Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps

Dannan, Fozi M.; Elaydi, Saber; Ponomarenko, Vadim

doi:10.1080/1023619031000078315

Cited by 27 publications

(19 citation statements)

References 9 publications

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“…In the recent paper [3], we have introduced what we call stability constants to study the stability of the origin of one-dimensional maps of the form (2) and also of periodic discrete dynamical systems with a common fixed point. A summary of results on this issue can also be found in [6]. The analysis of these constants plays also an important role in the study of the cyclicity, as the proof of our main result of this paper evidences.…”

Section: Introductionsupporting

confidence: 59%

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

Cima

Gasull

Mañosa

2018

Journal of Mathematical Analysis and Applications

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We introduce the concept of 2-cyclicity for families of one-dimensional maps with a non-hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus. This new concept is useful in order to study the number of 2-periodic orbits that can bifurcate from the fixed point. As an application we study the 2-cyclicity of some natural families of polynomial maps.Peer ReviewedPostprint (author's final draft

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Section: Introductionsupporting

confidence: 59%

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

Cima

Gasull

Mañosa

2018

Journal of Mathematical Analysis and Applications

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“…We start recalling some definitions, see [13,14,21]. (ii) Repeller if there exists ε 0 > 0 such that for any 0 < ε < ε 0 and for all x 0 = p such…”

Section: Examples and Some Definitionsmentioning

confidence: 99%

Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

Cima¹,

Gasull²,

Mañosa³

2018

Discrete &Amp; Continuous Dynamical Systems - A

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“…By Theorem 2.3 in [21], since -(1 -β) < 0, the fixed point 0 of the map (11) is semiasymptotically stable from the right. Thus E 0 of model (3) is locally asymptotically stable in {(x, y) :…”

Section: Stability Of Equilibrium Pointsmentioning

confidence: 99%

Stability and bifurcation analysis for a single-species discrete model with stage structure

Zhao

Zhang

2018

Adv Differ Equ

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In this paper, a single-species discrete model with stage structure is investigated. By analyzing the corresponding characteristic equations, the local asymptotic stability of non-negative equilibrium points and the existence of flip bifurcation are discussed. Using the center manifold theory, the stability of the non-hyperbolic equilibrium point is obtained. Based on bifurcation theory, we obtain the direction and the stability of a flip bifurcation at the positive equilibrium with the birth rate as the bifurcation parameter. Finally, some numerical simulations, including phase portraits, chaotic bands with period windows, and Lyapunov exponent methods, are performed to validate the theoretical results, which extends the results in previous papers.

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Stability of Hyperbolic and Nonhyperbolic Fixed Points of One-dimensional Maps

Cited by 27 publications

References 9 publications

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

Bifurcation of 2-periodic orbits from non-hyperbolic fixed points

Parrondo's dynamic paradox for the stability of non-hyperbolic fixed points

Stability and bifurcation analysis for a single-species discrete model with stage structure

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