The inverse integrating factor and the Poincaré map

García, Isaac; Giacomini, Héctor; Grau, Maite

doi:10.1090/S0002-9947-10-05014-2

The inverse integrating factor and the Poincaré map
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by Isaac A. García, Héctor Giacomini and Maite Grau

Trans. Amer. Math. Soc. 362 (2010), 3591-3612

DOI: https://doi.org/10.1090/S0002-9947-10-05014-2

Published electronically: February 17, 2010

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Abstract:

This work is concerned with planar real analytic differential systems with an analytic inverse integrating factor defined in a neighborhood of a regular orbit. We show that the inverse integrating factor defines an ordinary differential equation for the transition map along the orbit. When the regular orbit is a limit cycle, we can determine its associated Poincaré return map in terms of the inverse integrating factor. In particular, we show that the multiplicity of a limit cycle coincides with the vanishing multiplicity of an inverse integrating factor over it. We also apply this result to study the homoclinic loop bifurcation. We only consider homoclinic loops whose critical point is a hyperbolic saddle and whose Poincaré return map is not the identity. A local analysis of the inverse integrating factor in a neighborhood of the saddle allows us to determine the cyclicity of this polycycle in terms of the vanishing multiplicity of an inverse integrating factor over it. Our result also applies in the particular case in which the saddle of the homoclinic loop is linearizable, that is, the case in which a bound for the cyclicity of this graphic cannot be determined through an algebraic method.

References

A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maĭer, Theory of bifurcations of dynamic systems on a plane, Halsted Press [John Wiley & Sons], New York-Toronto; Israel Program for Scientific Translations, Jerusalem-London, 1973. Translated from the Russian. MR 0344606
V. I. Arnol′d and Yu. S. Il′yashenko, Ordinary differential equations [Current problems in mathematics. Fundamental directions, Vol. 1, 7–149, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1985; MR0823489 (87e:34049)], Dynamical systems, I, Encyclopaedia Math. Sci., vol. 1, Springer, Berlin, 1988, pp. 1–148. Translated from the Russian by E. R. Dawson and D. O’Shea. MR 970794
Lucio R. Berrone and Hector J. Giacomini, On the vanishing set of inverse integrating factors, Qual. Theory Dyn. Syst. 1 (2000), no. 2, 211–230. MR 1808363, DOI 10.1007/BF02969478
A. D. Brjuno, Analytic form of differential equations. I, II, Trudy Moskov. Mat. Obšč. 25 (1971), 119–262; ibid. 26 (1972), 199–239 (Russian). MR 0377192
J. Chavarriga, H. Giacomini, and J. Giné, On a new type of bifurcation of limit cycles for a planar cubic system, Nonlinear Anal. 36 (1999), no. 2, Ser. A: Theory Methods, 139–149. MR 1668868, DOI 10.1016/S0362-546X(97)00663-9
Javier Chavarriga, Hector Giacomini, Jaume Giné, and Jaume Llibre, On the integrability of two-dimensional flows, J. Differential Equations 157 (1999), no. 1, 163–182. MR 1710019, DOI 10.1006/jdeq.1998.3621
J. Chavarriga, H. Giacomini, and M. Grau, Necessary conditions for the existence of invariant algebraic curves for planar polynomial systems, Bull. Sci. Math. 129 (2005), no. 2, 99–126 (English, with English and French summaries). MR 2123262, DOI 10.1016/j.bulsci.2004.09.002
L. A. Čerkas, Structure of the sequence function in the neighborhood of a separatrix cycle during perturbation of an analytic autonomous system on the plane, Differentsial′nye Uravneniya 17 (1981), no. 3, 469–478, 573 (Russian). MR 610508
C. Christopher, P. Mardešić, and C. Rousseau, Normalizable, integrable, and linearizable saddle points for complex quadratic systems in $\Bbb C^2$, J. Dynam. Control Systems 9 (2003), no. 3, 311–363. MR 1990240, DOI 10.1023/A:1024643521094
H. Dulac, Recherche sur les points singuliers des équations différentielles , J. Ecole Polytechnique 2 (1904), 1–25.
Isaac A. García and Douglas S. Shafer, Integral invariants and limit sets of planar vector fields, J. Differential Equations 217 (2005), no. 2, 363–376. MR 2168828, DOI 10.1016/j.jde.2005.06.022
Armengol Gasull, Jaume Giné, and Maite Grau, Multiplicity of limit cycles and analytic $m$-solutions for planar differential systems, J. Differential Equations 240 (2007), no. 2, 375–398. MR 2351182, DOI 10.1016/j.jde.2007.05.037
H. Giacomini, J. Llibre, and M. Viano, On the nonexistence, existence and uniqueness of limit cycles, Nonlinearity 9 (1996), no. 2, 501–516. MR 1384489, DOI 10.1088/0951-7715/9/2/013
Hector Giacomini, Mireille Viano, and Jaume Llibre, Semistable limit cycles that bifurcate from centers, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 11, 3489–3498. MR 2031154, DOI 10.1142/S0218127403008600
Maoan Han and Katie Jiang, Normal forms of integrable systems at a resonant saddle, Ann. Differential Equations 14 (1998), no. 2, 150–155. MR 1634022
Maoan Han and Huaiping Zhu, The loop quantities and bifurcations of homoclinic loops, J. Differential Equations 234 (2007), no. 2, 339–359. MR 2300659, DOI 10.1016/j.jde.2006.11.009
M. A. Jebrane and A. Mourtada, Cyclicité finie des lacets doubles non triviaux, Nonlinearity 7 (1994), no. 5, 1349–1365 (French, with English and French summaries). MR 1294547
Ahmed Jebrane, Pavao Mardešić, and Michèle Pelletier, A generalization of Françoise’s algorithm for calculating higher order Melnikov functions, Bull. Sci. Math. 126 (2002), no. 9, 705–732 (English, with English and French summaries). MR 1941082, DOI 10.1016/S0007-4497(02)01138-7
Jibin Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), no. 1, 47–106. MR 1965270, DOI 10.1142/S0218127403006352
Jaume Llibre and Chara Pantazi, Polynomial differential systems having a given Darbouxian first integral, Bull. Sci. Math. 128 (2004), no. 9, 775–788. MR 2099106, DOI 10.1016/j.bulsci.2004.04.001
R. Roussarie, On the number of limit cycles which appear by perturbation of separatrix loop of planar vector fields, Bol. Soc. Brasil. Mat. 17 (1986), no. 2, 67–101. MR 901596, DOI 10.1007/BF02584827
J.-P. Françoise and R. Roussarie (eds.), Bifurcations of planar vector fields, Lecture Notes in Mathematics, vol. 1455, Springer-Verlag, Berlin, 1990. MR 1094374, DOI 10.1007/BFb0085387
A. Seidenberg, Reduction of singularities of the differential equation $A\,dy=B\,dx$, Amer. J. Math. 90 (1968), 248–269. MR 220710, DOI 10.2307/2373435