• Project 1: Population Dynamics
  • The overall theme of this project is a study of various population models (e.g. predator/prey systems, competing species) with possible control mechanisms (such as harvesting) from a dynamical systems point of view.
  • Starting point is Chapter 11 in the textbook, with more details in Chapter 3 of J.D. Murray, "Mathematical Biology, I. An Introduction", 3rd edition, Springer Verlag, 2002.
  • Intended Project Outcome: Find examples of dynamical systems which exhibit the following phenomena: coexistence equilibria, extinction, limit cycle periodic behavior. (see section 11.4 in the textbook or problem section of Chapter 3 in the Murray reference). For each such example, determine parameter domain of stability, bifurcation diagrams; discuss the underlying mathematical theory and its relevance to the physical context.

• Project 2: Oscillatory vs. Excitable Systems
• The overall theme of this project is a study of two distinct types of phenomena: I. oscillatory behavior in electrical/biological/chemical systems and II. excitable behavior in neurodynamics (Hodgkins - Huxley system, FitzHugh-Nagumo system)
• Starting point is Chapter 12 in the textbook, with more details in Chapter 7 of J.D. Murray, "Mathematical Biology, I. An Introduction", 3rd edition, Springer Verlag 2002.
• Intended Project Outcome: Describe dynamical systems that exhibit each of the two behaviours (oscilllatory vs excitable) and justify them using the mathematical tools developed in the course. Highlight the specific phenomena present in these systems (e.g. threshold phenomenon, Hopf Bifurcation). See section 12.5 in the textbook and the problem section of Chapter 7 in the Murray reference.

• Project 3: Motion in Central Force Fields
  • In this project the aim is to study various systems appearing in classical mechanics, when the forces are so called central forces. One typical example is the Newton's law of motion for objects in a gravitational field.
  • Starting point is Chapter 13 in the textbook, with more details in Chapter 7 of Jerry B. Marion, S. T. Thornton, "Classical Dynamics of particles and Systems. 3rd edition, Academic Press 1988.
  • Intended Project Outcome: Describe the various models for motion in a central force field and its trajectories. Examples are in Sections 13.8 and Sections 13.9 in the textbook or in the problem section of Chapter 7 in the Marion-Thornton reference.

TIMELINE:

• March 9th - Every graduate student is expected to choose a topic and let me know in writing (email is fine).
• April 6th - An initial report (2 page minimum) is expected. This should contain the description of the project, choice of models and an outline of the analysis to be performed.
• May 4th - The preliminary final report is due (6-8 pages). In class 15-minute presentations will be scheduled May 4 & 6th.
• May 11th - The final report on the project (about 6-8 pages) is due.

FORMAT: Preferably, the report should be typed in LaTeX (or equivalent type-setting software). A template will be made available soon. When in doubt about the format, please ask the instructor.