Algorithmic Derivation of Centre Conditions

Pearson, J. M.; Lloyd, N. G.; Christopher, Colin

doi:10.1137/s0036144595283575

Cited by 72 publications

(53 citation statements)

References 19 publications

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“…Since the eigenvalues at the singular point located at the origin of system (1) are λ ± i, the origin is either a weak focus or a center if λ = 0, see for instance [1,15].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems

Llibre

Valls²

2009

Nonlinear Differ. Equ. Appl.

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“…Since the eigenvalues at the singular point located at the origin of system (1) are λ ± i, the origin is either a weak focus or a center if λ = 0, see for instance [1,15].…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems

Llibre

Valls²

2009

Nonlinear Differ. Equ. Appl.

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“…Of course, given such a bound for N , it is then easy to compute the invariant algebraic curves of the system and also describe its elementary or Liouvillian first integrals (modulo any exponential factors) see for instance Man and Maccallum [31], Christopher [6],and Pearson, Lloyd and Christopher [33].…”

Section: An Open Question For Planar Polynomial Vector Fieldsmentioning

confidence: 99%

On the Darboux Integrability of Polynomial Differential Systems

Llibre

Zhang

2011

Qual. Theory Dyn. Syst.

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Abstract. This is a survey on recent results on the Darboux integrability of polynomial vector fields in R n or C n with n ≥ 2. We also provide an open question and some applications based on the existence of such first integrals. IntroductionIn many branches of applied mathematics, physics and, in general, in applied sciences appear nonlinear ordinary differential equations. If a differential equation or vector field defined on a real or complex manifold has a first integral, then its study can be reduced by one dimension. Therefore a natural question is: Given a vector field on a manifold, how to recognize if this vector field has a first integral defined on an open and dense subset of the manifold? In general this question has no a good answer up to now.In this survey we provide sufficient conditions for the existence of a first integral for polynomial vector fields in R n or C n with n ≥ 2. An open question which essentially goes back to Poincaré is presented. Finally some applications of the existence of this kind of first integrals to physical problems, centers, foci, limit cycles and invariant hyperplanes are mentioned. Darboux theory of integrabilitySince any polynomial differential system in R n can be thought as a polynomial differential system inside C n we shall work only in C n . If our initial differential system is in R n , once we get a complex first integral of this system inside C n the real and the imaginary parts of it are real first integrals. Moreover if that complex first integral is rational, the same occur for its real and imaginary parts. In short in the rest of the paper we shall work in C n .In this section we study the existence of first integrals for polynomial vector fields in C n through the Darboux theory of integrability. The algebraic theory of integrability is a classical one, which is related with the first part of the Hilbert's 16th problem [19]. This kind of integrability is usually called Darboux integrability, and it provides a link between the integrability of polynomial vector fields and the number of invariant algebraic hypersurfaces that they have (see Darboux [13] and Poincaré [36]). This theory shows the fascinating relationships between integrability (a topological phenomenon) and the existence of exact algebraic invariant hypersurfaces formed by solutions for the polynomial vector field. This theory is now known as the Darboux theory of integrability, see for more details the Chapter 8 of [16].2010 Mathematics Subject Classification. 34A34, 34C05, 34C14.

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“…. , n−1 (see for instance [16] and [19]). Another method is to construct a Poincaré formal power series in polar coordinates and the Poincaré-Lyapunov constants can be computed from recursive linear formulas as definite integrals of trigonometric polynomials (see for example [1] and [5]).…”

Section: Introductionmentioning

confidence: 99%