American Institute of Mathematical Sciences

September  2014, 34(9): 3667-3681. doi: 10.3934/dcds.2014.34.3667

Weak-Painlevé property and integrability of general dynamical systems

 1 College of Mathematics, Jilin University, Changchun 130012, China 2 College of Mathematics, & Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012

Received  July 2013 Revised  December 2013 Published  March 2014

The purpose of this paper is to investigate the connection between singular property and integrability for general dynamical systems. We will firstly present some methods to test the Painlevé property and weak-Painlevé property, then we will show the equivalence between the weak-Painlevé property and certain formal integrability for general dynamical systems.
Citation: Wenlei Li, Shaoyun Shi. Weak-Painlevé property and integrability of general dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3667-3681. doi: 10.3934/dcds.2014.34.3667
References:
 [1] M. Adler and P. Van Moerbeke, The complex geometry of the Kowalevski-Painlevé analysis, Invent. math., 97 (1989), 3-51. doi: 10.1007/BF01850654. [2] M. Ayoul and N. T. Zung, Galoisian obstructions to non-Hamiltonian integrability, Comptes Rendus Mathematique, 348 (2013), 1323-1326. arXiv:0901.4586. doi: 10.1016/j.crma.2010.10.024. [3] A. Baider, R. C. Churchill, D. L. Rod and M. F. Singer, On the infinitesimal geometry of integrable systems, Fields Inst. Commun., 7 (1996), 5-56. [4] D. Boucher and J. A. Weil, Application of the Morales-Ramis Theorem to Test the Non-Complete Integrability of the Planar Three-Body Problem, IRMA Lectures in Mathematics and Theoretical Physics, 2002. [5] T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264. doi: 10.1103/PhysRevA.25.1257. [6] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982. [7] V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Rigid Body About a Fixed Point, State Publishing House, Moscou. 1953. [8] A. Goriely, Integrability and Non-integrability of Dynamical Systems, World Scientific Publishing Co. Singapore. 2001. doi: 10.1142/9789812811943. [9] A. Goriely, Investigation of Painlevé property under time singularities transformations, J. Math. Phys., 33 (1992), 2728-2742. doi: 10.1063/1.529593. [10] A. Goriely, Integrability, patial integrability and nonintegrability for systems of orsdinary differential equaitons, J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484. [11] A. Goriely, Painlevé analysis and normal form, Physics D., 152-153 (2001), 124-144. doi: 10.1016/S0167-2789(01)00165-8. [12] J. J. Kovacic, An algoriithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4. [13] S. Kowalveskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. doi: 10.1007/BF02592182. [14] V. V. Kozlov, Symmeetries, Topology, and Resonances in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1995. [15] M. D. Kruskal and P. A. Clarkson, The Painlevé-Kowalevski and Poly-Painlevé test for integrability, Stud. Appl. Math., 86 (1992), 87-165. [16] M. D. Kruskal, A. Ramani and B. Grammaticos, Singularity and its relation to complete, partial and non-integrability, Partially integrable evlotion equations in physics. Kluwer Academic Publishers, Dordrechr, 310 (1990), 321-372. [17] Y. Kosmann-Schwarzbach, B. Grammaticos and K. M. Tamizhmani, Integrability of Nonlinear Systems, Lect. Notes Phys. 638. Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/b94605. [18] W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12. doi: 10.1007/s10569-010-9315-1. [19] W. L. Li and S. Y. Shi, Galoisian obstruction to the integrability of general dynamical systems, J. Differential Equations, 252 (2012), 5518-5534. doi: 10.1016/j.jde.2012.01.004. [20] W. L. Li and S. Y. Shi, Non-integrability of generalized Yang-Mills Hamiltonian system, Discrete and Continuous Dynamical Systems, 33 (2013), 1645-1655. doi: 10.3934/dcds.2013.33.1645. [21] A. M. Lyapounov, On a certain property of the differential equations of the problem of motion of a heavy rigid body having a fixed point, Soobshch. Kharkov Math. Obshch, Ser. 2, 4 (1894), 120-140. Collected Works, V. 5, Izdat. Akad. SSSR, Moscow, 1954, (in Russian) [22] A. J. Maciejewski and M. Przybylska, All meromorphically integrable 2D Hamiltonian systems with homogeneous potential of degree 3, Phy. Lett. A., 327 (2004), 461-473. doi: 10.1016/j.physleta.2004.05.042. [23] A. R. Magid, Lecture on Differential Galois theory, American Mathematical Society. Providence. Rhode Island, 1994. [24] J. J. Morales-Ruiz, Técnias Algebraicas Para El Estedio De La Integrabilidad De Sistemas Hamiltonianos, Ph.D. Thesis, University of Barcelona, 1989. [25] J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162. doi: 10.1006/jdeq.1994.1006. [26] J. J. Morales Ruiz, Differencial Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, 179. Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8718-2. [27] J. J. Morales-Ruiz and C. Simó, Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Differential Equations, 129 (1996), 111-135. doi: 10.1006/jdeq.1996.0113. [28] J. J. Morales Ruiz, A remark about the painlevé transcendents, Séminaires. Congrés, 14 (2006), 229-235. [29] J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some $N$-body problems, Discrete Contin. Dyn. Syst., 24 (2009), 1225-1273. doi: 10.3934/dcds.2009.24.1225. [30] J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Annales Scientifiques de l'école Normale Supéieure, 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002. [31] A. Ramani and B. Grammaticos, The Painlevé property and singularity analysis of integrable and non-integrable systems, Physics Reports., 180 (1989), 159-245. doi: 10.1016/0370-1573(89)90024-0. [32] A. Ramani, B. Dorizzi and B. Grammaticos, Painlevé conjecture revisited, Phys. Rev. Lett., 49 (1982), 1539-1541. doi: 10.1103/PhysRevLett.49.1539. [33] A. Ramani, B. Grammaticos and S Tremblay, Integrable systems without the Painlevé property, J. Phys. A: Math. Gen., 33 (2000), 3045-3052. doi: 10.1088/0305-4470/33/15/311. [34] A. Ramani, B. Dorizzi, B. Grammaticos and T. Bountis, Integrability and the Painlevé property for low-dimensional systems, J. Math. Phys., 25 (1984), 878-883. doi: 10.1063/1.526240. [35] A. F. Rañada, A. Ramani, B. Dorizzi and B. Grammaticos, The weak-Painlevé property as a criterion for the integrability of dynamical systems, J. Math. Phys., 26 (1985), 708-710. doi: 10.1063/1.526611. [36] A. K. Roy-Chowdhury, Painlevé Analysis and Its Applications, Chapman and Hall$/$CRC, 1999. [37] A. Tsygvintsev, The meromorphic nonintegrability of the three-body problem, C. R. Acad. Sci. Paris Sér.I Math, 331 (2000), 241-244. doi: 10.1016/S0764-4442(00)01623-2. [38] K. Umeno, Non-integrable character of Hamiltonian systems with global and symmetric coupling, Physica D., 82 (1995), 11-35. doi: 10.1016/0167-2789(94)00217-E. [39] M. Van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, 2003. [40] S. Wojciechowski, Superintegrability of the calogero-moser system, Phys. Lett. A, 95 (1983), 279-281. doi: 10.1016/0375-9601(83)90018-X. [41] H. Yoshida, Necessary condition for the non-existence of algebraic first integral I, II, Celestial Mech., 31 (1983), 363-379, 381-399. [42] H. Yoshida, A criterion for non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9. [43] H. Yoshida, Non-integrability of the truncated Toda lattice Hamiltonian at any order, Commun. Math. Phys, 116 (1988), 529-538. doi: 10.1007/BF01224900. [44] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II, Funct. Anal. Appl., 16 (1983), 181-189; 6-17.

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References:
 [1] M. Adler and P. Van Moerbeke, The complex geometry of the Kowalevski-Painlevé analysis, Invent. math., 97 (1989), 3-51. doi: 10.1007/BF01850654. [2] M. Ayoul and N. T. Zung, Galoisian obstructions to non-Hamiltonian integrability, Comptes Rendus Mathematique, 348 (2013), 1323-1326. arXiv:0901.4586. doi: 10.1016/j.crma.2010.10.024. [3] A. Baider, R. C. Churchill, D. L. Rod and M. F. Singer, On the infinitesimal geometry of integrable systems, Fields Inst. Commun., 7 (1996), 5-56. [4] D. Boucher and J. A. Weil, Application of the Morales-Ramis Theorem to Test the Non-Complete Integrability of the Planar Three-Body Problem, IRMA Lectures in Mathematics and Theoretical Physics, 2002. [5] T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264. doi: 10.1103/PhysRevA.25.1257. [6] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982. [7] V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Rigid Body About a Fixed Point, State Publishing House, Moscou. 1953. [8] A. Goriely, Integrability and Non-integrability of Dynamical Systems, World Scientific Publishing Co. Singapore. 2001. doi: 10.1142/9789812811943. [9] A. Goriely, Investigation of Painlevé property under time singularities transformations, J. Math. Phys., 33 (1992), 2728-2742. doi: 10.1063/1.529593. [10] A. Goriely, Integrability, patial integrability and nonintegrability for systems of orsdinary differential equaitons, J. Math. Phys., 37 (1996), 1871-1893. doi: 10.1063/1.531484. [11] A. Goriely, Painlevé analysis and normal form, Physics D., 152-153 (2001), 124-144. doi: 10.1016/S0167-2789(01)00165-8. [12] J. J. Kovacic, An algoriithm for solving second order linear homogeneous differential equations, J. Symb. Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4. [13] S. Kowalveskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe, Acta Math., 12 (1889), 177-232. doi: 10.1007/BF02592182. [14] V. V. Kozlov, Symmeetries, Topology, and Resonances in Hamiltonian Mechanics, Springer-Verlag, Berlin, 1995. [15] M. D. Kruskal and P. A. Clarkson, The Painlevé-Kowalevski and Poly-Painlevé test for integrability, Stud. Appl. Math., 86 (1992), 87-165. [16] M. D. Kruskal, A. Ramani and B. Grammaticos, Singularity and its relation to complete, partial and non-integrability, Partially integrable evlotion equations in physics. Kluwer Academic Publishers, Dordrechr, 310 (1990), 321-372. [17] Y. Kosmann-Schwarzbach, B. Grammaticos and K. M. Tamizhmani, Integrability of Nonlinear Systems, Lect. Notes Phys. 638. Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/b94605. [18] W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12. doi: 10.1007/s10569-010-9315-1. [19] W. L. Li and S. Y. Shi, Galoisian obstruction to the integrability of general dynamical systems, J. Differential Equations, 252 (2012), 5518-5534. doi: 10.1016/j.jde.2012.01.004. [20] W. L. Li and S. Y. Shi, Non-integrability of generalized Yang-Mills Hamiltonian system, Discrete and Continuous Dynamical Systems, 33 (2013), 1645-1655. doi: 10.3934/dcds.2013.33.1645. [21] A. M. Lyapounov, On a certain property of the differential equations of the problem of motion of a heavy rigid body having a fixed point, Soobshch. Kharkov Math. Obshch, Ser. 2, 4 (1894), 120-140. Collected Works, V. 5, Izdat. Akad. SSSR, Moscow, 1954, (in Russian) [22] A. J. Maciejewski and M. Przybylska, All meromorphically integrable 2D Hamiltonian systems with homogeneous potential of degree 3, Phy. Lett. A., 327 (2004), 461-473. doi: 10.1016/j.physleta.2004.05.042. [23] A. R. Magid, Lecture on Differential Galois theory, American Mathematical Society. Providence. Rhode Island, 1994. [24] J. J. Morales-Ruiz, Técnias Algebraicas Para El Estedio De La Integrabilidad De Sistemas Hamiltonianos, Ph.D. Thesis, University of Barcelona, 1989. [25] J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162. doi: 10.1006/jdeq.1994.1006. [26] J. J. Morales Ruiz, Differencial Galois Theory and Non-Integrability of Hamiltonian Systems, Progress in Mathematics, 179. Birkhäuser Verlag, Basel, 1999. doi: 10.1007/978-3-0348-8718-2. [27] J. J. Morales-Ruiz and C. Simó, Non-integrability criteria for Hamiltonians in the case of Lamé normal variational equations, J. Differential Equations, 129 (1996), 111-135. doi: 10.1006/jdeq.1996.0113. [28] J. J. Morales Ruiz, A remark about the painlevé transcendents, Séminaires. Congrés, 14 (2006), 229-235. [29] J. J. Morales-Ruiz and S. Simon, On the meromorphic non-integrability of some $N$-body problems, Discrete Contin. Dyn. Syst., 24 (2009), 1225-1273. doi: 10.3934/dcds.2009.24.1225. [30] J. J. Morales-Ruiz, J. P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Annales Scientifiques de l'école Normale Supéieure, 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002. [31] A. Ramani and B. Grammaticos, The Painlevé property and singularity analysis of integrable and non-integrable systems, Physics Reports., 180 (1989), 159-245. doi: 10.1016/0370-1573(89)90024-0. [32] A. Ramani, B. Dorizzi and B. Grammaticos, Painlevé conjecture revisited, Phys. Rev. Lett., 49 (1982), 1539-1541. doi: 10.1103/PhysRevLett.49.1539. [33] A. Ramani, B. Grammaticos and S Tremblay, Integrable systems without the Painlevé property, J. Phys. A: Math. Gen., 33 (2000), 3045-3052. doi: 10.1088/0305-4470/33/15/311. [34] A. Ramani, B. Dorizzi, B. Grammaticos and T. Bountis, Integrability and the Painlevé property for low-dimensional systems, J. Math. Phys., 25 (1984), 878-883. doi: 10.1063/1.526240. [35] A. F. Rañada, A. Ramani, B. Dorizzi and B. Grammaticos, The weak-Painlevé property as a criterion for the integrability of dynamical systems, J. Math. Phys., 26 (1985), 708-710. doi: 10.1063/1.526611. [36] A. K. Roy-Chowdhury, Painlevé Analysis and Its Applications, Chapman and Hall$/$CRC, 1999. [37] A. Tsygvintsev, The meromorphic nonintegrability of the three-body problem, C. R. Acad. Sci. Paris Sér.I Math, 331 (2000), 241-244. doi: 10.1016/S0764-4442(00)01623-2. [38] K. Umeno, Non-integrable character of Hamiltonian systems with global and symmetric coupling, Physica D., 82 (1995), 11-35. doi: 10.1016/0167-2789(94)00217-E. [39] M. Van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, 2003. [40] S. Wojciechowski, Superintegrability of the calogero-moser system, Phys. Lett. A, 95 (1983), 279-281. doi: 10.1016/0375-9601(83)90018-X. [41] H. Yoshida, Necessary condition for the non-existence of algebraic first integral I, II, Celestial Mech., 31 (1983), 363-379, 381-399. [42] H. Yoshida, A criterion for non-existence of an additional integral in Hamiltonian systems with a homogeneous potential, Physica D, 29 (1987), 128-142. doi: 10.1016/0167-2789(87)90050-9. [43] H. Yoshida, Non-integrability of the truncated Toda lattice Hamiltonian at any order, Commun. Math. Phys, 116 (1988), 529-538. doi: 10.1007/BF01224900. [44] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II, Funct. Anal. Appl., 16 (1983), 181-189; 6-17.
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