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Invariant foliations for random dynamical systems
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 |
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. |
show all references
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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