On the Poincaré Problem

Walcher, Sebastian

doi:10.1006/jdeq.2000.3801

Cited by 46 publications

(63 citation statements)

References 25 publications

(60 reference statements)

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“…With Bezout's theorem, one may thus hope to obtain degree bounds for the possible semi-invariants. This approach works well for the stationary points at infinity; see [27] for technicalities and a stronger version of the following statement, and for details on stationary points at infinity. Here it suffices to know that stationary points at infinity correspond to invariant straight lines of the homogeneous term (P (m) , Q (m) ) of highest degree of the vector field in (2).…”

Section: The Direct Problemsmentioning

confidence: 99%

“…Generally, the problem to find all invariant algebraic curves of a given vector field is still unresolved. A relatively successful strategy, which will be briefly outlined, is to use local information at the stationary points; see [27]: Consider a local analytic (or formal) vector field X with stationary point 0, and non-nilpotent linearization, thus without loss of generality P (x, y) = λx + · · · Q(x, y) = µy + · · · with µ = 0.…”

Section: The Direct Problemsmentioning

confidence: 99%

“…We provide a partial picture of the local setting; more and more detailed information is available; see e.g. [27], Theorem 2.3. The first statement in the following Proposition is classical; see e.g.…”

Section: The Direct Problemsmentioning

confidence: 99%

“…For instance, a stationary point of a (real) polynomial vector field with inverse polynomial integrating factor f −1 is a center if it is a center by linearization (see e.g. [27]). …”

Section: Introductionmentioning

confidence: 99%

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Inverse Problems in Darboux’ Theory of Integrability

et al. 2012

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Abstract. The Darboux theory of integrability for planar polynomial differential equations is a classical field, with connections to Lie symmetries, differential algebra and other areas of mathematics. In the present paper we introduce the concepts, problems and inverse problems, and we outline some recent results on inverse problems. We also prove a new result, viz. a general finiteness theorem for the case of prescribed integrating factors. A number of relevant examples and applications is included.

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Section: The Direct Problemsmentioning

confidence: 99%

Section: The Direct Problemsmentioning

confidence: 99%

Section: The Direct Problemsmentioning

confidence: 99%

“…For instance, a stationary point of a (real) polynomial vector field with inverse polynomial integrating factor f −1 is a center if it is a center by linearization (see e.g. [27]). …”

Section: Introductionmentioning

confidence: 99%

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Inverse Problems in Darboux’ Theory of Integrability

et al. 2012

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“…A natural question which goes back to Poincaré [36] is: to give an effective procedure to find N . Partial answers to this question were given by Cerveau and Lins Neto [3], Carnicer [2], Campillo and Carnicer [1], and Walcher [42]. These results depend on either restricting the nature of the polynomial differential system, or more specifically on the singularities of its invariant algebraic curves.…”

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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Analytic normal forms and symmetries of strict feedforward control systems

Tall

Respondek

2010

Intl J Robust & Nonlinear

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This paper deals with the problem of convergence of normal forms of control systems. We identify an n‐dimensional subclass of control systems, called special strict feedforward form, shortly (SSFF), possessing a normal form which is a smooth (resp. analytic) counterpart of the formal normal form of Kang. We provide a constructive algorithm and illustrate by several examples including the Kapitsa pendulum and the cart–pole system. The second part of the paper is concerned about symmetries of single‐input control systems. We show that any symmetry of a smooth system in SSFF is conjugated to a scaling translation and any 1‐parameter family of symmetries is conjugated to a family of scaling translations along the first variable. We compute explicitly those symmetries by finding the conjugating diffeomorphism. We illustrate our results by computing the symmetries of the cart–pole system. Copyright © 2009 John Wiley & Sons, Ltd.

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On the Poincaré Problem

Cited by 46 publications

References 25 publications

Inverse Problems in Darboux’ Theory of Integrability

Inverse Problems in Darboux’ Theory of Integrability

On the Darboux Integrability of Polynomial Differential Systems

Analytic normal forms and symmetries of strict feedforward control systems

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