# 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

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topCima, 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

top- [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] 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] D. Eisenbud and H. Levine, An algebraic formula for the degree of a ${C}^{\infty}$ map germ, Ann. of Math. 106 (1977), 19-44. Zbl0398.57020
- [4] W. Fulton, Algebraic Curves. An Introduction to Algebraic Geometry, Benjamin, New York, 1969. Zbl0181.23901
- [5] A. M. Lyapunov, Stability of Motion, Math. Sci. Engrng. 30, Academic Press, New York, 1966. Zbl0161.06303
- [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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