The first nonzero Melnikov function for a family of good divides. (English) Zbl 1430.37058
The author studies Hamiltonian systems \(dF=0\) and their perturbations \(dF+\varepsilon \omega =0\) by polynomial 1-forms \(\omega\). The displacement function
\(\Delta (t,\varepsilon )=\sum _{j=\mu}^{\infty}\varepsilon ^jM_j(t)\) along a cycle \(\gamma (t)\) on a given level set \(F^{-1}(t)\) is considered, where \(M_{\mu}\) is the first nonzero Melnikov function.
Y. S. Ilyashenko [Mat. Sb. (N.S.), 78, 360–373, (1969)] and J. P. Francoise [Ergodic Theory Dyn. Syst. 16, No. 1, 87–96 (1996; Zbl 0852.34008)] have shown that \(M_{\mu}\) is an abelian integral. L. Gavrilov [Ann. Fac. Sci. Toulouse, Math. (6) 14, No. 4, 663–682 (2005; Zbl 1104.34024)] has shown that in general it is an iterated integral of length at most \(\mu\).
The author continues the study of linear deformations of a family of non-generic Hamiltonian systems fulfilling certain geometric conditions. It turns out that the first nonzero Melnikov function is an iterated integral of length at most two.
Y. S. Ilyashenko [Mat. Sb. (N.S.), 78, 360–373, (1969)] and J. P. Francoise [Ergodic Theory Dyn. Syst. 16, No. 1, 87–96 (1996; Zbl 0852.34008)] have shown that \(M_{\mu}\) is an abelian integral. L. Gavrilov [Ann. Fac. Sci. Toulouse, Math. (6) 14, No. 4, 663–682 (2005; Zbl 1104.34024)] has shown that in general it is an iterated integral of length at most \(\mu\).
The author continues the study of linear deformations of a family of non-generic Hamiltonian systems fulfilling certain geometric conditions. It turns out that the first nonzero Melnikov function is an iterated integral of length at most two.
Reviewer: Vladimir P. Kostov (Nice)
MSC:
| 37J06 | General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants |
| 34M35 | Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 14D05 | Structure of families (Picard-Lefschetz, monodromy, etc.) |
References:
| [1] | A’Campo, N., Le groupe de monodromie du déploiement des singularités isolées de courbes planes I (French), Math. Ann., 213, 1-32 (1975) · Zbl 0316.14011 · doi:10.1007/BF01883883 |
| [2] | Arnold, VI, Arnold’s Problems (2004), Berlin: Springer, Berlin |
| [3] | Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps, vol. II. In: Monodromy and Asymptotics of Integrals. Birkhäuser Boston Inc., Boston (1988) · Zbl 1297.32001 |
| [4] | Benditkis, S.; Novikov, D., On the number of zeros of Melnikov functions, Ann. Fac. Sci. Toulouse Math. Sér. 6, 20, 3, 465-491 (2011) · Zbl 1243.34042 · doi:10.5802/afst.1314 |
| [5] | Binyamini, G.; Novikov, D.; Yakovenko, S., On the number of zeros of Abelian integrals, Invent. Math., 181, 2, 227-289 (2010) · Zbl 1207.34039 · doi:10.1007/s00222-010-0244-0 |
| [6] | Chen, KT, Iterated path integrals, Bull. Am. Math. Soc., 83, 5, 831-879 (1977) · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6 |
| [7] | Dumortier, F.; Roussarie, R., Abelian integrals and limit cycles, J. Differ. Equ., 227, 1, 116-165 (2006) · Zbl 1111.34028 · doi:10.1016/j.jde.2005.08.015 |
| [8] | Françoise, JP, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergod. Theory Dyn. Syst., 16, 1, 87-96 (1996) · Zbl 0852.34008 · doi:10.1017/S0143385700008725 |
| [9] | Gavrilov, L., Higher order Poincaré-Pontryagin functions and iterated path integrals, Ann. Fac. Sci. Toulouse Math. (6), 14, 4, 663-682 (2005) · Zbl 1104.34024 · doi:10.5802/afst.1107 |
| [10] | Gavrilov, L.: On the Topology of Polynomials in Two Complex Variables, Preprint No. 45. Laboratoire de Topologie et Geometrie, UPS, Toulouse (1994) (non-published) |
| [11] | Gavrilov, L.; Iliev, ID, The displacement map associated to polynomial unfoldings of planar Hamiltonian vector fields, Am. J. Math., 127, 6, 1153-1190 (2005) · Zbl 1093.34015 · doi:10.1353/ajm.2005.0039 |
| [12] | Grothendieck, A.: On the de Rham cohomology of algebraic varieties. Publ. Math. l’I.H.É.S. 29, 95-103 (1966) · Zbl 0145.17602 |
| [13] | Harris, B.: Iterated integrals and cycles on algebraic manifolds. In: Nankai Tracts in Mathematics, vol. 7. World Scientific Publishing Co., Inc., River Edge (2004) · Zbl 1063.14010 |
| [14] | Ilyashenko, YS, The appearance of limit cycles under a perturbation of the equation \(dw/dz=R_z/R_w\), where \(R(z, w)\) is a polynomial (Russian), Mat. Sb. (N.S.), 78, 120, 360-373 (1969) · Zbl 0183.36501 |
| [15] | Ilyashenko, Y.S., Yakovenko, S.: Lecture on analytic differential equations. In: Graduate Studies in Mathematics. American Mathematical Society, Providence (2008). ISBN 978-0-8218-3667-5 · Zbl 1186.34001 |
| [16] | Jebrane, A.; Mardesic, P.; Pelletier, M., A generalization of Franoises algorithm for calculating higher order Melnikov functions, Bull. Sci. Math., 126, 705-732 (2002) · Zbl 1029.34081 · doi:10.1016/S0007-4497(02)01138-7 |
| [17] | Mardesic, P.; Novikov, D.; Ortiz-Bobadilla, L.; Pontigo-Herrera, J., Bounding the length of iterated integrals of the first nonzero Melnikov function, Mosc. Math. J., 18, 2, 367-386 (2018) · Zbl 1418.34075 · doi:10.17323/1609-4514-2018-18-2-367-386 |
| [18] | Mardesic, P., Novikov, D., Ortiz-Bobadilla, L., Pontigo-Herrera, J.: Infinite orbit depth and length of Melnikov functions (submitted) · Zbl 1418.34075 |
| [19] | Murasugi, K.: Knot Theory and Its Applications. Birkhauser, Basel (1996). ISBN 978-0-8176-4719-3 · Zbl 0864.57001 |
| [20] | Pelletier, M.; Uribe, M., Principal Poincaré-Pontryagin function associated to some families of Morse real polynomials, Nonlinearity, 27, 2, 257-269 (2014) · Zbl 1293.34048 · doi:10.1088/0951-7715/27/2/257 |
| [21] | Pontigo-Herrera, J., Tangential center problem for a family of non-generic Hamiltonians, J. Dyn. Control Syst., 23, 3, 597-622 (2017) · Zbl 1395.34089 · doi:10.1007/s10883-016-9343-6 |
| [22] | Uribe, M., Principal Poincaré-Pontryagin function associated to polynomial perturbations of a product of \((d+1)\) straight lines, J. Differ. Equ., 246, 4, 1313-1341 (2009) · Zbl 1173.34026 · doi:10.1016/j.jde.2008.11.014 |
| [23] | Yakovenko, S.: A geometric proof of the Bautin theorem. In: Concerning the Hilbert 16th Problem, pp. 203-219. American Mathematical Society, Translation Series 2, vol. 165. Advanced Mathematical Sciences, vol. 23. American Mathematical Society, Providence (1995) · Zbl 0828.34026 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
