Non-integrability of measure preserving maps via Lie symmetries

Cima, Anna; Gasull, Armengol; Mañosa, Vı́ctor

doi:10.1016/j.jde.2015.06.019

Cited by 8 publications

(7 citation statements)

References 21 publications

(36 reference statements)

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“…The period function is important to study theoretical properties of planar ordinary differential equations and their perturbations, see for instance [9, pp. 369-370]; to understand some mathematical models in physics or ecology, see [14,17,39,45] and the references therein; in the description of the dynamics of some discrete dynamical systems, see [6,11,12]; or for counting the solutions of some boundary value problems, see [7,8]. When the system is Hamiltonian, with Hamiltonian function H and γ h ⊂ {H = h}, it is natural to consider s = h and write T = T (h).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The local period function for Hamiltonian systems with applications

Buzzi¹,

Carvalho²,

Gasull³

2020

Preprint

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In the first part of the paper we develop a constructive procedure to obtain the Taylor expansion, in terms of the energy, of the period function for a non-degenerated center of any planar analytic Hamiltonian system. We apply it to several examples, including the whirling pendulum and a cubic Hamiltonian system. The knowledge of this Taylor expansion of the period function for this system is one of the key points to study the number of zeroes of an Abelian integral that controls the number of limit cycles bifurcating from the periodic orbits of a planar Hamiltonian system that is inspired by a physical model on capillarity. Several other classical tools, like for instance Chebyshev systems are applied to study this number of zeroes. The approach introduced can also be applied in other situations.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

The local period function for Hamiltonian systems with applications

Buzzi¹,

Carvalho²,

Gasull³

2020

Preprint

Self Cite

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“…From a more mathematical point of view, it is important in the study of the bifurcations from a continuum of periodic orbits giving rise to isolated ones, see [8, pp. 369-370], in the description of the dynamics of some discrete dynamical systems, see [5,10,11] or for counting the solutions of some boundary value problems, see [6,7].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global behaviour of the period function of the sum of two quasi-homogeneous vector fields

Alvarez

Gasull

Prohens

2017

Journal of Mathematical Analysis and Applications

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We study the global behaviour of the period function on the period annulus of degenerate centers for two families of planar polynomial vector fields. These families are the quasi-homogeneous vector fields and the vector fields given by the sum of two quasi-homogeneous Hamiltonian ones. In the first case we prove that the period function is globally decreasing, extending previous results that deal either with the Hamiltonian quasi-homogeneous case or with the general homogeneous situation. In the second family, and after adding some more additional hypotheses, we show that the period function of the origin is either decreasing or has at most one critical period and that both possibilities may happen. This result also extends some previous results that deal with the situation where both vector fields are homogenous and the origin is a non-degenerate center.

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“…Some results about using KAM theory are given in References [9][10][11][12][13][14][15][16][17][18][19][20][21]. Other techniques have been used (not only KAM theory) to study the Lyness' equation (see References [9][10][11]17,[22][23][24][25][26][27]).…”

Section: Introductionmentioning

confidence: 99%

Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

Ibrahim

Nurkanović

2019

Mathematics

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By using the Kolmogorov-Arnold-Moser (KAM) theory, we investigate the stability of two elliptic equilibrium points (zero equilibrium and negative equilibrium) of the difference equation t n+1 = αt n + βt 2 n − t n−1 , n = 0, 1, 2, . . . , where are t −1 , t 0 , α ∈ R, α = 0, β > 0. By using the symmetries we find the periodic solutions with some periods. Finally, some numerical examples are given to verify our theoretical results. MSC: 39A10; 39A11; 37E40; 37J40; 37N25In studying the global dynamics of (1) and (2), with non-negative initial conditions and non-negative parameters, the authors used the theory of monotonic maps.There is extensive literature on polynomial difference equations in the complex domain in the last 30 years by mathematicians. First investigations on polynomial difference equations with non-negative parameters and initial conditions were for a special case

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Non-integrability of measure preserving maps via Lie symmetries

Cited by 8 publications

References 21 publications

The local period function for Hamiltonian systems with applications

The local period function for Hamiltonian systems with applications

Global behaviour of the period function of the sum of two quasi-homogeneous vector fields

Kolmogorov-Arnold-Moser Theory and Symmetries for a Polynomial Quadratic Second Order Difference Equation

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