Pitchfork bifurcation for non-autonomous interval maps

D’Aniello, Emma; Oliveira, Henrique M.

doi:10.1080/10236190802258669

Cited by 13 publications

(11 citation statements)

References 9 publications

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“…for µ = 2 we have the cusp, for µ = 3 the swallowtail and for µ = 4 the butterfly [4,8,9,14,15,23]. The transversality condition (see pages 66, 297, 298 and 303 of [23]) at the solution of the above conditions is given by the condition on the non-singularity of the Jacobian matrix of the map (f, f x , f xx , .…”

Section: Conditions For the A µ Class Of Bifurcations In Autonomous Smentioning

confidence: 99%

“…repeating this process and using the Faà di Bruno Formula (13) and the bifurcation equations (9), knowing that the lower order terms in derivatives relative to x (order less than j − 1) do not contribute to the diagonal of A, we have for 1 ≤ j ≤ µ…”

Section: Alternating Mapsmentioning

confidence: 99%

“…The same query was posed in [9,11] and positively solved in the particular cases dealt in those works for degeneracy conditions until the cusp, i.e., µ = 2.…”

mentioning

confidence: 99%

“…Hence, if f has negative Schwarzian derivative then g must have positive Schwarzian derivative. As in [9], while working on pitchfork bifurcation, we can state the two following propositions for the A 3 degenerate bifurcation. The proofs are similar.…”

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confidence: 99%

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Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

Oliveira¹

2016

TMNA

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Bifurcations of the Aµ class in Arnold's classification, in nonautonomous p-periodic difference equations generated by parameter depending families with p maps, are studied. It is proved that the conditions of degeneracy, non-degeneracy and unfolding are invariant relative to cyclic order of compositions for any natural number µ. The main tool for the proofs is local topological conjugacy. Invariance results are essential to the proper definition of the bifurcations of class Aµ, and lower codimension bifurcations associated, using all the possible cyclic compositions of the fiber families of maps of the p-periodic difference equation. Finally, we present two actual examples of class A 3 or swallowtail bifurcation occurring in period two difference equations for which the bifurcation conditions are invariant.Date: February, 2015 -AMS: Primary:37G15; Secondary: 39A28.

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Section: Conditions For the A µ Class Of Bifurcations In Autonomous Smentioning

confidence: 99%

Section: Alternating Mapsmentioning

confidence: 99%

“…The same query was posed in [9,11] and positively solved in the particular cases dealt in those works for degeneracy conditions until the cusp, i.e., µ = 2.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

Oliveira¹

2016

TMNA

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“…The special case of nonautonomous periodic discrete systems was, thoroughly, investigated by Elaydi and Sacker [18], and the references there in, by Kloeden [24], Silva [32] and Franco, Silva, and Simões [21]. D'Aniello and Steele ([12], [13]) investigated the limit sets of 2-periodic (alternating) systems, and D'Aniello and Oliveira [11] investigated attracting periodic orbits of these systems.…”

mentioning

confidence: 99%

The structure of <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets of asymptotically non-autonomous discrete dynamical systems

D’Aniello¹,

Elaydi²

2020

Discrete &Amp; Continuous Dynamical Systems - B

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We consider a discrete non-autonomous semi-dynamical system generated by a family of continuous maps defined on a locally compact metric space. It is assumed that this family of maps uniformly converges to a continuous map. Such a non-autonomous system is called an asymptotically autonomous system. We extend the dynamical system to the metric one-point compactification of the phase space. This is done via the construction of an associated skew-product dynamical system. We prove, among other things, that the omega limit sets are invariant and invariantly connected. We apply our results to two populations models, the Ricker model with no Allee effect and Elaydi-Sacker model with the Allee effect, where it is assumed that the reproduction rate changes with time due to habitat fluctuation.

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Towards a Theory of Periodic Difference Equations and Its Application to Population Dynamics

Elaydi

Luís

Oliveira

2011

Dynamics, Games and Science I

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This survey contains the most updated results on the dynamics of periodic difference equations or discrete dynamical systems this time. Our focus will be on stability theory, bifurcation theory, and on the effect of periodic forcing on the welfare of the population (attenuance versus resonance). Moreover, the survey alludes to two more types of dynamical systems, namely, almost periodic difference equations and stochastic difference equations.

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Pitchfork bifurcation for non-autonomous interval maps

Cited by 13 publications

References 9 publications

Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

Invariance of bifurcation equations for high degeneracy bifurcations of non-autonomous periodic maps

The structure of <inline-formula><tex-math id="M1">\begin{document}$ \omega $\end{document}</tex-math></inline-formula>-limit sets of asymptotically non-autonomous discrete dynamical systems

Towards a Theory of Periodic Difference Equations and Its Application to Population Dynamics

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