Periodic solutions of nonlinear autonomous functional differential equations

Nussbaum, Roger D.

doi:10.1007/bfb0064325

Cited by 35 publications

(10 citation statements)

References 43 publications

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“…Also, f is an odd function. By results of Kaplan and Yorke [11], and Nussbaum [16], we expect solutions of period 4 for ~> re/2. A Fortran program (about 120 statements+Gauss Routine) has been executed on a PDP 11-40.…”

Section: Examplementioning

confidence: 91%

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Hadeler

1980

Numer. Math.

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Section: Examplementioning

confidence: 91%

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Hadeler

1980

Numer. Math.

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“…[Kaplan & Yorke, 1974;Nussbaum, 1978;Walther, 1978]) and numerically [Hadeler, 1980;Doedel & Leung, 1982;Luzyanina et al, 1997]. It was proven that at α = π/2 a family of periodic solutions bifurcates from the zero solution and that stable periodic solutions of period T = 4 exist for α > π/2.…”

Section: Equations and Periodic Solutionsmentioning

confidence: 96%

Computing Floquet Multipliers for Functional Differential Equations

Luzyanina

Engelborghs

2002

Int. J. Bifurcation Chaos

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Floquet multipliers determine the local asymptotic stability of a periodic solution and, in the context of parameter dependence, determine also its bifurcations. This paper deals with numerical aspects of the computation of the Floquet multipliers for three classes of functional differential equations: ordinary differential equations (ODEs), differential equations with constant delay (DDEs) and differential equations with state-dependent delay (sd-DDEs). Using a collocation approach for computing periodic solutions, we obtain an approximation of the (corresponding) monodromy operator, a monodromy matrix. The eigenvalues of this matrix form an approximation to the Floquet multipliers. The accuracy of the computed multipliers is an important issue in bifurcation analysis of a dynamical system. As far as we know, no prior work on the study of the convergence and accuracy of computed Floquet multipliers for DDEs and sd-DDEs exists. We analyze the dependency of the accuracy of the computed multipliers on the parameters and on the type of collocation approximation. In particular, we show that the accuracy of the computed trivial multiplier is not always comparable to the accuracy of the computed periodic solution and the accuracy of the other computed multipliers.

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“…Following the same procedure of [1,11], we use Theorem 2.2 below, due to Nussbaum, see [9] or [4, 11.2, Theorem 2.1]. Definition 2.3.…”

Section: Lemma 24mentioning

confidence: 99%

Periodic and backset solutions of differential delay systems with self-supporting condition

Gadotti

Táboas

2006

Journal of Differential Equations

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Periodic solutions of nonlinear autonomous functional differential equations

Cited by 35 publications

References 43 publications

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Computing Floquet Multipliers for Functional Differential Equations

Periodic and backset solutions of differential delay systems with self-supporting condition

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