Elements of invariant nonlinear analysis
7404-MAT-ENANI
1. Definitions of the Brouwer degree.
2. The concept of local and global bifurcation points.
3. Global Bifurcation Theorem.
4. Definition of the Hamiltonian and the Hamiltonian system.
5. Functional associated with the Hamiltonian system.
6. Definition of local and global bifurcation for autonomous Hamiltonian systems.
7. Global Bifurcation Theorem for autonomous Hamiltonian systems.
Type of assessment: oral examination
Total student workload
30 hours — lecture
30 hours — studying literature and materials indicated by the instructors and consultations with the instructor
30 hours — individual work, exam preparation
Total 90 hours.
3 ECTS points
Learning outcomes - knowledge
After completing the course, the PhD student:
W1: understands and knows the basic definition of Brouwer's degree. (P8S_WG)
W2: knows an alternative definition of the Brouwer degree for special types of mappings. (P8S_WG)
W3: knows the concepts of local and global bifurcation of solutions of autonomous Hamiltonian systems. (P8S_WG)
W4: knows the Global Bifurcation Theorem. (P8S_WG)
W5: knows the concept of the Hamiltonian and the Hamiltonian system. (P8S_WG)
W7: knows the reformulated definitions of local and global bifurcation for an associated functional. (P8S_WG)
W8: knows the Global Bifurcation Theorem for autonomous Hamiltonian systems. (P8S_WG)
Learning outcomes - skills
After completing the course, the PhD student:
U1: is able to apply the appropriate definition of Brouwer degree. (P8S_UW)
U2: is able to calculate Brouwer degree for specific types of mappings. (P8S_UW)
U3: verifies necessary and sufficient conditions for the existence of local and global bifurcation points. (P8S_UW)
U4: is able to parameterize the Hamiltonian system. (P8S_UW)
U5: is able to associate a functional with the Hamiltonian system under study and calculate the eigenvalues of the Hessian of this functional. (P8S_UW)
U6: is able to apply the Global Bifurcation Theorem and verify its assumptions. (P8S_UW)
Learning outcomes - social competencies
After completing the course, the PhD student achieves the following results:
K1: understands the appropriate formulation of questions and problems, uses professional terminology correctly. (P8S_KK)
K2: analyzes the problem correctly using the principles of logic. (P8S_KK)
K3: conveys the acquired knowledge in an understandable way. (P8S_KK)
Observation/demonstration teaching methods
- display
Expository teaching methods
- informative (conventional) lecture
Exploratory teaching methods
- classic problem-solving
- practical
Prerequisites
Completed course in mathematical analysis, algebra and differential equations.
Course coordinators
Assessment criteria
Oral exam: W1, W2, W3, W4, W5, W6, W7, W8, U1, U2, U3, U4, U5, U6, K1, K2, K3
Bibliography
1. K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, Heidelberg, 1985.
2. R. F. Brown, A topological introduction to nonlinear analysis. Third edition. Springer, Cham, 2014.
3. J. Mawhin & M. Willem, Critical point theory and Hamiltonian systems, Springer-Verlag, Berlin Heidelberg New York, 1989.
4. K.R. Meyer. G.R. Hall & D. Offin, Introduction to Hamiltonian dynamical systems nd the N-body problem, Applied Mathematical Sciences 90, Springer, 2009.
5. M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems, Springer, 1996.
Additional information
Additional information (registration calendar, class conductors,
localization and schedules of classes), might be available in the USOSweb system: