An $S^1$-equivariant degree and the Fuller index

Dylawerski, Grzegorz; Gęba, Kazimierz; Jodel, Jerzy; Marzantowicz, Wacław

doi:10.4064/ap-52-3-243-280

Cited by 35 publications

(31 citation statements)

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“…and [21] established the following global bifurcation result by applying the S 1 -degree theory due to Dylawerski, Geba, Jodel and Marzantowicz [12].…”

Section: A Topological Global Hopf Bifurcation Theoremmentioning

confidence: 99%

Symmetric functional differential equations and neural networks with memory

1998

Trans. Amer. Math. Soc.

404

134

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Abstract. We establish an analytic local Hopf bifurcation theorem and a topological global Hopf bifurcation theorem to detect the existence and to describe the spatial-temporal pattern, the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry. We apply these general results to obtain the coexistence of multiple large-amplitude wave solutions for the delayed Hopfield-Cohen-Grossberg model of neural networks with a symmetric circulant connection matrix.

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“…and [21] established the following global bifurcation result by applying the S 1 -degree theory due to Dylawerski, Geba, Jodel and Marzantowicz [12].…”

Section: A Topological Global Hopf Bifurcation Theoremmentioning

confidence: 99%

Symmetric functional differential equations and neural networks with memory

1998

Trans. Amer. Math. Soc.

404

134

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“…As mentioned in Section 2, Theorem 2.3 can be established by an argument similar to that of Erbe et al [19], using the S 1 -degree and bifurcation theory of Dylawerski et al [18] and Geba and Marzantowicz [21]. In this section, we will take (2.9) (2.10) as an example to demonstrate the main steps of the proofs of Theorems 2.3 and 2.4.…”

Section: Periodic Traveling Wavesmentioning

confidence: 99%

Asymptotic and Periodic Boundary Value Problems of Mixed FDEs and Wave Solutions of Lattice Differential Equations

Zou

1997

Journal of Differential Equations

119

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We discuss the existence and approximation of solutions of asymptotic or periodic boundary value problems of mixed functional differential equations. Our approach is via monotone iteration and non-standard ordering in the profile set for asymptotic boundary value problems and via S 1 -degree and equivariant bifurcation theory for periodic boundary value problems. Applications will be given to wave fronts and to slowly oscillatory spatially periodic traveling waves of lattice delay differential equations arising from population genetics, population dynamics, and neural networks.1997 Academic Press

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“…[14,16,18,19,21,27], see also [2,8,15,24,28,29,30,31,34]), which are important tools of the equivariant analysis, provide an effective alternative to such methods as Conley index, Morse theory, minimax techniques and singularity theory. The main difficulty related to the usage of the equivariant degree seems to be its complicated construction relying on the notions from the equivariant topology, homotopy theory and algebraic topology.…”

Section: Introductionmentioning

confidence: 99%