# 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 & Continuous Dynamical Systems - A, 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. doi: 10.1007/BF01850654. Google Scholar [2] M. Ayoul and N. T. Zung, Galoisian obstructions to non-Hamiltonian integrability,, Comptes Rendus Mathematique, 348 (2013), 1323. doi: 10.1016/j.crma.2010.10.024. Google Scholar [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. Google Scholar [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). Google Scholar [5] T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property,, Phys. Rev. A., 25 (1982), 1257. doi: 10.1103/PhysRevA.25.1257. Google Scholar [6] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer, (1982). Google Scholar [7] V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Rigid Body About a Fixed Point,, State Publishing House, (1953). Google Scholar [8] A. Goriely, Integrability and Non-integrability of Dynamical Systems,, World Scientific Publishing Co. Singapore. 2001., (2001). doi: 10.1142/9789812811943. Google Scholar [9] A. Goriely, Investigation of Painlevé property under time singularities transformations,, J. Math. Phys., 33 (1992), 2728. doi: 10.1063/1.529593. Google Scholar [10] A. Goriely, Integrability, patial integrability and nonintegrability for systems of orsdinary differential equaitons,, J. Math. Phys., 37 (1996), 1871. doi: 10.1063/1.531484. Google Scholar [11] A. Goriely, Painlevé analysis and normal form,, Physics D., 152-153 (2001), 152. doi: 10.1016/S0167-2789(01)00165-8. Google Scholar [12] J. J. Kovacic, An algoriithm for solving second order linear homogeneous differential equations,, J. Symb. Comput., 2 (1986), 3. doi: 10.1016/S0747-7171(86)80010-4. Google Scholar [13] S. Kowalveskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177. doi: 10.1007/BF02592182. Google Scholar [14] V. V. Kozlov, Symmeetries, Topology, and Resonances in Hamiltonian Mechanics,, Springer-Verlag, (1995). Google Scholar [15] M. D. Kruskal and P. A. Clarkson, The Painlevé-Kowalevski and Poly-Painlevé test for integrability,, Stud. Appl. Math., 86 (1992), 87. Google Scholar [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, 310 (1990), 321. Google Scholar [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. Google Scholar [18] W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System,, Celest. Mech. Dynam. Astr., 109 (2010), 1. doi: 10.1007/s10569-010-9315-1. Google Scholar [19] W. L. Li and S. Y. Shi, Galoisian obstruction to the integrability of general dynamical systems,, J. Differential Equations, 252 (2012), 5518. doi: 10.1016/j.jde.2012.01.004. Google Scholar [20] W. L. Li and S. Y. Shi, Non-integrability of generalized Yang-Mills Hamiltonian system,, Discrete and Continuous Dynamical Systems, 33 (2013), 1645. doi: 10.3934/dcds.2013.33.1645. Google Scholar [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, 4 (1894), 120. Google Scholar [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. doi: 10.1016/j.physleta.2004.05.042. Google Scholar [23] A. R. Magid, Lecture on Differential Galois theory,, American Mathematical Society. Providence. Rhode Island, (1994). Google Scholar [24] J. J. Morales-Ruiz, Técnias Algebraicas Para El Estedio De La Integrabilidad De Sistemas Hamiltonianos,, Ph.D. Thesis, (1989). Google Scholar [25] J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory,, J. Differential Equations, 107 (1994), 140. doi: 10.1006/jdeq.1994.1006. Google Scholar [26] J. J. Morales Ruiz, Differencial Galois Theory and Non-Integrability of Hamiltonian Systems,, Progress in Mathematics, (1999). doi: 10.1007/978-3-0348-8718-2. Google Scholar [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. doi: 10.1006/jdeq.1996.0113. Google Scholar [28] J. J. Morales Ruiz, A remark about the painlevé transcendents,, Séminaires. Congrés, 14 (2006), 229. Google Scholar [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. doi: 10.3934/dcds.2009.24.1225. Google Scholar [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. doi: 10.1016/j.ansens.2007.09.002. Google Scholar [31] A. Ramani and B. Grammaticos, The Painlevé property and singularity analysis of integrable and non-integrable systems,, Physics Reports., 180 (1989), 159. doi: 10.1016/0370-1573(89)90024-0. Google Scholar [32] A. Ramani, B. Dorizzi and B. Grammaticos, Painlevé conjecture revisited,, Phys. Rev. Lett., 49 (1982), 1539. doi: 10.1103/PhysRevLett.49.1539. Google Scholar [33] A. Ramani, B. Grammaticos and S Tremblay, Integrable systems without the Painlevé property,, J. Phys. A: Math. Gen., 33 (2000), 3045. doi: 10.1088/0305-4470/33/15/311. Google Scholar [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. doi: 10.1063/1.526240. Google Scholar [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. doi: 10.1063/1.526611. Google Scholar [36] A. K. Roy-Chowdhury, Painlevé Analysis and Its Applications,, Chapman and Hall$/$CRC, (1999). Google Scholar [37] A. Tsygvintsev, The meromorphic nonintegrability of the three-body problem,, C. R. Acad. Sci. Paris Sér.I Math, 331 (2000), 241. doi: 10.1016/S0764-4442(00)01623-2. Google Scholar [38] K. Umeno, Non-integrable character of Hamiltonian systems with global and symmetric coupling,, Physica D., 82 (1995), 11. doi: 10.1016/0167-2789(94)00217-E. Google Scholar [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). Google Scholar [40] S. Wojciechowski, Superintegrability of the calogero-moser system,, Phys. Lett. A, 95 (1983), 279. doi: 10.1016/0375-9601(83)90018-X. Google Scholar [41] H. Yoshida, Necessary condition for the non-existence of algebraic first integral I, II,, Celestial Mech., 31 (1983), 363. Google Scholar [42] H. Yoshida, A criterion for non-existence of an additional integral in Hamiltonian systems with a homogeneous potential,, Physica D, 29 (1987), 128. doi: 10.1016/0167-2789(87)90050-9. Google Scholar [43] H. Yoshida, Non-integrability of the truncated Toda lattice Hamiltonian at any order,, Commun. Math. Phys, 116 (1988), 529. doi: 10.1007/BF01224900. Google Scholar [44] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II,, Funct. Anal. Appl., 16 (1983), 181. Google Scholar

show all references

##### References:
 [1] M. Adler and P. Van Moerbeke, The complex geometry of the Kowalevski-Painlevé analysis,, Invent. math., 97 (1989), 3. doi: 10.1007/BF01850654. Google Scholar [2] M. Ayoul and N. T. Zung, Galoisian obstructions to non-Hamiltonian integrability,, Comptes Rendus Mathematique, 348 (2013), 1323. doi: 10.1016/j.crma.2010.10.024. Google Scholar [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. Google Scholar [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). Google Scholar [5] T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property,, Phys. Rev. A., 25 (1982), 1257. doi: 10.1103/PhysRevA.25.1257. Google Scholar [6] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Springer, (1982). Google Scholar [7] V. V. Golubev, Lectures on the Integration of the Equations of Motion of a Rigid Body About a Fixed Point,, State Publishing House, (1953). Google Scholar [8] A. Goriely, Integrability and Non-integrability of Dynamical Systems,, World Scientific Publishing Co. Singapore. 2001., (2001). doi: 10.1142/9789812811943. Google Scholar [9] A. Goriely, Investigation of Painlevé property under time singularities transformations,, J. Math. Phys., 33 (1992), 2728. doi: 10.1063/1.529593. Google Scholar [10] A. Goriely, Integrability, patial integrability and nonintegrability for systems of orsdinary differential equaitons,, J. Math. Phys., 37 (1996), 1871. doi: 10.1063/1.531484. Google Scholar [11] A. Goriely, Painlevé analysis and normal form,, Physics D., 152-153 (2001), 152. doi: 10.1016/S0167-2789(01)00165-8. Google Scholar [12] J. J. Kovacic, An algoriithm for solving second order linear homogeneous differential equations,, J. Symb. Comput., 2 (1986), 3. doi: 10.1016/S0747-7171(86)80010-4. Google Scholar [13] S. Kowalveskaya, Sur le probleme de la rotation d'un corps solide autour d'un point fixe,, Acta Math., 12 (1889), 177. doi: 10.1007/BF02592182. Google Scholar [14] V. V. Kozlov, Symmeetries, Topology, and Resonances in Hamiltonian Mechanics,, Springer-Verlag, (1995). Google Scholar [15] M. D. Kruskal and P. A. Clarkson, The Painlevé-Kowalevski and Poly-Painlevé test for integrability,, Stud. Appl. Math., 86 (1992), 87. Google Scholar [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, 310 (1990), 321. Google Scholar [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. Google Scholar [18] W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System,, Celest. Mech. Dynam. Astr., 109 (2010), 1. doi: 10.1007/s10569-010-9315-1. Google Scholar [19] W. L. Li and S. Y. Shi, Galoisian obstruction to the integrability of general dynamical systems,, J. Differential Equations, 252 (2012), 5518. doi: 10.1016/j.jde.2012.01.004. Google Scholar [20] W. L. Li and S. Y. Shi, Non-integrability of generalized Yang-Mills Hamiltonian system,, Discrete and Continuous Dynamical Systems, 33 (2013), 1645. doi: 10.3934/dcds.2013.33.1645. Google Scholar [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, 4 (1894), 120. Google Scholar [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. doi: 10.1016/j.physleta.2004.05.042. Google Scholar [23] A. R. Magid, Lecture on Differential Galois theory,, American Mathematical Society. Providence. Rhode Island, (1994). Google Scholar [24] J. J. Morales-Ruiz, Técnias Algebraicas Para El Estedio De La Integrabilidad De Sistemas Hamiltonianos,, Ph.D. Thesis, (1989). Google Scholar [25] J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory,, J. Differential Equations, 107 (1994), 140. doi: 10.1006/jdeq.1994.1006. Google Scholar [26] J. J. Morales Ruiz, Differencial Galois Theory and Non-Integrability of Hamiltonian Systems,, Progress in Mathematics, (1999). doi: 10.1007/978-3-0348-8718-2. Google Scholar [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. doi: 10.1006/jdeq.1996.0113. Google Scholar [28] J. J. Morales Ruiz, A remark about the painlevé transcendents,, Séminaires. Congrés, 14 (2006), 229. Google Scholar [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. doi: 10.3934/dcds.2009.24.1225. Google Scholar [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. doi: 10.1016/j.ansens.2007.09.002. Google Scholar [31] A. Ramani and B. Grammaticos, The Painlevé property and singularity analysis of integrable and non-integrable systems,, Physics Reports., 180 (1989), 159. doi: 10.1016/0370-1573(89)90024-0. Google Scholar [32] A. Ramani, B. Dorizzi and B. Grammaticos, Painlevé conjecture revisited,, Phys. Rev. Lett., 49 (1982), 1539. doi: 10.1103/PhysRevLett.49.1539. Google Scholar [33] A. Ramani, B. Grammaticos and S Tremblay, Integrable systems without the Painlevé property,, J. Phys. A: Math. Gen., 33 (2000), 3045. doi: 10.1088/0305-4470/33/15/311. Google Scholar [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. doi: 10.1063/1.526240. Google Scholar [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. doi: 10.1063/1.526611. Google Scholar [36] A. K. Roy-Chowdhury, Painlevé Analysis and Its Applications,, Chapman and Hall$/$CRC, (1999). Google Scholar [37] A. Tsygvintsev, The meromorphic nonintegrability of the three-body problem,, C. R. Acad. Sci. Paris Sér.I Math, 331 (2000), 241. doi: 10.1016/S0764-4442(00)01623-2. Google Scholar [38] K. Umeno, Non-integrable character of Hamiltonian systems with global and symmetric coupling,, Physica D., 82 (1995), 11. doi: 10.1016/0167-2789(94)00217-E. Google Scholar [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). Google Scholar [40] S. Wojciechowski, Superintegrability of the calogero-moser system,, Phys. Lett. A, 95 (1983), 279. doi: 10.1016/0375-9601(83)90018-X. Google Scholar [41] H. Yoshida, Necessary condition for the non-existence of algebraic first integral I, II,, Celestial Mech., 31 (1983), 363. Google Scholar [42] H. Yoshida, A criterion for non-existence of an additional integral in Hamiltonian systems with a homogeneous potential,, Physica D, 29 (1987), 128. doi: 10.1016/0167-2789(87)90050-9. Google Scholar [43] H. Yoshida, Non-integrability of the truncated Toda lattice Hamiltonian at any order,, Commun. Math. Phys, 116 (1988), 529. doi: 10.1007/BF01224900. Google Scholar [44] S. L. Ziglin, Branching of solutions and non-existence of first integrals in Hamiltonian mechanics I, II,, Funct. Anal. Appl., 16 (1983), 181. Google Scholar
 [1] Roderick S. C. Wong, H. Y. Zhang. On the connection formulas of the third Painlevé transcendent. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 541-560. doi: 10.3934/dcds.2009.23.541 [2] Xueting Tian, Shirou Wang, Xiaodong Wang. Intermediate Lyapunov exponents for systems with periodic orbit gluing property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 1019-1032. doi: 10.3934/dcds.2019042 [3] Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161 [4] Lidong Wang, Hui Wang, Guifeng Huang. Minimal sets and $\omega$-chaos in expansive systems with weak specification property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1231-1238. doi: 10.3934/dcds.2015.35.1231 [5] Yifu Feng, Min Zhang. A $p$-spherical section property for matrix Schatten-$p$ quasi-norm minimization. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-11. doi: 10.3934/jimo.2018159 [6] Andy Hammerlindl. Integrability and Lyapunov exponents. Journal of Modern Dynamics, 2011, 5 (1) : 107-122. doi: 10.3934/jmd.2011.5.107 [7] Pablo Amster, Colin Rogers. On a Ermakov-Painlevé II reduction in three-ion electrodiffusion. A Dirichlet boundary value problem. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3277-3292. doi: 10.3934/dcds.2015.35.3277 [8] Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543 [9] Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581 [10] Pablo Sánchez, Jaume Sempere. Conflict, private and communal property. Journal of Dynamics & Games, 2016, 3 (4) : 355-369. doi: 10.3934/jdg.2016019 [11] Kazumine Moriyasu, Kazuhiro Sakai, Kenichiro Yamamoto. Regular maps with the specification property. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2991-3009. doi: 10.3934/dcds.2013.33.2991 [12] Konstantinos Drakakis, Scott Rickard. On the generalization of the Costas property in the continuum. Advances in Mathematics of Communications, 2008, 2 (2) : 113-130. doi: 10.3934/amc.2008.2.113 [13] Bo Su. Doubling property of elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 143-147. doi: 10.3934/cpaa.2008.7.143 [14] Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162 [15] Rehana Naz, Fazal M. Mahomed. Characterization of partial Hamiltonian operators and related first integrals. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 723-734. doi: 10.3934/dcdss.2018045 [16] Elena Celledoni, Brynjulf Owren. Preserving first integrals with symmetric Lie group methods. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 977-990. doi: 10.3934/dcds.2014.34.977 [17] Jinjun Li, Min Wu. Generic property of irregular sets in systems satisfying the specification property. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 635-645. doi: 10.3934/dcds.2014.34.635 [18] Björn Gebhard. A note concerning a property of symplectic matrices. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2135-2137. doi: 10.3934/cpaa.2018101 [19] Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037 [20] Fang Zhang, Yunhua Zhou. On the limit quasi-shadowing property. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2861-2879. doi: 10.3934/dcds.2017123

2018 Impact Factor: 1.143