis Theorem 4. Application to steady state problems 5. Application to evolution problems
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Learning activities and teaching methods
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Lecture, Dialogic Lecture (Discussion, Dialog, Brainstorming), Demonstration
- Preparation for the Exam
- 45 hours per semester
- Attendace
- 65 hours per semester
- Semestral Work
- 15 hours per semester
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Learning outcomes
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Understand the language of dynamical systems and solution methods for nonlinear ODEs and PDEs based on fixed-point principles and monotonicity.
Comprehension Understand the mathematical tools of nonlinear partial differential equations.
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Prerequisites
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Classical and modern theory of ODEs and PDEs, Lebesgue's theory, calculus.
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Assessment methods and criteria
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Oral exam, Written exam
seminar work
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Recommended literature
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C. Robinson. (1999). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos. CRC Press.
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E. Feireisl. (2004). Dynamics of viscous compressible fluids. Oxford: Oxford University Press.
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J. Lukeš. (2001). Zápisky z funkcionální analýzy. MatFyzPress.
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L.C. Evans. (1998). Partial differential equations. AMS.
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P. Drábek, J. Milota. (2004). Lectures on Nonlinear Analysis. Plzeň.
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Roubíček T. (2008). Nonlinear Partial Differential Equations with Applications. Birkhauser.
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V. M. Aleksejev, V. M. Tichomirov, S. V. Fomin. (1991). Matematická teorie optimálních procesů. Academia, Praha.
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