Isochronicity via normal form

Algaba, Antonio; Freire, Emilio; Gamero, E.

doi:10.1007/bf02969475

Cited by 44 publications

(72 citation statements)

References 10 publications

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“…To be useful we need to be able to compute m; and control its integral. The existence of U and m satisfying ½X ; U ¼ mX ; for sufficiently regular vector fields X with a non-degenerate centre at p is already known, see for instance [1]. Note also that our expression of T 0 given in (1), and based on the knowledge of U; is simpler that the one obtained in [7].…”

Section: Assume Thatmentioning

confidence: 89%

First derivative of the period function with applications

Freire

Gasull

Guillamón

2004

Journal of Differential Equations

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Given a centre of a planar differential system, we extend the use of the Lie bracket to the determination of the monotonicity character of the period function. As far as we know, there are no general methods to study this function, and the use of commutators and Lie bracket was restricted to prove isochronicity. We give several examples and a special method which simplifies the computations when a first integral is known. r

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Section: Assume Thatmentioning

confidence: 89%

First derivative of the period function with applications

Freire

Gasull

Guillamón

2004

Journal of Differential Equations

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“…We conjecture that in fact all centers of Lie´nard systems are of the form above. Apart from the results of Sabatini, the conjecture is also known to hold in other special cases (Algaba et al [1]). …”

Section: Introductionmentioning

confidence: 85%

On the classification of Liénard systems with amplitude-independent periods

Christopher

Devlin

2004

Journal of Differential Equations

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We consider conditions under which the second-order differential equationx þ f ðxÞ 'x þ gðxÞ ¼ 0 has a family of periodic orbits with constant period. This condition is equivalent to seeking conditions under which the two-dimensional autonomous system 'x ¼ y; ' y ¼ ÀgðxÞ À f ðxÞy has a center with constant period: i.e., an isochronous center. In turn, this is equivalent to the latter system being locally linearizable. We give a simple necessary and sufficient condition for this when f and g are analytic functions. In the case where when f and g are polynomials, we show that this reduces to a finitely determinable system of equations. A complete classification is given of all such systems of degree 34 or less. r 2004 Published by Elsevier Inc.

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“…The following result, proved in [1], characterizes centers of analytic vector fields in the plane in terms of Lie brackets.…”

Section: Theorem 3 a Weak Focus Of An Analytic System Is A Center Ifmentioning

confidence: 95%

“…These two different criteria to detect isochronicity suggest the authors if there exists, in general, an equivalence between linearization and commutation for other types of singular points. In the paper of Algaba et al [1], the characterization of a center of an analytic differential system in terms of Lie brackets is given. From Poincaré it is known the equivalence between integrability of an analytic differential system in a neighborhood of a nondegenerate singular point of focus-center type and the existence of a center.…”

Section: Introductionmentioning

confidence: 99%

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Linearizability and integrability of vector fields via commutation

Giné

Grau

2006

Journal of Mathematical Analysis and Applications

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In this paper, we consider complex smooth and analytic vector fields X in a neighborhood of a nondegenerate singular point. It is proved the equivalence between linearizability and commutation, i.e., the existence of a commuting vector field Y such that the Lie brackets [X , Y] ≡ 0. For complex smooth and analytic vector fields in the plane and in a neighborhood of a nondegenerate singular point, it is also proved the equivalence between integrability and the existence of a smooth vector field Y, such that Y is a normalizer of X , i.e., [X , Y] = µX .  2005 Elsevier Inc. All rights reserved.

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Isochronicity via normal form

Cited by 44 publications

References 10 publications

First derivative of the period function with applications

First derivative of the period function with applications

On the classification of Liénard systems with amplitude-independent periods

Linearizability and integrability of vector fields via commutation

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