A survey on local integrability and its regularity. (English) Zbl 1538.34003
Summary: In this survey paper, we summarize our results and also some related ones on local integrability of analytic autonomous differential systems near an equilibrium. The results are on necessary conditions related to existence of local analytic or meromorphic first integrals, on existence of analytic normalization of local analytically integrable system, and also on some sufficient conditions for existence of local analytic first integrals. Among which the results are also on regularity of the local first integrals, including analytic and Gevrey smoothness. We also present some open questions for further investigation.
MSC:
| 34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34A05 | Explicit solutions, first integrals of ordinary differential equations |
References:
| [1] | Bibikov Yu.N. Local Theory of Nonlinear Analytic Ordinary Differential Equations. Lecture Notes in Math., vol. 702, Springer-Verlag, Berlin, 1979. · Zbl 0404.34005 |
| [2] | Chen J., Yi Y., Zhang X. First integrals and normal forms for germs of analytic vector fields. J. Differential Equations 245 (2008), 1167-1184. · Zbl 1166.34018 |
| [3] | Darboux G. Mémoire sur leséquations différentielles algébriques du premier ordre et du premier degré (Mélanges). Bull. Sci. Math. 2éme série 2 (1878) 60-96, 123-144, 151-200. · JFM 10.0214.01 |
| [4] | Darboux G. De l’emploi des solutions particulières algébriques dans l’intégration des systèmes d’équations différentielles algébriques. C. R. Math. Acad. Sci. Paris 86 (1878), 1012-1014. · JFM 10.0214.04 |
| [5] | YANTAO YANG AND XIANG ZHANG |
| [6] | Chow S.N., Li C., Wang D. Normal Forms and Bifurcation of Planar Vector Fields. Cam-bridge University Press, Cambridge, 1994. · Zbl 0804.34041 |
| [7] | Cong W., Llibre J., Zhang X. Generalized rational first integrals of analytic differential systems J. Differ. Equ. 251 (2011), 2770-2788. · Zbl 1232.34004 |
| [8] | Du Z., Romanovski V., Zhang X. Varieties and analytic normalizations of partially inte-grable systems. J. Differential Equations 260 (2016), 6855-6871. · Zbl 1357.34072 |
| [9] | Furta S.D. On non-integrability of general systems of differential equations. Z. angew Math. Phys. 47 (1996), 112-131. · Zbl 0845.34012 |
| [10] | Ito H. Convergence of Birkhoff normal forms for integrable systems. Comment. Math. Helv. 64 (1989), 412-461. · Zbl 0686.58021 |
| [11] | Ito H. Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Math. Ann. 292 (1992), 411-444. · Zbl 0735.58022 |
| [12] | Ito H. Birkhoff normalization and superintegrability of Hamilton systems. Ergodic Theory Dynam. Systems 29 (2009), 1853-1880. · Zbl 1194.37086 |
| [13] | Li W. Normal Form Theory and Applications(in Chinese), Science Press, Beijing, 2000. |
| [14] | Li W., Llibre J., Zhang X. Local first integrals of differential systems and diffeomorphisms. Z. Angew. Math. Phys. 54 (2003), 235-255. · Zbl 1043.37010 |
| [15] | Liu Y.R., Li J., Huang W. Planar dynamical systems. Selected classical problems. De Gruyter, Berlin; Science Press New York, Ltd., New York, 2014. · Zbl 1319.34004 |
| [16] | Llibre J., Pantazi C., Walcher S. First integrals of local analytic differential systems. Bull. Sci. Math. 136 (2012), 342-359. · Zbl 1245.34004 |
| [17] | Llibre J., Walcher S., Zhang X. Local Darboux first integrals of analytic differential sys-tems. Bull. Sci. Math. 138 (2014), 71-88. · Zbl 1291.34068 |
| [18] | Poincaré H. Sur l’intégration deséquations différentielles du premier order et du premier degré I. Rend. Circ. Mat. Palermo 5 (1891), 161-191. |
| [19] | Romanovski V.G., Shafer D.S. The center and cyclicity problems: a computational algebra approach. Birkhäuser Boston, Ltd., Boston, MA, 2009. · Zbl 1192.34003 |
| [20] | Schlomiuk D. Algebraic particular integrals, integrability and the problem of the center. Trans. Amer. Math. Soc. 338 (1993), no. 2, 799-841. · Zbl 0777.58028 |
| [21] | Shi S. On the nonexistence of rational first integrals for nonlinear systems and semiquasiho-mogeneous systems. J. Math. Anal. Appl. 335 (2007), 125-134. · Zbl 1132.34004 |
| [22] | Shi S., Li Y. Non-integrability for general nonlinear systems. Z. Angew. Math. Phys. 52 (2001), 191-200. · Zbl 1099.34505 |
| [23] | Stolovitch L. Smooth Gevrey normal forms of vector fields near a fixed point. Ann. Inst. Fourier 63 (2013), 241-267. · Zbl 1273.37033 |
| [24] | Wu H., Xu X., Zhang D. On the ultradifferentiable normalization. Math. Z. 299 (2021), 751-779. · Zbl 1484.34100 |
| [25] | Wu K., Zhang X. Analytic normalization of analytically integrable differential systems near a periodic orbit. J. Differential Equations 256 (2014), 3552-3567. · Zbl 1297.34055 |
| [26] | Xu S., Wu H., Zhang X. On the local Gevrey integrability, preprint, 2023. |
| [27] | Ye Y. Theory of Limit Cycles. Transl. Math. Monograph, Vol. 66, Amer. Math. Soc., 1986. · Zbl 0588.34022 |
| [28] | Ye Y. Qualitative Theory of Polynomial Differential Systems (in Chinese). Shanghai Science Technical Publisher, Shanghai, 1995. · Zbl 0854.34003 |
| [29] | Zhang X. Analytic normalization of analytic integrable systems and the embedding flows. J. Differential Equations 244 (2008), 1080-1092. · Zbl 1141.34004 |
| [30] | Zhang X. Analytic integrable systems: Analytic normalization and embedding flows. J. Dif-ferential Equations 254 (2013), 3000-3022. · Zbl 1335.34072 |
| [31] | Zhang X. A note on local integrability of differential systems. J. Differential Equations 263 (2017), 7309-7321. · Zbl 1385.34004 |
| [32] | Zhang X. Integrability of Dynamical Systems: Algebra and Analysis. Developments in Math-ematics, Vol. 47, Springer-Natural, Singapore, 2017. · Zbl 1373.37053 |
| [33] | Zhang X. Regularity and convergence of local first integrals of analytic differential systems. J. Differential Equations 294 (2021), 40-59. · Zbl 1481.37061 |
| [34] | Zhang Z., Ding T., Huang W., Dong Z. Qualitative Theory of Differential Equations. Transl. Math. Monograph, Vol. 101, Amer. Math. Soc., 1992. · Zbl 0779.34001 |
| [35] | Ziglin S.L. Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics II, Functional. Anal. Appl. 17 (1983), no. 1, 8-23. · Zbl 0524.58015 |
| [36] | Ziglin S.L. Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics I, Functional. Anal. Appl. 16 (1982), no. 3, 30-41. · Zbl 0524.58015 |
| [37] | Zung N.T. Convergence versus integrability in Birkhoff normal form. Ann. Math. 161 (2005), 141-156. · Zbl 1076.37045 |
| [38] | Zung N.T. Convergence versus integrability in Poincaré-Dulac normal form. Math. Res. Lett. 9 (2002), 217-228. · Zbl 1019.34084 |
| [39] | Yantao Yang College of Mathematics and Computer Science, Yanan University, Yanan 716000, Shaanxi, People’s Republic of China E-mail: yadxyyt@163.com |
| [40] | Xiang Zhang School of Mathematical Sciences, MOE-LSC, and CAM-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China E-mail: xzhang@sjtu.edu.cn |
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