Graduate Lecture in Nonlinear Analysis

Numbering Code G-SCI11 90412 LJ55 Year/Term 2022 ・ Second semester
Number of Credits 2 Course Type Lecture
Target Year Master's students Target Student
Language Japanese Day/Period
Instructor name MIYAJI TOMOYUKI (Graduate School of Science Associate Professor)
Karel SVADLENKA (Graduate School of Science Associate Professor)
Outline and Purpose of the Course This course focuses on the qualitative theory of ordinary differential equations and the fundamentals of the calculus of variations. It is aimed at mastering the basic theories and methods through which nonlinear ordinary differential equations and partial differential equations are mathematically analyzed.
Course Goals Understand the basic theory of nonlinear differential equations and learn methods of analysis of specific equations.
Schedule and Contents A total of 15 lectures (including feedback) will be conducted on the following topics:
1. Nonlinear analysis of ordinary differential equations [3-4 lectures]
・ Ordinary differential equations and dynamical systems
・ Structural stability of dynamical systems
・ Hyperbolicity of equilibria and structural stability
・ Stable manifold theorem for hyperbolic equilibria
2. Bifurcation theory of ordinary differential equations [3-4 lectures]
・ What is bifurcation in the context of a dynamical system?
・ Center manifold theorem (including calculation of specific examples)
・ Bifurcations of codimension 1 (saddle-node bifurcation, Hopf bifurcation)
・ Bifurcations of codimension 2
・ * Numerical continuation of bifurcation curves
3. Mathematical analysis through variational methods [6-8 lectures]
・ Relationship between solutions to variational problems and the Euler-Lagrange equation
・ Necessary and sufficient conditions for extrema
・ Existence theorem via the direct method of the calculus of variations (Tonelli's theorem)
・ Introduction to variational analysis of multiple integrals
・ Basics of regularity theory
・ Specific examples related to physical phenomena and their analysis (minimal surfaces, constraints, etc.)
・ * Numerical calculation of optimization problems
・ * Evolution problems with variational structure (gradient flows, Hamilton's principle)
(*: topics that will be covered only if there is sufficient time)

Miyaji is in charge of topics 1 and 2, and Svadlenka is in charge of topic 3.
The lecturer-in-charge will decide how time is allocated to the above topics, keeping in mind the students' level of understanding. Moreover, instructions on the progress of the lecture will be provided, so that students have sufficient time to prepare for lectures in advance.
Course Requirements We assume that the students have basic background in mathematics up to second-year such as knowledge of Calculus (including Advanced calculus), Linear algebra, Set theory and topology, and have mastered the subject Introduction to nonlinear analysis taught in the second semester of the second year. In addition, taking Differential equations in the first semester of the third year and Functional analysis in the second semester of the third year is recommended since it will be beneficial to the understanding of the material of this course.
Study outside of Class (preparation and review) As the lecture progresses, we will provide appropriate instructions on what to prepare and review so that the students can fully understand the content of the lecture.
Textbooks Textbooks/References No textbook will be used.
References, etc. C. Chicone "Ordinary Differential Equations with Applications" (Springer)
J. Guckenheimer and P. Holmes "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields" (Springer)
B. Dacorogna "Introduction to the Calculus of Variations" (Imperial College Press)
F. Clarke "Functional Analysis, Calculus of Variations and Optimal Control" (Springer)
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