Lin's method and homoclinic bifurcations for functional differential equations of mixed type

We show that the fast travelling pulses of the discrete FitzHugh-Nagumo system in the weak-recovery regime are nonlinearly stable. The spectral conditions that need to be verified involve linear operators that are associated to functional differential equations of mixed type. Such equations are ill-posed and do not admit a semi-flow, which precludes the use of standard Evans-function techniques. Instead, we construct the potential eigenfunctions directly by using exponential dichotomies, Fredholm techniques and an infinitedimensional version of the Exchange Lemma.AMS 2010 Subject Classification: 34A33, 34K26, 34D35, 34K08.

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“…Arguing as in [24, §3] 6 , we may show that the inhomogeneous equations LV = f can be solved on the half-lines R ± . Proceeding as in [24 …”

Section: Proof Of Proposition 21mentioning

confidence: 99%

“…Indeed, we will first consider the MFDE 24) in which the function A qf , which is related to the quasi-fronts described in Proposition 4.2, is given by…”

Section: Region Rmentioning

confidence: 99%

Stability of pulse solutions for the discrete FitzHugh–Nagumo system

Hupkes

Sandstede

2012

Trans. Amer. Math. Soc.

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“…Moreover, it has been shown in the proof of theorem 6 in [15] that any fixed point of (21) defines a weak solution of the equation (19). This weak solution is actually a classical solution for our special choice of V s,+ .…”

Section: An Outline Of the Proofmentioning

confidence: 79%

“…As a technical point, it then would be very desirable to have more than continuous dependence on the flight times {ω k } k∈Z which in the case of hyperbolic steady states (i.e. m = 0) has recently been proved in [19]. We should point out that using theorem 3 we are also able to detect globally defined solutions U, whose profile can be far away from the original profile of H pointwisely.…”

Section: Theoremmentioning

confidence: 99%

“…This shows that the m additional parameters are a consequence from the absence of hyperbolicity at the asymptotic steady state; at least, if we work with the integral equation (21), (22). But only in this setting we are able to guarantee that the constructed fixed points V ± i are solutions of (19) and not just a modified equation, where the nonlinearity G is replaced by a modified nonlinearity G mod , which possesses a globally sufficient small Lipschitz-constant. This shows that we can think of the parameters η i 1 , .…”

Section: An Outline Of the Proofmentioning

confidence: 99%

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The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

Georgi¹

2009

Indiana Univ. Math. J.

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In this article we consider Hamiltonian lattice differential equations and investigate the existence of travelling waves near solitary wave solutions. We focus on the case where the solitary wave profile induces a homoclinic solution of the associated traveling wave equation, which is a typical scenario in the Fermi-Pasta-Ulam lattice. We will then show the existence of finitely many scalar bifurcation equations, such that zeros of these equations correspond to multi-pulses or periodic traveling waves of the original lattice equation that are located near the primary solitary wave. Compared to previous works, we will have to overcome technical complications which result from the lack of hyperbolicity of the asymptotic steady state.We use our results to prove the existence of periodic travelling waves accompanying a family of stable solitary waves in the Fermi-Pasta-Ulam lattice, where properties of the latter waves have been recently investigated by Pego and Friesecke [8]. As we will show, these waves persist under small reversible perturbations of the FPU lattice, where they induce a family of solitary waves.

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Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

Hupkes

Morelli

Schouten-Straatman

et al. 2020

Springer Proceedings in Mathematics &Amp; Statistics

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We survey some recent results on traveling waves and pattern formation in spatially discrete bistable reaction-diffusion equations. We start by recalling several classic results concerning the existence, uniqueness and stability of travelling wave solutions to the discrete Nagumo equation with nearest-neighbour interactions, together with the Fredholm theory behind some of the proofs. We subsequently discuss extensions involving wave connections between periodic equilibria, long-range interactions and planar lattices. We show how some of the results can be extended to the two-component discrete FitzHugh-Nagumo equation, which can be analyzed using singular perturbation theory. We conclude by studying the behaviour of the Nagumo equation when discretization schemes are used that involve both space and time, or that are non-uniform but adaptive in space.

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Lin's method and homoclinic bifurcations for functional differential equations of mixed type

Cited by 18 publications

References 15 publications

Stability of pulse solutions for the discrete FitzHugh–Nagumo system

Stability of pulse solutions for the discrete FitzHugh–Nagumo system

The dynamical behaviour near solitary waves in Hamiltonian lattice differential equations

Traveling Waves and Pattern Formation for Spatially Discrete Bistable Reaction-Diffusion Equations

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