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Chaos in delay differential equations with applications in population dynamics
Non-integrability of generalized Yang-Mills Hamiltonian system
1. | College of Mathematics, & Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun 130012, China |
2. | College of Mathematics, Jilin University, Changchun 130012, China |
References:
[1] |
P. B. Acosta-Humanez, D. Blazquez-Sanz and C. V. Contreras, On Hamiltonian potentials with quartic polynomial normal variational equations, Nonlinear Studies The International Journal, 16 (2009), 299-314. |
[2] |
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. |
[3] |
F. Baldassarri, On Algebraic solution of Lamé's differential equation, J. Differential Equations, 41 (1981), 44-58. |
[4] |
G. Baumann, W. G. Glöckle and T. F. Nonnenmacher, Sigular point analysis and integrals of motion for coupled nonlinear Schrödinger equations, Proc. R. Soc. Lond. A, 434 (1991), 263-278. |
[5] |
D. Boucher and J. A. Weil, About nonintegrability in the Friedmann-Robertson-Walker cosmological model, Brazilian Journal of Physics, 37 (2007), 398-405.
doi: 10.1007/s10765-007-0152-8. |
[6] |
T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264. |
[7] |
R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15-48. |
[8] |
L. A. A. Cohelo, J. E. F. Skea and T. J. Stuchi, On the non-integrability of a class of Hamiltonian cosmological models, Brazilian Journal of Physics, 35 (2005). |
[9] |
B. Dwork, Differential operators with nilponent $p$-curvature, Amer. J. Math., 112 (1990), 749-786.
doi: 10.2307/2374806. |
[10] |
A. Elipe, J. Hietarinta and S. Tompaidis, Comment on paper by S. Kasperczuk, Celest. Mech 58:387-391(1994), Celest. Mech. Dynam. Astr., 62 (1995), 191-192.
doi: 10.1007/BF00692087. |
[11] |
R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions, Phys. Rev. D., 13 (1976), 2739-2761. |
[12] |
G. H. Halphen, Traité des fonctions elliptiques VOl. I, II, Gauthier-Villars, Paris, (1888). |
[13] |
J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154.
doi: 10.1016/0370-1573(87)90089-5. |
[14] |
S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system, Celest. Mech. Dynam. Astr., 58 (1994), 387-391. |
[15] |
W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12. |
[16] |
A. J. Maciejewski, M. Przybylska, T. Stachowiak and M. Szydlowski, Global integrability of cosmological scalar fields, J. Phys. A., 41 (2008), 26 pp. 465101. |
[17] |
A. J. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A., 37 (2004), 2579-2597. |
[18] |
A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901. |
[19] |
S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Soviet Phys. JETP., 38 (1974), 248-253. |
[20] |
J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos," Ph.D. Thesis, University of Barcelona, 1989. |
[21] |
J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162. |
[22] |
J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser Verlag, Basel, 1999. |
[23] |
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. |
[24] |
J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergod. Th & Dynam. Sys., 25 (2005), 1237-1256. |
[25] |
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. |
[26] |
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. |
[27] |
E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations," Oxford Univ. Press, London, 1936. |
[28] |
R. Rajaraman and E. J. Weinberg, Internal symmetry and the semi-classical method in quantum field theory, Phys. Rev. D., 11 (1975), 2950-2966. |
[29] |
Van der Put M and M. F. Singer, "Galois Theory of Linear Differential Equations," volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, 2003. |
[30] |
P. Vanhaecke, A special case of the Garnier system, (1,4)-polarised Abelian surfaces and their moduli, Compositio Math., 29 (1994), 157-203.
doi: 10.1016/0165-0270(94)90123-6. |
[31] |
E. T. Whittaker and E. T. Watson, "A Course of Modern Analysis," Cambrige Univ. Press, Cambrige, 1969. |
[32] |
V. E. Zakharv, M. F. Ivanov and L. I. Shoor, On anomalously slow stochastization in certain two-dimensional models of field theory, Zh. Eksp. Teor. Fiz. Lett., 30 (1979), 39-44. |
[33] |
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] |
P. B. Acosta-Humanez, D. Blazquez-Sanz and C. V. Contreras, On Hamiltonian potentials with quartic polynomial normal variational equations, Nonlinear Studies The International Journal, 16 (2009), 299-314. |
[2] |
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. |
[3] |
F. Baldassarri, On Algebraic solution of Lamé's differential equation, J. Differential Equations, 41 (1981), 44-58. |
[4] |
G. Baumann, W. G. Glöckle and T. F. Nonnenmacher, Sigular point analysis and integrals of motion for coupled nonlinear Schrödinger equations, Proc. R. Soc. Lond. A, 434 (1991), 263-278. |
[5] |
D. Boucher and J. A. Weil, About nonintegrability in the Friedmann-Robertson-Walker cosmological model, Brazilian Journal of Physics, 37 (2007), 398-405.
doi: 10.1007/s10765-007-0152-8. |
[6] |
T. Bountis, H. Segur and F. Vivaldi, Integrable Hamiltonian systems and the Painleve property, Phys. Rev. A., 25 (1982), 1257-1264. |
[7] |
R. C. Churchill, D. L. Rod and M. F. Singer, Group-theoretic obstructions to integrability, Ergod. Th & Dynam. Sys. (1), 5 (1995), 15-48. |
[8] |
L. A. A. Cohelo, J. E. F. Skea and T. J. Stuchi, On the non-integrability of a class of Hamiltonian cosmological models, Brazilian Journal of Physics, 35 (2005). |
[9] |
B. Dwork, Differential operators with nilponent $p$-curvature, Amer. J. Math., 112 (1990), 749-786.
doi: 10.2307/2374806. |
[10] |
A. Elipe, J. Hietarinta and S. Tompaidis, Comment on paper by S. Kasperczuk, Celest. Mech 58:387-391(1994), Celest. Mech. Dynam. Astr., 62 (1995), 191-192.
doi: 10.1007/BF00692087. |
[11] |
R. Fridberg, T. D. Lee and R. Padjen, Class of scalar-field solutions in three space dimensions, Phys. Rev. D., 13 (1976), 2739-2761. |
[12] |
G. H. Halphen, Traité des fonctions elliptiques VOl. I, II, Gauthier-Villars, Paris, (1888). |
[13] |
J. Hietarinta, Direct methods for the search of the second invariant, Phys. Rep., 147 (1987), 87-154.
doi: 10.1016/0370-1573(87)90089-5. |
[14] |
S. Kasperczuk, Integrability of the Yang-Mills Hamiltonian system, Celest. Mech. Dynam. Astr., 58 (1994), 387-391. |
[15] |
W. L. Li and S. Y. Shi, Non-integrability of Hénon-Heiles System, Celest. Mech. Dynam. Astr., 109 (2010), 1-12. |
[16] |
A. J. Maciejewski, M. Przybylska, T. Stachowiak and M. Szydlowski, Global integrability of cosmological scalar fields, J. Phys. A., 41 (2008), 26 pp. 465101. |
[17] |
A. J. Maciejewski, M. Przybylska and J. A. Weil, Non-integrability of the generalized spring-pendulum problem, J. Phys. A., 37 (2004), 2579-2597. |
[18] |
A. J. Maciejewski and M. Przybylska, Darboux points and integrability of Hamiltonian systems with homogeneous polynomial potential, J. Math. Phys., 46 (2005), 062901. |
[19] |
S. V. Manakov, On the theory of two-dimensional stationary self-focusing of electromagnetic waves, Soviet Phys. JETP., 38 (1974), 248-253. |
[20] |
J. J. Morales-Ruiz, "Técnicas Algebraicas Para el Estudio de la Integrabilidad de Sistemas Hamiltonianos," Ph.D. Thesis, University of Barcelona, 1989. |
[21] |
J. J. Morales-Ruiz and C. Simó, Picard-Vessiot theory and Ziglin's theory, J. Differential Equations, 107 (1994), 140-162. |
[22] |
J. J. Morales-Ruiz, "Differential Galois Theory and Non-Integrability of Hamiltonian Systems," Birkhäuser Verlag, Basel, 1999. |
[23] |
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. |
[24] |
J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergod. Th & Dynam. Sys., 25 (2005), 1237-1256. |
[25] |
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. |
[26] |
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. |
[27] |
E. G. C. Poole, "Introduction to the Theory of Linear Differential Equations," Oxford Univ. Press, London, 1936. |
[28] |
R. Rajaraman and E. J. Weinberg, Internal symmetry and the semi-classical method in quantum field theory, Phys. Rev. D., 11 (1975), 2950-2966. |
[29] |
Van der Put M and M. F. Singer, "Galois Theory of Linear Differential Equations," volume 328 of Grundlehren der mathematischen Wissenshaften. Springer. Heidelberg, 2003. |
[30] |
P. Vanhaecke, A special case of the Garnier system, (1,4)-polarised Abelian surfaces and their moduli, Compositio Math., 29 (1994), 157-203.
doi: 10.1016/0165-0270(94)90123-6. |
[31] |
E. T. Whittaker and E. T. Watson, "A Course of Modern Analysis," Cambrige Univ. Press, Cambrige, 1969. |
[32] |
V. E. Zakharv, M. F. Ivanov and L. I. Shoor, On anomalously slow stochastization in certain two-dimensional models of field theory, Zh. Eksp. Teor. Fiz. Lett., 30 (1979), 39-44. |
[33] |
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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