Center manifolds for periodic functional differential equations of mixed type

Hupkes, Hermen Jan; Lunel, Sjoerd M. Verduyn

doi:10.1016/j.jde.2008.02.026

Cited by 23 publications

(21 citation statements)

References 26 publications

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“…As exhibited in detail in [8], lattice differential equations have many modelling applications in a wide range of scientific disciplines. As a consequence they are attracting a considerable amount of interest, both from an applied as well as a theoretical perspective.…”

Section: Introductionmentioning

confidence: 99%

“…One of the driving forces in these investigations is the desire to apply the powerful tools that are currently available for ODEs to the infinite dimensional setting of (1.1). The constructions in [10,11] concerning finite dimensional center manifolds, which describe the behaviour of solutions to (1.1) in the vicinity of equilibria and periodic solutions, should be seen in this light.…”

Section: Introductionmentioning

confidence: 99%

“…In previous work [9,10,11], the absence of a variation-of-constants formula was circumvented by utilizing variants of the Greens function that was constructed by Mallet-Paret for autonomous MFDEs [14]. Continuing in this spirit, we will use the Fredholm theory developed in [14] for nonautonomous MFDEs to construct inverses for inhomogeneous MFDEs on half-lines.…”

Section: Introductionmentioning

confidence: 99%

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

Hupkes

Lunel²

2009

Indiana Univ. Math. J.

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We extend Lin's method for use in the setting of parameter-dependent nonlinear functional differential equations of mixed type (MFDEs). We show that the presence of M -homoclinic and M -periodic solutions that bifurcate from a prescribed homoclinic connection can be detected by studying a finite dimensional bifurcation equation. As an application, we describe the codimension two orbit-flip bifurcation in the setting of MFDEs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Hupkes

Lunel²

2009

Indiana Univ. Math. J.

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“…As a consequence, the LDE (1.6) admits spatial anisotropy in the sense that the wave speed c depends on the angle θ of propagation through the lattice. Numerical illustrations of this fact can be found in [25,14,27].…”

Section: Lattice Differential Equationsmentioning

confidence: 98%

Negative Diffusion and Traveling Waves in High Dimensional Lattice Systems

Hupkes¹,

Vleck²

2013

SIAM J. Math. Anal.

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Abstract. We consider bistable reaction diffusion systems posed on rectangular lattices in two or more spatial dimensions. The discrete diffusion term is allowed to have positive spatially periodic coefficients, and the two spatially periodic equilibria are required to be well ordered. We establish the existence of traveling wave solutions to such pure lattice systems that connect the two stable equilibria. In addition, we show that these waves can be approximated by traveling wave solutions to systems that incorporate both local and nonlocal diffusion. In certain special situations our results can also be applied to reaction diffusion systems that include (potentially large) negative coefficients. Indeed, upon splitting the lattice suitably and applying separate coordinate transformations to each sublattice, such systems can sometimes be transformed into a periodic diffusion problem that fits within our framework. In such cases, the resulting traveling structure for the original system has a separate wave profile for each sublattice and connects spatially periodic patterns that need not be well ordered. There is no direct analogue of this procedure that can be applied to reaction diffusion systems with continuous spatial variables.

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“…; x n ðtÞ T A R n ; the A i and B j are n-by-n real matrices and the r i and t j are positive real numbers. Systems of this kind can arise in the study on travelling waves in domains with nonlocal interactions initiated in [1], [2], and more recently developed in [3], [4]. Equation (1) can be looked in the more general framework of the mixed functional di¤erential system…”

Section: Introductionmentioning

confidence: 99%