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Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo System

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Abstract:
The FitzHugh-Nagumo equations are known to admit fast traveling pulses that have monotone tails and arise as the concatenation of Nagumo fronts and backs in an appropriate singular limit, where a parameter $\epsilon$ goes to zero. These pulses are known to be nonlinearly stable with respect to the underlying PDE. Numerical studies indicate that the FitzHugh-Nagumo system exhibits stable traveling pulses with oscillatory tails. In this work, the existence and stability of such pulses is proved analytically in the singular perturbation limit near parameter values where the FitzHugh-Nagumo system exhibits folds. 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. For the stability result, similar to the case of monotone tails, stability is decided by the location of a nontrivial eigenvalue near the origin of the PDE linearization about the traveling pulse. We prove that this real eigenvalue is always negative. However, the expression that governs the sign of this eigenvalue for oscillatory pulses differs from that for monotone pulses, and we show indeed that the nontrivial eigenvalue in the monotone case scales with $\epsilon$, while the relevant scaling in the oscillatory case is $\epsilon^{2/3}$. Finally a mechanism is proposed that explains the transition from single to double pulses that was observed in earlier numerical studies, and this transition is constructed analytically using geometric singular perturbation theory and blow-up techniques.
Notes:
Thesis (Ph.D. -- Brown University (2016)

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Citation

Carter, Paul A., "Fast Pulses with Oscillatory Tails in the FitzHugh-Nagumo System" (2016). Mathematics Theses and Dissertations. Brown Digital Repository. Brown University Library. https://doi.org/10.7301/Z0707ZVP

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