Polynomial algebra of constants of the Lotka-Volterra system

Jean Moulin Ollagnier; Andrzej Nowicki

Colloquium Mathematicae (1999)

  • Volume: 81, Issue: 2, page 263-270
  • ISSN: 0010-1354

Abstract

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Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form d = x ( C y + z ) ∂ ∂ x + y ( A z + x ) ∂ ∂ y + z ( B x + y ) ∂ ∂ z , called the Lotka-Volterra derivation, where A,B,C ∈ k.

How to cite

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Moulin Ollagnier, Jean, and Nowicki, Andrzej. "Polynomial algebra of constants of the Lotka-Volterra system." Colloquium Mathematicae 81.2 (1999): 263-270. <http://eudml.org/doc/210738>.

@article{MoulinOllagnier1999,
abstract = {Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form $d = x(Cy+z)\frac\{∂\}\{∂x\} + y(Az+x)\frac\{∂\}\{∂y\} + z(Bx+y)\frac\{∂\}\{∂z\}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.},
author = {Moulin Ollagnier, Jean, Nowicki, Andrzej},
journal = {Colloquium Mathematicae},
language = {eng},
number = {2},
pages = {263-270},
title = {Polynomial algebra of constants of the Lotka-Volterra system},
url = {http://eudml.org/doc/210738},
volume = {81},
year = {1999},
}

TY - JOUR
AU - Moulin Ollagnier, Jean
AU - Nowicki, Andrzej
TI - Polynomial algebra of constants of the Lotka-Volterra system
JO - Colloquium Mathematicae
PY - 1999
VL - 81
IS - 2
SP - 263
EP - 270
AB - Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form $d = x(Cy+z)\frac{∂}{∂x} + y(Az+x)\frac{∂}{∂y} + z(Bx+y)\frac{∂}{∂z}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.
LA - eng
UR - http://eudml.org/doc/210738
ER -

References

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  1. [1] B. Grammaticos, J. Moulin Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in ℝ 3 : the Lotka-Volterra system, Phys. A 163 (1990), 683-722. Zbl0714.34005
  2. [2] J. Moulin Ollagnier, Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math. 121 (1997), 463-476. Zbl0887.34002
  3. [3] J. Moulin Ollagnier, Rational integration of the Lotka-Volterra system, ibid. 123 (1999), 437-466. 
  4. [4] J. Moulin Ollagnier, A. Nowicki and J.-M. Strelcyn, On the non-existence of constants of derivations: the proof of a theorem of Jouanolou and its development, ibid. 119 (1995), 195-233. Zbl0855.34010
  5. [5] A. Nowicki, Polynomial derivations and their rings of constants, N. Copernicus University Press, Toruń, 1994. 
  6. [6] A. Nowicki, On the non-existence of rational first integrals for systems of linear differential equations, Linear Algebra Appl. 235 (1996), 107-120. Zbl0843.34013
  7. [7] A. Nowicki and M. Nagata, Rings of constants for k-derivations in k [ x 1 , . . . , x n ] , J. Math. Kyoto Univ. 28 (1988), 111-118. Zbl0665.12024
  8. [8] A. Nowicki and J.-M. Strelcyn, Generators of rings of constants for some diagonal derivations in polynomial rings, J. Pure Appl. Algebra 101 (1995), 207-212. Zbl0832.12002

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