On the absence of a real-analytic first integral for ABC flow when A=B

1998 ◽  
Vol 8 (1) ◽  
pp. 272-273 ◽  
Author(s):  
S. L. Ziglin
Keyword(s):  
First Integral ◽  
Real Analytic ◽  
Analytic First Integral

Analytic nilpotent centers with analytic first integral

2010 ◽  
Vol 72 (9-10) ◽  
pp. 3732-3738 ◽  
Author(s):  
Isaac A. García ◽  
Jaume Giné
Keyword(s):  
First Integral ◽  
Analytic First Integral

Jets of differential equations having integrable extensions

1998 ◽  
Vol 18 (6) ◽  
pp. 1527-1544
Author(s):  
MASSIMO VILLARINI
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
First Integral ◽  
Jet Space ◽  
Analytic First Integral

We characterize the set of $n$-jets admitting an extension which is a germ of a differential equation with an analytic first integral, and compute its codimension in the $n$-jet space. Some applications in the case of the centre-focus problem are given.

scholarly journals Analytic integrability of quadratic–linear polynomial differential systems

2009 ◽  
Vol 31 (1) ◽  
pp. 245-258 ◽  
Author(s):  
JAUME LLIBRE ◽  
CLÀUDIA VALLS
Keyword(s):  
Singular Point ◽  
Explicit Expression ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Analytic Integrability ◽  
Linear Polynomial ◽  
Analytic First Integral

AbstractFor the quadratic–linear polynomial differential systems with a finite singular point, we classify the ones which have a global analytic first integral, and provide the explicit expression of their first integrals.

A note on analytic integrability of planar vector fields

2012 ◽  
Vol 23 (5) ◽  
pp. 555-562 ◽  
Author(s):  
A. ALGABA ◽  
C. GARCÍA ◽  
M. REYES
Keyword(s):  
Vector Field ◽  
Integrable Systems ◽  
Analytic Functions ◽  
Vector Fields ◽  
First Integral ◽  
Analytic Integrability ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
Analytic First Integral ◽  
Image Position

We give a new characterisation of integrability of a planar vector field at the origin. This allows us to prove that the analytic systemswhereh,K, Ψ and ξ are analytic functions defined in the neighbourhood ofOwithK(O) ≠ 0 or Ψ(O) ≠ 0 andn≥ 1, have a local analytic first integral at the origin. We show new families of analytically integrable systems that are held in the above class. In particular, this class includes all the nilpotent and generalised nilpotent integrable centres that we know.

scholarly journals Singular Levi-flat hypersurfaces in complex projective space induced by curves in the Grassmannian

2015 ◽  
Vol 26 (05) ◽  
pp. 1550036 ◽  
Author(s):  
Jiří Lebl
Keyword(s):  
Algebraic Curve ◽  
Complex Projective Space ◽  
First Integral ◽  
Holomorphic Foliation ◽  
Singular Set ◽  
Analytic Subvariety ◽  
Codimension One ◽  
Real Analytic ◽  
Complex Cone ◽  
Complex Projective

Let H ⊂ ℙn be a real-analytic subvariety of codimension one induced by a real-analytic curve in the Grassmannian G(n + 1, n). Assuming H has a global defining function, we prove H is Levi-flat, the closure of its smooth points of top dimension is a union of complex hyperplanes, and its singular set is either of dimension 2n - 2 or dimension 2n - 4. If the singular set is of dimension 2n - 4, then we show the hypersurface is algebraic and the Levi-foliation extends to a singular holomorphic foliation of ℙn with a meromorphic (rational of degree 1) first integral. In this case, H is in some sense simply a complex cone over an algebraic curve in ℙ1. Similarly if H has a degenerate singularity, then H is also algebraic. If the dimension of the singular set is 2n - 2 and is nondegenerate, we show by construction that the hypersurface need not be algebraic nor semialgebraic. We construct a Levi-flat real-analytic subvariety in ℙ2 of real codimension 1 with compact leaves that is not contained in any proper real-algebraic subvariety of ℙ2. Therefore a straightforward analogue of Chow's theorem for Levi-flat hypersurfaces does not hold.

ON THE DEGENERATE CENTER PROBLEM

2011 ◽  
Vol 21 (05) ◽  
pp. 1383-1392 ◽  
Author(s):  
JAUME GINÉ
Keyword(s):  
First Integral ◽  
New Method ◽  
Center Problem ◽  
Linear Type ◽  
Analytic First Integral

In this work, it is proved that any degenerate center is limit of a [Formula: see text] linear type center and when the degenerate center has an analytic first integral then it is limit of an analytic linear type center. A new method to detect integrability developed in [Giné & Santallusia, 2011] is applied to the degenerate center problem. Moreover, a review of the most important recent contributions to the degenerate center problem is given.

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