Search

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 oscillatory tails. Upon numerical continuation in parameter space, it has been observed that the oscillations in the tails of the pulses grow into a secondary excursion resembling a second copy of the primary pulse. In this paper, we outline in detail the geometric mechanism responsible for this single-to-double-pulse transition, and we construct the transition analytically using geometric singular perturbation theory and blow-up techniques. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Abstract: We expand upon a general framework for studying the bifurcation diagrams of localized spatially oscillatory structures. Building on work by Beck et al., the present work provides rigorous analytical results on the effects of perturbations to systems exhibiting snaking behavior. Starting with a reversible variational system possessing an additional symmetry, we elucidate the distinct effects of breaking symmetry and breaking variational structure, and characterize the resulting changes in both the bifurcation diagram and the solutions themselves. We show how to predict the branch reorganization and drift speeds induced by any particular given perturbative term, and illustrate our results via numerical continuation. We further demonstrate the utility of our methods in understanding the effects of particular perturbations breaking reversibility. Our approach yields an analytical explanation for previous numerical results on the effects of perturbations in the one-dimensional cubic–quintic Swift–Hohenberg model and allows us to make predictions on the effects of perturbations in more general settings, including planar systems. While our numerical results involve the Swift–Hohenberg model system, we emphasize the general applicability of the analytical results. [Copyright &y& Elsevier]

Abstract: The bifurcation structure of localized stationary radial patterns of the Swift–Hohenberg equation is explored when a continuous parameter is varied that corresponds to the underlying space dimension whenever is an integer. In particular, we investigate how 1D pulses and 2-pulses are connected to planar spots and rings when is increased from 1 to 2. We also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3. [Copyright &y& Elsevier]

Stable localized roll structures have been observed in many physical problems and model equations, notably in the one-dimensional (1D) Swift--Hohenberg equation. Reflection-symmetric localized rolls are often found to lie on two "snaking" solution branches so that the spatial width of the localized rolls increases when moving along each branch. Recent numerical results by Burke and Knobloch indicate that the two branches are connected by infinitely many "ladder" branches of asymmetric localized rolls. In this paper, these phenomena are investigated analytically. It is shown that both snaking of symmetric pulses and the ladder structure of asymmetric states can be predicted completely from the bifurcation structure of fronts that connect the trivial state to rolls. It is also shown that isolas of asymmetric states may exist, and it is argued that the results presented here apply to 2D stationary states that are localized in one spatial direction. [ABSTRACT FROM AUTHOR]