Paul A. Carter

Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo System

Numerical studies indicate that the FitzHugh--Nagumo system exhibits stable traveling pulses with oscillatory tails. In this paper, the existence of such pulses is proved analytically in the singular perturbation limit near parameter values where the FitzHugh--Nagumo system exhibits folds. In addition, the stability of these pulses is investigated numerically, and a mechanism is proposed that explains the transition from single to double pulses that was observed in earlier numerical studies. The existence proof utilizes geometric blow-up techniques combined with the exchange lemma: the main challenge is to understand the passage near two fold points on the slow manifold where normal hyperbolicity fails.

Multijam Solutions in Traffic Models with Velocity-Dependent Driver Strategies

The optimal-velocity follow-the-leader model is augmented with an equation that allows each driver to adjust their target headway according to the velocity difference between the driver and the car in front. In this more detailed model, which is investigated on a ring, stable and unstable multipulse or multijam solutions emerge. Analytical investigations using truncated Fourier analysis are confirmed and complemented by a detailed numerical bifurcation analysis. In addition to standard rotating waves, time-modulated waves are found.

Unpeeling a Homoclinic Banana in the FitzHugh--Nagumo System

The FitzHugh--Nagumo equations are known to admit fast traveling pulse solutions with monotone tails. It is also known that this system admits traveling pulses with exponentially decaying oscillato...

Relaxing the Integral Test: A Challenge for the Advanced Calculus Student

Summary Illustrative, elementary counterexamples are hard to come by. In this note we propose an elementary, closed-form counterexample to a generalization of the classic integral test where the condition of monotonicity is relaxed. The analysis only uses techniques accessible to a second-semester calculus student.