Open problems on the algebraic limit cycles of planar polynomial vector fields

J. Llibre

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2008), pp. 19-26.

Résumé

We collect several open problems that have appeared in the study of the algebraic limit cycles of the real planar polynomial vector fields.

Bibliographie

[1] Bautin N. N., “Estimation of the number of algebraic limit cycles of the system $\dot x=P(x,y)$, $\dot y=Q(x,y)$, with algebraic right-hand sides”, Differentsial'nye Uravneniya, 16:2 (1980), 362–383 (in Russian) | MR

[2] Campillo A., Carnicer M. M., “Proximity inequalities and bounds for the degree of invariant curves by foliations of $\mathbf P_{\mathbb C}^2$”, Trans. Amer. Math. Soc., 349 (1997), 2211–2228 | DOI | MR | Zbl

[3] Carnicer M. M., “The Poincaré problem in the nondicritical case”, Annals of Math., 140 (1994), 289–294 | DOI | MR | Zbl

[4] Cerveau D., Lins Neto A., “Holomorphic foliations in $CP(2)$ having an invariant algebraic curve”, Ann. Inst. Fourier, 41:4 (1991), 883–903 | MR | Zbl

[5] Chavarriga J., Giacomini H., Llibre J., “Uniqueness of algebraic limit cycles for quadratic systems”, J. Math. Anal. and Appl., 261 (2001), 85–99 | DOI | MR | Zbl

[6] Chavarriga J., Llibre J., Sorolla J., “Algebraic limit cycles of degree 4 for quadratic systems”, J. Differential Equations, 200 (2004), 206–244 | DOI | MR | Zbl

[7] Christopher C., Llibre J., Świrszcs G., “Invariant algebraic curves of large degree for quadratic system”, J. Math. Anal. Appl., 303 (2005), 450–461 | DOI | MR | Zbl

[8] Christopher C., “Polynomial vector fields with prescribed algebraic limit cycles”, Geometria Dedicata, 88 (2001), 255–258 | DOI | MR | Zbl

[9] Christopher C., Llibre J., “Algebraic aspects of integrability for polynomial systems”, Qualitative Theory of Planar Differential Equations, 1:1 (1999), 71–95 | MR

[10] Christopher C., Llibre J., “Integrability via invariant algebraic curves for planar polynomial differential systems”, Annals of Differential Equations, 16 (2000), 5–19 | MR | Zbl

[11] Christopher C., Llibre J., “A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals”, Proc. Amer. Math. Soc., 130 (2002), 2025–2030 | DOI | MR | Zbl

[12] Darboux G., “Mémoire sur les équations différentielles algébriques du premier ordre et du premier degré (Mélanges)”, Bull. Sci. math. 2ème série, 2 (1878), 60–96 ; 123–144 ; 151–200 | Zbl

[13] Dolov M. V., Kuzmin R. V., “On limit cycles of a class of systems”, Differential Equations, 29:9 (1993), 1282–1285 | MR | Zbl

[14] Dolov M. V., Kuzmin R. V., “Limit cycles of systems with a given particular integral”, Differential Equations, 30:7 (1994), 1044–1050 | MR | Zbl

[15] Evdokimenco R. M., “Construction of algebraic trajectories and a qualitative investigation in the large of the behavior of the integral curves of a certain system of differential equations”, Differential Equations, 6 (1970), 1780–1791 (in Russian) | MR

[16] Evdokimenco R. M., “Behavior of the integral curves of a certain dynamical system”, Differential Equations, 12 (1976), 1557–1567 (in Russian) | MR

[17] Evdokimenco R. M., “Investigation in the large of a dynamical system in the presence of a given integral curve”, Differential Equations, 15 (1979), 215–221 (in Russian) | MR

[18] Filiptsov V. F., “Algebraic limit cycles”, Differential Equations, 9 (1973), 983–986 (in Russian) | MR

[19] Ilyashenko Yu., “Centennial History of Hilbert's 16th Problem”, Bull. Amer. Math. Soc., 39 (2002), 301–354 | DOI | MR | Zbl

[20] Bull. Amer. Math. Soc., 8 (1902), 437–479 | DOI | MR | Zbl | Zbl

[21] Jouanolou J. P., Equations de Pfaff algébriques, Lectures Notes in Mathematics, 708, Springer-Verlag, New York–Berlin, 1979 | MR | Zbl

[22] Jibin Li, “Hilbert's 16th problem and bifurcations of planar polynomial vector fields”, Int. J. Bif. Chaos, 13 (2003), 47–106 | DOI | MR | Zbl

[23] Yung-ching Liu, “On differential equation with algebraic limit cycle of second degree $dy/dx=(a_{10}x+a_{01}y+a_{30}x_3+a_{21}x_2y+a_{12}xy_2+a_{03}y_3)/(b_{10}x+b_{01}y+b_{30}x_3+b_{21}x_2y+b_{12}xy_2+b_{03}y_3)$”, Advancement in Math., 4 (1958), 143–149 | MR

[24] Deming Liu, “The cubic systems with cubic curve solution $y^2=ax^3+bx^2+cx+d$”, Northeast Math. J., 5:4 (1989), 437–448 | MR | Zbl

[25] Llibre J., “Integrability of polynomial differential systems”, Handbook of Differential Equations, Ordinary Differential Equations, Vol. 1, eds. A. Cañada, P. Drabek and A. Fonda, Elsevier, 2004, 437–533 | MR

[26] Llibre J., Rodríguez G., “Configurations of limit cycles and planar polynomial vector fields”, J. Differential Equations, 198 (2004), 374–380 | DOI | MR | Zbl

[27] Llibre J., Swirszcz G., “Relationships between limit cycles and algebraic invariant curves for quadratic systems”, J. Differential Equations, 229 (2006), 529–537 | DOI | MR | Zbl

[28] Llibre J., Swirszcz G., “Classification of quadratic systems admitting the existence of an algebraic limit cycle”, Bull. Sci. Math., 131 (2007), 405–421 | DOI | MR | Zbl

[29] Llibre J., Yulin Zhao, Algebraic limit cycles in polynomial differential systems, Preprint, 2007 | MR

[30] Moulin-Ollagnier J., “About a conjecture on quadratic vector fields”, Journal of Pure and Applied Algebra, 165 (2001), 227–234 | DOI | MR | Zbl

[31] Moulin-Ollagnier J., Nowicki A., Strelcyn J.M., “On the non-existence of constants of derivations: the proof of a theorem of Jouanolou and its development”, Bull. Sci. Math., 119 (1995), 195–233 | MR | Zbl

[32] Odani K., “The limit cycle of the van der Pol equation is not algebraic”, J. Differential Equations, 115 (1995), 146–152 | DOI | MR | Zbl

[33] Poincaré H., “Sur l'intégration des équations différentielles du premier ordre et du premier degré I and II”, Rendiconti del Circolo Matematico di Palermo, 5 (1891), 161–191 ; 11 (1987), 193–239 | DOI | DOI

[34] Yablonskii A. I., “Limit cycles of a certain differential equations”, Differential Equations, 2 (1966), 335–344 (in Russian) | MR

[35] Qin Yuan-Xun, “On the algebraic limit cycles of second degree of the differential equation $dy/dx=\sum_{0\le i+j\le 2}a_{ij}x^iy^j/\sum_{0\le i+j\le 2}b_{ij}x^iy^j$”, Acta Math. Sinica, 8 (1958), 23–35 | MR

[36] Walcher S., “On the Poincaré problem”, J. Differential Equations, 166 (2000), 51–78 | DOI | MR | Zbl