Limit cycles for vector fields with homogeneous components

A. Cima; A. Gasukk; F. Mañosas

Applicationes Mathematicae (1997)

  • Volume: 24, Issue: 3, page 281-287
  • ISSN: 1233-7234

Abstract

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We study planar polynomial differential equations with homogeneous components. This kind of equations present a simple and well known dynamics when the degrees (n and m) of both components coincide. Here we consider the case n ≠ m and we show that the dynamics is more complicated. In fact, we prove that such systems can exhibit periodic orbits only when nm is odd. Furthermore, for nm odd we give examples of such differential equations with at least (n+m)/2 limit cycles.

How to cite

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Cima, A., Gasukk, A., and Mañosas, F.. "Limit cycles for vector fields with homogeneous components." Applicationes Mathematicae 24.3 (1997): 281-287. <http://eudml.org/doc/219169>.

@article{Cima1997,
abstract = {We study planar polynomial differential equations with homogeneous components. This kind of equations present a simple and well known dynamics when the degrees (n and m) of both components coincide. Here we consider the case $n\ne m$ and we show that the dynamics is more complicated. In fact, we prove that such systems can exhibit periodic orbits only when nm is odd. Furthermore, for nm odd we give examples of such differential equations with at least (n+m)/2 limit cycles.},
author = {Cima, A., Gasukk, A., Mañosas, F.},
journal = {Applicationes Mathematicae},
keywords = {homogeneous function; limit cycle; vector field; planar polynomial differential equations; periodic orbits; limit cycles},
language = {eng},
number = {3},
pages = {281-287},
title = {Limit cycles for vector fields with homogeneous components},
url = {http://eudml.org/doc/219169},
volume = {24},
year = {1997},
}

TY - JOUR
AU - Cima, A.
AU - Gasukk, A.
AU - Mañosas, F.
TI - Limit cycles for vector fields with homogeneous components
JO - Applicationes Mathematicae
PY - 1997
VL - 24
IS - 3
SP - 281
EP - 287
AB - We study planar polynomial differential equations with homogeneous components. This kind of equations present a simple and well known dynamics when the degrees (n and m) of both components coincide. Here we consider the case $n\ne m$ and we show that the dynamics is more complicated. In fact, we prove that such systems can exhibit periodic orbits only when nm is odd. Furthermore, for nm odd we give examples of such differential equations with at least (n+m)/2 limit cycles.
LA - eng
KW - homogeneous function; limit cycle; vector field; planar polynomial differential equations; periodic orbits; limit cycles
UR - http://eudml.org/doc/219169
ER -

References

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  1. [1] J. Argémi, Sur les points singuliers multiples de systèmes dynamiques dans R 2 , Ann. Mat. Pura Appl. (4) 79 (1968), 35-69. Zbl0203.39802
  2. [2] A. Cima, A. Gasull and F. Ma nosas, Cyclicity of a family of vector fields, J. Math. Anal. Appl. 196 (1995), 921-937. Zbl0851.34027
  3. [3] D. Eisenbud and H. Levine, An algebraic formula for the degree of a C ∞ map germ, Ann. of Math. 106 (1977), 19-44. Zbl0398.57020
  4. [4] W. Fulton, Algebraic Curves. An Introduction to Algebraic Geometry, Benjamin, New York, 1969. Zbl0181.23901
  5. [5] A. M. Lyapunov, Stability of Motion, Math. Sci. Engrng. 30, Academic Press, New York, 1966. Zbl0161.06303
  6. [6] L. S. Pontryagin, On dynamical systems close to the Hamiltonian ones, Zh. Eksperiment. Teoret. Fiz. 4 (1934), 883-885 (in Russian). 

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