Possible topics of project:

1. FitzHugh-Nagumo model in neural science  (original paper in 1961)
   
2. Van der Pol oscillator
3. Canard explosion
4. Smallest chemical reaction system with Hopf bifurcation
5. Smallest chemical reaction system with bistability
6. Something from John Tyson's lectures
7. Bifurcation analysis in a bioinformatics model
8. Bistability in coral reef model
9. Unraveling Complex Systems
10. Backward bifurcation in epidemic model
11. Two-body problem in general relativity (see 6.5.7 in Strogatz)
    
12. Fireflies (Strogatz section 4.5)
13. Josephson Junction (Strogatz section 4.6)
14. survey the known Lyapunov functions

Projects:

Nathan Abraham

Two Body Problem in General Relativity

The classical two body problem can be expressed as a reduced mass orbiting a central force at the center of mass.  For an eccentricity e: 0<=e<1 the body moves in a fixed stable periodic orbit as described by Newton's equations. General relativity introduces corrections to the radial force resulting in elliptical orbits that precess due to a non-radial component of force. The effects of general relativity on a two body system will be explored with analytic and numerical techniques to determine the behavior of the system, including degradation of orbital period and gravitational radiation.

Ross Alexander

Fireflies

Matt Devendorf

Bistability of coral reef model

Amy Fillipeck

Expectations and Exchange Rate Dynamics Revisited

Rudiger Dornbusch’s 1976 paper on exchange rate overshooting is one of the most influential papers in international macroeconomics.  He derived a variant on Mundell and Fleming’s IS-LM model for an open economy with perfect capital mobility that assumed rational expectations, meaning that market participants have perfect foresight. The traditional Mundell Fleming model could not explain the excess volatility observed over a short period of time between countries’ exchange rates. However, Dornbusch’s model predicts that even when agents have perfect foresight, in response to an unanticipated monetary supply expansion, the exchange rate will initially overshoot its long-run level, so that it declines during the adjustment period.

I will begin with a brief survey of the literature. Then, I will lay the groundwork for the Dornbusch model by stating the assumptions, the resulting equations, and their general economic justifications. Using the assumed equations, I will then re-derive Dornbusch’s resulting linear system:

where p is the price level, s is the nominal exchange rate, v and  are constants (although derived from numerous other parameters) that signify the speed of adjustment, and an “overbar” signifies a fixed steady-state value.  I will then perform equilibrium stabilization analysis using the literature’s consensus for the values of v and . Similarly, I will perform a sensitivity analysis to determine the values of the parameters for which the model is stable. Finally, I will use the model to explain the natural phenomenon of overshooting in the exchange rate market.

Dornbusch, Rudiger. 1976. "Expectations and Exchange Rate Dynamics". The Journal of Political Economy. 84 (6): 1161-1176.

Jonathan Fischer

Analysis of the Dynamics of the Van der Pol Oscillator

This project will comprehensively analyze the behavior and characteristics of the Van der Pol oscillator system. I will present a brief history of the model and then will proceed to model the qualitative behavior of the system by solving for equilibria, bifurcations, and the existence limit cycles. Finally, I will demonstrate a real-world example and compare the behavior of the system with that of the typical mass-spring system.

Abby Ng

Modeling tumor growth - the Gompertz equation

Albert Ng-Sui-Hing

The Fitzhugh-Nagumo model for Modeling Action Potential Behavior

The Hodgkin-Huxley model is a set of equations originally designed to modeling action potentials measured from squid axons.  The Fitzhugh-Nagumo model was designed as a two dimensional simplification of the Hodgkin-Huxley model to allow for an easier analysis of behavior, at the cost of some of the flexibility of the original equations.  As a simplification this model makes a number of assumptions and several limitations, but it still remains one of the most used models for describing neuronal firing behavior even in recent years.

Tigist Tamir

Fission Yeast Cell Model

The cell cycle is a biological process by which a single cell replicates its genetic material and divides into two daughter cells. This process is highly regulated by biochemical interactions among several cell cycle molecules. Biologist have proposed a verbal model describing the steps in this process, however the accuracy of these verbal models have to be verified. Constructing a feasible mathematical model for this system is an important step to our understanding of dynamics of molecular regulatory networks in the cell cycle. Furthermore, using mathematical model to study a system improves the verbal model proposed by researchers. John Tyson and colleagues utilized fission yeast cell, a well studied model organism, and Ordinary differential equations (ODE) to draw a simplified mathematical model. The study mainly focused on constructing a two dimensional ODE for S-phase of the cell cycle, and reducing parameters by understanding the relationship among each of the molecules. Finally, equilibrium points of the system were characterized via bifurcation theorem. This paper seeks to elucidate how the model was simplified to two dimensions and study the stability of steady states. In addition, the model will be interpreted using the estimated parameters from previous publications. Therefore, mathematical models of cell cycle phases serve to check the reality of proposed biological models.