Course Syllabus

Announcements and other information about the class can be found here.

Course description. The focus of this course is bifurcation theory and averaging methods for nonlinear dynamical systems, including infinite-dimensional dynamical systems and partial differential equations. Topics covered include the Lyapunov-Schmidt reduction, Leray-Schauder degree-theoretic methods, some of the local and global bifurcation results due to Crandall, Rabinowitz, Krasnoselski, and Dancer, the Hopf bifurcation, center manifolds and normal forms, and geometric approaches to averaging theory. Several topics from functional analysis, such as Fredholm operators and spectral theory for bounded operators, will be covered in class to facilitate the understanding of the above material.

Useful references. Although there is no required text, the following, among others, may be useful in parts of the course:

  • Hansjörg Kielhöfer, Bifurcation theory: An introduction with applications to partial differential equations, 2nd edition, Springer.
  • Shui-Nee Chow and Jack Hale, Methods of bifurcation theory, Springer.
  • Antonio Ambrosetti and Andrea Malchiodi, Nonlinear analysis and semilinear elliptic problems, Cambridge University Press.
  • Jan Sanders, Ferdinand Verhulst, and James Murdock, Averaging methods in nonlinear dynamical systems, 2nd edition, Springer.

Grading policy. The final grade will be based on a combination of homework assignments and in-class presentations. Discussion of homework assignments with other students is encouraged but what you hand in should be your own work. There will be no final exam.

Accommodations. Brown University is committed to full inclusion of all students. Please inform me early in the term if you have a disability or other conditions that might require accommodations or modification of any of these course procedures. You may speak with me after class or during office hours. For more information, please contact Student and Employee Accessibility Services at 401-863-9588 or

Students in need of short-term academic advice or support can contact one of the deans in the Dean of the College office.

Announcements and other information about the class can be found here. A PDF copy of the syllabus can be found here: APMA_2200.pdf

Course Summary:

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