Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

Denette, Erin; Kulenović, M. R. S.; Pilav, Esmir

doi:10.3390/math4010020

Cited by 4 publications

(2 citation statements)

References 18 publications

(11 reference statements)

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“…As we can see in [3,4,5,7], symmetries are important in the study of area-preserving maps because they yield special dynamic behavior. A transformation R of the plane is a time reversal symmetry for T if R −1 • T • R = T −1 .…”

Section: Symmetriesmentioning

confidence: 99%

Birkhoff Normal Forms, Kam Theory, Periodicity and Symmetries for Certain Rational Difference Equation With Cubic Terms

Nurkanović¹,

Nurkanović²

2016

Sarajevo J. Math.

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By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of the positive elliptic equilibrium point of the difference equation xn+1 = Ax 3 n + B axn−1 , n = 0, 1, 2,. .. where the parameters A, B, a and the initial conditions x−1, x0 are positive numbers. The specific feature of this difference equation is the fact that we were not able to use the invariant to prove stability or to find feasible periods of the solutions.

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Section: Symmetriesmentioning

confidence: 99%

Birkhoff Normal Forms, Kam Theory, Periodicity and Symmetries for Certain Rational Difference Equation With Cubic Terms

Nurkanović¹,

Nurkanović²

2016

Sarajevo J. Math.

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“…See also [21] for the results on the stability of Lyness equation with period two coefficient by using KAM theory. In [1,7] authors consider the rational second-order difference equation…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of a certain class of difference equations by using KAM theory

et al. 2019

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By using KAM theory we investigate the stability of equilibrium points of the class of difference equations of the form x n+1 = f (x n) x n-1 , n = 0, 1,. .. , f : (0, +∞) → (0, +∞), f is sufficiently smooth and the initial conditions are x-1 , x 0 ∈ (0, +∞). We establish when an elliptic fixed point of the associated map is non-resonant and non-degenerate, and we compute the first twist coefficient α 1. Then we apply the results to several difference equations.

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Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

Kalabušić

Pilav

2021

Int. J. Biomath.

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Using the Kolmogorov–Arnold–Mozer (KAM) theory, we investigate the stability of May’s host–parasitoid model’s solutions with proportional stocking upon the parasitoid population. We show the existence of the extinction, boundary, and interior equilibrium points. When the host population’s intrinsic growth rate and the releasement coefficient are less than one, both populations are extinct. There are an infinite number of boundary equilibrium points, which are nonhyperbolic and stable. Under certain conditions, there appear 1:1 nonisolated resonance fixed points for which we thoroughly described dynamics. Regarding the interior equilibrium point, we use the KAM theory to prove its stability. We give a biological meaning of obtained results. Using the software package Mathematica, we produce numerical simulations to support our findings.

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Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map

Cited by 4 publications

References 18 publications

Birkhoff Normal Forms, Kam Theory, Periodicity and Symmetries for Certain Rational Difference Equation With Cubic Terms

Birkhoff Normal Forms, Kam Theory, Periodicity and Symmetries for Certain Rational Difference Equation With Cubic Terms

Stability analysis of a certain class of difference equations by using KAM theory

Stability of May’s Host–Parasitoid model with variable stocking upon parasitoids

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