×

Local integrability of Poincaré-Dulac normal forms. (English) Zbl 1416.37053

This paper studies the analytic integrability of Poincaré-Dulac normal forms. The author considers dynamical systems of the form \(\dot{x} = f(x)\) for \(x\in\mathbb{C}^n\), where \(n\ge 1\), \(f:\mathbb{C}^n\rightarrow\mathbb{C}^n\) and \(f(0) = 0\). Throughout it is assumed that \(f\) has a power series expansion of the form \(f(x)=\sum_{n=1}^{\infty} f^{i}\), where \(f^i\) is a homogeneous vector field of degree \(i\) for \(i=1, 2,\dots\). If \(A = Df(0)\), the matrix \(A\) can be decomposed as \(A = S + N\), where \(S\) is semi-simple and \(N\) is nilpotent. This leads to a decomposition \(f^1=f^s(x)+f^n(x)\) where \(f^s(x) = Sx\) and \(f^n(x) = Nx\). The vector field \(f\) is said to be in Poincaré-Dulac normal form if the Lie bracket of \(f^s\) and \(f\), written \(Df(x) f^s(x) - Df^s(x)f(x)\), vanishes for every \(x\in\mathbb{C}^n\).
The author reviews previous results for analytic integrability of the Poincaré-Dulac and Birkhoff normal forms [A. D. Bryuno, Tr. Mosk. Mat. O.-va 25, 119–262 (1971; Zbl 0263.34003); Trans. Mosc. Math. Soc. 26, 199–239 (1974; Zbl 0283.34013); O. I. Bogoyavlenskij, Commun. Math. Phys. 196, No. 1, 19–51 (1998; Zbl 0931.37028); Nguyen Tien Zung, Math. Res. Lett. 9, No. 2–3, 217–228 (2002; Zbl 1019.34084); Ann. Math. (2) 161, No. 1, 141–156 (2005; Zbl 1076.37045)], and notes that there is a general transformation that produces a normal form for the original dynamical system but that transformation is not generally analytic.
The main result of the paper is a sufficient condition for a dynamical system (having a form as described above) in Poincaré-Dulac normal form to be analytically integrable, and that condition is also necessary if certain technical criteria are met. The integrability conditions depend on the resonance degrees of the dynamical system. The author shows that systems with resonance degrees of 0 or 1 are integrable, but may not be if the resonance degrees are greater than 1.
The author applies his theoretical results to a dynamical system with a codimension-two fold-Hopf bifurcation.

MSC:

37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arnol’d, V. I., Geometrical Methods in the Theory of Ordinary Differential Equations, 2nd ed., Grundlehren Math. Wiss., vol. 250, New York: Springer, 1988.
[2] Ayoul, M. and Zung, N.T., Galoisian Obstructions to Non-Hamiltonian Integrability, C. R. Math. Acad. Sci. Paris, 2010, vol. 348, nos. 23-24, pp. 1323-1326. · Zbl 1210.37076 · doi:10.1016/j.crma.2010.10.024
[3] Bogoyavlenski, O. I., Extended Integrability and Bi-Hamiltonian Systems, Comm. Math. Phys., 1998, vol. 196, no. 1, pp. 19-51. · Zbl 0931.37028 · doi:10.1007/s002200050412
[4] Brjuno, A.D., Analytic Form of Differential Equations: 1, Trans. Moscow Math. Soc., 1971, vol. 25, pp. 131-288; see also: Tr. Mosk. Mat. Obs., 1971, vol. 25, pp. 119-262.
[5] Brjuno, A.D., Analytic Form of Differential Equations: 2, Trans. Moscow Math. Soc., 1972, vol. 26, pp. 199-239; see also: Tr. Mosk. Mat. Obs., 1972, vol. 26, pp. 199-239.
[6] Bruno, A.D., Local Methods in Nonlinear Differential Equations, Berlin: Springer, 1989. · doi:10.1007/978-3-642-61314-2
[7] Christov, O., Non-Integrability of First Order Resonances in Hamiltonian Systems in Three Degrees of Freedom, Celestial Mech. Dynam. Astronom., 2012, vol. 112, no. 2, pp. 149-167. · Zbl 1266.70038 · doi:10.1007/s10569-011-9389-4
[8] Chow, Sh.-N., Li, C. Z., and Wang, D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge: Cambridge Univ. Press, 1994. · Zbl 0804.34041 · doi:10.1017/CBO9780511665639
[9] Chen, J., Yi, Y., and Zhang, X., First Integrals and Normal Forms for Germs of Analytic Vector Fields, J. Differential Equations, 2008, vol. 245, no. 5, pp. 1167-1184. · Zbl 1166.34018 · doi:10.1016/j.jde.2008.06.002
[10] Du, Z., Romanovski, V.G., and Zhang, X., Varieties and Analytic Normalizations of Partially Integrable Systems, J. Differential Equations, 2016, vol. 260, no. 9, pp. 6855-6871. · Zbl 1357.34072 · doi:10.1016/j.jde.2016.01.009
[11] Duistermaat, J. J., Nonintegrability of the 1: 1: 2-Resonance, Ergodic Theory Dynam. Systems, 1984, vol. 4, no. 4, pp. 553-568. · Zbl 0537.58026 · doi:10.1017/S0143385700002649
[12] Ito, H., Convergence of Birkhoff Normal Forms for Integrable Systems, Comment. Math. Helv., 1989, vol. 64, no. 3, pp. 412-461. · Zbl 0686.58021 · doi:10.1007/BF02564686
[13] Ito, H., Integrability of Hamiltonian Systems and Birkhoff Normal Forms in the Simple Resonance Case, Math. Ann., 1992, vol. 292, no. 3, pp. 411-444. · Zbl 0735.58022 · doi:10.1007/BF01444629
[14] Ito, H., Birkhoff Normalization and Superintegrability of Hamiltonian Systems, Ergodic Theory Dynam. Systems, 2009, vol. 29, no. 6, pp. 1853-1880. · Zbl 1194.37086 · doi:10.1017/S0143385708000965
[15] Kappeler, T., Kodama, Y., and Nemethi, A., On the Birkhoff Normal Form of a Completely Integrable Hamiltonian System near a Fixed Point with Resonance, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1998, vol. 26, no. 4, pp. 623-661. · Zbl 0919.58030
[16] Kuznetsov, Yu.A., Elements of Applied Bifurcation Theory, 3rd ed., Appl. Math. Sci., vol. 112, New York: Springer, 2004. · Zbl 1082.37002
[17] Llibre, J., Pantazi, Ch., and Walcher, S., First Integrals of Local Analytic Differential Systems, Bull. Sci. Math., 2012, vol. 136, no. 3, pp. 342-359. · Zbl 1245.34004 · doi:10.1016/j.bulsci.2011.10.003
[18] Morales-Ruiz, J. J., Ramis, J.-P., and Simó, C., Integrability of Hamiltonian Systems and Differential Galois Groups of Higher Variational Equations, Ann. Sci. École Norm. Sup. (4), 2007, vol. 40, no. 6, pp. 845-884. · Zbl 1144.37023 · doi:10.1016/j.ansens.2007.09.002
[19] Siegel, C. L., Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtsl ösung, Nachr. Akad. Wiss. Göttingen, math.-phys. Kl., 1952, vol. 1952, pp. 21-30. · Zbl 0047.32901
[20] Stolovitch, L., Singular Complete Integrability, Inst. Hautes Études Sci. Publ. Math., 2000, no. 91, pp. 133-210. · Zbl 0997.32024 · doi:10.1007/BF02698742
[21] Stolovitch, L., Normalisation holomorphe d’algebres de type Cartan de champs de vecteurs holomorphes singuliers, Ann. of Math. (2), 2005, vol. 161, no. 2, pp. 589-612. · Zbl 1080.32019 · doi:10.4007/annals.2005.161.589
[22] Vey, J., Sur certains systèmes dynamiques séparables, Amer. J. Math., 1978, vol. 100, no. 3, pp. 591-614. · Zbl 0384.58012 · doi:10.2307/2373841
[23] Vey, J., Algèbres commutatives de champs de vecteurs isochores, Bull. Soc. Math. France, 1979, vol. 107, no. 4, pp. 423-432. · Zbl 0426.58022 · doi:10.24033/bsmf.1904
[24] Walcher, S., On Differential Equations in Normal Form, Math. Ann., 1991, vol. 291, no. 2, pp. 293-314. · Zbl 0754.34032 · doi:10.1007/BF01445209
[25] Walcher, S., Symmetries and Convergence of Normal Form Transformations, in Proc. of the 6th Conf. on Celestial Mechanics, Monogr. Real Acad. Ci. Exact. Fís.-Quím. Nat. Zaragoza, vol. 25, Zaragoza: Real Acad. Ci. Exact., Fís. Quím. Nat. Zar., 2004, pp. 251-268.
[26] Yagasaki, K., Nonintegrability of the Unfolding of the Fold-Hopf Bifurcation, Nonlinearity, 2018, vol. 31, no. 2, pp. 341-350. · Zbl 1387.37051 · doi:10.1088/1361-6544/aa92e8
[27] Zung, Nguyen Tien, Convergence versus Integrability in Poincaré-Dulac Normal Form, Math. Res. Lett., 2002, vol. 9, nos. 2-3, pp. 217-228. · Zbl 1019.34084 · doi:10.4310/MRL.2002.v9.n2.a8
[28] Zung, Nguyen Tien, Convergence versus Integrability in Birkhoff Normal Form, Ann. of Math. (2), 2005, vol. 161, no. 1, pp. 141-156. · Zbl 1076.37045 · doi:10.4007/annals.2005.161.141
[29] Zung, Nguyen Tien, Non-Degenerate Singularities of Integrable Dynamical Systems, Ergodic Theory Dynam. Systems, 2015, vol. 35, no. 3, pp. 994-1008. · Zbl 1369.37067 · doi:10.1017/etds.2013.65
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.