Geodesic
A survey on local integrability and its regularity
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2023), pp. 29-41.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{BASM_2023_1_a3,
     author = {Yantao Yang and Xiang Zhang},
     title = {A survey on local integrability and its regularity},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {29--41},
     publisher = {mathdoc},
     number = {1},
     year = {2023},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/BASM_2023_1_a3/}
}
TY  - JOUR
AU  - Yantao Yang
AU  - Xiang Zhang
TI  - A survey on local integrability and its regularity
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2023
SP  - 29
EP  - 41
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/BASM_2023_1_a3/
LA  - en
ID  - BASM_2023_1_a3
ER  - 
%0 Journal Article
%A Yantao Yang
%A Xiang Zhang
%T A survey on local integrability and its regularity
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2023
%P 29-41
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/BASM_2023_1_a3/
%G en
%F BASM_2023_1_a3
Yantao Yang; Xiang Zhang. A survey on local integrability and its regularity. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2023), pp. 29-41. https://geodesic-test.mathdoc.fr/item/BASM_2023_1_a3/

[1] Bibikov Yu.N., Local Theory of Nonlinear Analytic Ordinary Differential Equations, Lecture Notes in Math., 702, Springer-Verlag, Berlin, 1979 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

[4] Darboux G., “De l'emploi des solutions particuliè res algébriques dans l'intégration des systèmes d'équations différentielles algébriques”, C. R. Math. Acad. Sci. Paris, 86 (1878), 1012–1014

[5] Chow S.N., Li C., Wang D., Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994 | MR | Zbl

[6] Cong W., Llibre J., Zhang X., “Generalized rational first integrals of analytic differential systems”, J. Differ. Equ., 251 (2011), 2770–2788 | DOI | MR | Zbl

[7] Du Z., Romanovski V., Zhang X., “Varieties and analytic normalizations of partially integrable systems”, J. Differential Equations, 260 (2016), 6855–6871 | DOI | MR | Zbl

[8] Furta S.D., “On non-integrability of general systems of differential equations”, Z. angew Math. Phys., 47 (1996), 112–131 | DOI | MR | Zbl

[9] Ito H., “Convergence of Birkhoff normal forms for integrable systems”, Comment. Math. Helv., 64 (1989), 412–461 | DOI | MR | Zbl

[10] Ito H., “Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case”, Math. Ann., 292 (1992), 411–444 | DOI | MR | Zbl

[11] Ito H., “Birkhoff normalization and superintegrability of Hamilton systems”, Ergodic Theory Dynam. Systems, 29 (2009), 1853–1880 | DOI | MR | Zbl

[12] Li W., Normal Form Theory and Applications, Science Press, Beijing, 2000 (in Chinese)

[13] Li W., Llibre J., Zhang X., “Local first integrals of differential systems and diffeomorphisms”, Z. Angew. Math. Phys., 54 (2003), 235–255 | DOI | MR | Zbl

[14] Liu Y.R., Li J., Huang W., Planar dynamical systems. Selected classical problems, De Gruyter, Berlin; Science Press New York, Ltd., New York, 2014 | MR | Zbl

[15] Llibre J., Pantazi C., Walcher S., “First integrals of local analytic differential systems”, Bull. Sci. Math., 136 (2012), 342–359 | DOI | MR | Zbl

[16] Llibre J., Walcher S., Zhang X., “Local Darboux first integrals of analytic differential systems”, Bull. Sci. Math., 138 (2014), 71–88 | DOI | MR | Zbl

[17] Poincaré H., “Sur l'intégration des équations différentielles du premier order et du premier degré I”, Rend. Circ. Mat. Palermo, 5 (1891), 161–191 | DOI

[18] Romanovski V.G., Shafer D.S., The center and cyclicity problems: a computational algebra approach, Birkhäuser Boston, Ltd., Boston, MA, 2009 | MR | Zbl

[19] Schlomiuk D., “Algebraic particular integrals, integrability and the problem of the center”, Trans. Amer. Math. Soc., 338:2 (1993), 799–841 | DOI | MR | Zbl

[20] Shi S., “On the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems”, J. Math. Anal. Appl., 335 (2007), 125–134 | DOI | MR | Zbl

[21] Shi S., Li Y., “Non-integrability for general nonlinear systems”, Z. Angew. Math. Phys., 52 (2001), 191–200 | DOI | MR | Zbl

[22] Stolovitch L., “Smooth Gevrey normal forms of vector fields near a fixed point”, Ann. Inst. Fourier, 63 (2013), 241–267 | DOI | MR | Zbl

[23] Wu H., Xu X., Zhang D., “On the ultradifferentiable normalization”, Math. Z., 299 (2021), 751–779 | DOI | MR | Zbl

[24] Wu K., Zhang X., “Analytic normalization of analytically integrable differential systems near a periodic orbit”, J. Differential Equations, 256 (2014), 3552–3567 | DOI | MR | Zbl

[25] Xu S., Wu H., Zhang X., On the local Gevrey integrability, Preprint, 2023

[26] Ye Y., Theory of Limit Cycles, Transl. Math. Monograph., 66, Amer. Math. Soc., 1986 | MR | Zbl

[27] Ye Y., Qualitative Theory of Polynomial Differential Systems, Shanghai Science Technical Publisher, Shanghai, 1995 (in Chinese) | Zbl

[28] Zhang X., “Analytic normalization of analytic integrable systems and the embedding flows”, J. Differential Equations, 244 (2008), 1080–1092 | DOI | MR | Zbl

[29] Zhang X., “Analytic integrable systems: Analytic normalization and embedding flows”, J. Differential Equations, 254 (2013), 3000–3022 | DOI | MR | Zbl

[30] Zhang X., “A note on local integrability of differential systems”, J. Differential Equations, 263 (2017), 7309–7321 | DOI | MR | Zbl

[31] Zhang X., Integrability of Dynamical Systems: Algebra and Analysis, Developments in Mathematics, 47, Springer-Natural, Singapore, 2017 | DOI | MR | Zbl

[32] Zhang X., “Regularity and convergence of local first integrals of analytic differential systems”, J. Differential Equations, 294 (2021), 40–59 | DOI | MR | Zbl

[33] Zhang Z., Ding T., Huang W., Dong Z., Qualitative Theory of Differential Equations, Transl. Math. Monograph, 101, Amer. Math. Soc., 1992 | MR | Zbl

[34] Ziglin S.L., “Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics II”, Functional. Anal. Appl., 17:1 (1983), 8–23 | DOI | MR | Zbl

[35] Ziglin S.L., “Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics I”, Functional. Anal. Appl., 16:3 (1982), 30–41 | MR

[36] Zung N.T., “Convergence versus integrability in Birkhoff normal form”, Ann. Math., 161 (2005), 141–156 | DOI | MR | Zbl

[37] Zung N.T., “Convergence versus integrability in Poincaré-Dulac normal form”, Math. Res. Lett., 9 (2002), 217–228 | DOI | MR | Zbl