Periodicities in Nonlinear Difference Equations

Grove, E.A.; Ladas, G.

doi:10.1201/9781420037722

Cited by 244 publications

(246 citation statements)

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“…This paper, rather of being a survey on global periodicity, wants to show the techniques that we have developed to detect globally periodic maps inside a parametric families of maps. The reader is addressed to [3,28,36], and the references therein, to have a general overview on the problem.…”

Section: Introductionmentioning

confidence: 99%

Different Approaches to the Global Periodicity Problem

Cima

Gasull

Mañosa

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

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Let F be a real or complex n-dimensional map. It is said that F is globally periodic if there exists some p ∈ N + such that F p (x) = x for all x, whereThe minimal p satisfying this property is called the period of F. Given a m-dimensional parametric family of maps, say F λ , a problem of current interest is to determine all the values of λ such that F λ is globally periodic, together with their corresponding periods. The aim of this paper is to show some techniques that we use to face this question, as well as some recent results that we have obtained. We will focus on proving the equivalence of the problem with the complete integrability of the dynamical system induced by the map F, and related issues; on the use of the local linearization given by the Bochner Theorem; and on the use the Normal Form theory. We also present some open questions in this setting.

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Section: Introductionmentioning

confidence: 99%

Different Approaches to the Global Periodicity Problem

Cima

Gasull

Mañosa

et al. 2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…We will use the following theorem, which is essentially a slight modification of Theorem 1.16 of [5]. The proof of Theorem 1.16 of [5] can be easily adapted to this case. .…”

Section: Global Behaviour Of Solutions Of System (14)mentioning

confidence: 99%

On the stability of some systems of exponential difference equations

Psarros

Papaschinopoulos

Schinas

2018

OpMath

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Abstract. In this paper we prove the stability of the zero equilibria of two systems of difference equations of exponential type, which are some extensions of an one-dimensional biological model. The stability of these systems is investigated in the special case when one of the eigenvalues is equal to -1 and the other eigenvalue has absolute value less than 1, using centre manifold theory. In addition, we study the existence and uniqueness of positive equilibria, the attractivity and the global asymptotic stability of these equilibria of some related systems of difference equations.

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“…Obviously all solution sequences of (1) can be embedded in solution sequences of (14). Notice that the right hand side of this sytem can be rewritten in pseudo-linear form:…”

Section: Where X(t) = (J (T) N (T) B(t)) and T γ (X(t)) Denotes Thmentioning

confidence: 99%

Discrete three-stage population model: persistence and global stability results

Ackleh

Leenheer

2008

Journal of Biological Dynamics

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A general three-stage discrete-time population model is studied. The inherent net reproductive number for this model is derived. Global stability of the origin is established provided that the inherent net reproductive number is less than one. If it is larger than one the existence of a unique positive fixed point is proved and the persistence of the system is established. Finally, for certain parameter ranges global stability of the positive fixed point is proved.

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Periodicities in Nonlinear Difference Equations

Cited by 244 publications

References 0 publications

Different Approaches to the Global Periodicity Problem

Different Approaches to the Global Periodicity Problem

On the stability of some systems of exponential difference equations

Discrete three-stage population model: persistence and global stability results

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