# American Institute of Mathematical Sciences

November  2014, 34(11): 4827-4854. doi: 10.3934/dcds.2014.34.4827

## Linearised higher variational equations

 1 Department of Mathematics, University of Portsmouth, Lion Gate Bldg, Lion Terrace, Portsmouth PO1 3HF, United Kingdom

Received  March 2013 Revised  April 2014 Published  May 2014

This work explores the tensor and combinatorial constructs underlying the linearised higher-order variational equations $\mathrm{LVE}_{\psi}^k$ of a generic autonomous system along a particular solution $\psi$. The main result of this paper is a compact yet explicit and computationally amenable form for said variational systems and their monodromy matrices. Alternatively, the same methods are useful to retrieve, and sometimes simplify, systems satisfied by the coefficients of the Taylor expansion of a formal first integral for a given dynamical system. This is done in preparation for further results within Ziglin-Morales-Ramis theory, specifically those of a constructive nature.
Citation: Sergi Simon. Linearised higher variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827
##### References:
 [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984, Reprint of the 1972 edition, Selected Government Publications. [2] A. Aparicio-Monforte, Méthodes Effectives Pour L'intégrabilité des Systèmes Dynamiques, Ph.D. thesis, Université de Limoges, December, 2010. [3] A. Aparicio Monforte and J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems， Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15. doi: 10.1090/conm/549/10850. [4] _______ and _______, A reduced form for linear differential systems and its application to integrability of Hamiltonian systems}, J. Symbolic Comput., 47 (2012), 192-213. doi: 10.1016/j.jsc.2011.09.011. [5] A. Aparicio-Monforte, M. Barkatou, S. Simon and J.-A. Weil, Formal first integrals along solutions of differential systems I, ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 19-26, ACM, New York, 2011. doi: 10.1145/1993886.1993896. [6] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés, vol. 8, Société Mathématique de France, Paris, 2001. [7] M. Barkatou, On rational solutions of systems of linear differential equations, J. Symbolic Comput., 28 (1999), 547-567. doi: 10.1006/jsco.1999.0314. [8] U. Bekbaev, A matrix representation of composition of polynomial maps,, , (). [9] ________, A radius of absolute convergence for power series in many variables,, , (). [10] ________, Matrix representations for symmetric and antisymmetric multi-linear maps,, , (). [11] ________, An inversion formula for multivariate power series,, , (). [12] E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419. doi: 10.2307/2300300. [13] A. Blokhuis and J. J. Seidel, An introduction to multilinear algebra and some applications, Philips J. Res., 39 (1984), 111-120. [14] H. Cartan, Calcul Différentiel, Hermann, Paris, 1967. [15] J. Casasayas, A. Nunes and N. B. Tufillaro, Swinging Atwood's machine: Integrability and dynamics, J. Phys., 51 (1990), 1693-1702. doi: 10.1051/jphys:0199000510160169300. [16] W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9. [17] I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, 2008. [18] S. Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0041-0. [19] K. Makino and M. Berz, Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by preconditioning, Int. J. Differ. Equ. Appl., 10 (2005), 353-384 (2006). [20] R. Martínez and C. Simó, Non-integrability of the degenerate cases of the swinging Atwood's machine using higher order variational equations, Discrete Contin. Dyn. Syst., 29 (2011), 1-24. doi: 10.3934/dcds.2011.29.1. [21] _______ and _______, Non-integrability of Hamiltonian systems through high order variational equations: summary of results and examples, Regul. Chaotic Dyn., 14 (2009), 323-348. doi: 10.1134/S1560354709030010. [22] J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. [23] J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal., 8 (2001), 33-95. [24] J. J. Morales-Ruiz, J.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002. [25] J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergodic Theory Dynam. Systems, 25 (2005), 1237-1256. doi: 10.1017/S0143385704001038. [26] O. Pujol, J.-P. Pérez, J.-P. Ramis, C. Simó, S. Simon and J.-A. Weil, Swinging Atwood machine: Experimental and numerical results, and a theoretical study, Phys. D, 239 (2010), 1067-1081. doi: 10.1016/j.physd.2010.02.017. [27] S. Ramanujan, Notebooks, (2 volumes) Tata Institute of Fundamental Research, Bombay, 1957. [28] S. Simon, Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian, Journal of Nonlinear Mathematical Physics, 21 (2014), 1-16. doi: 10.1080/14029251.2014.894710. [29] M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. [30] N. B. Tufillaro, Integrable motion of a swinging Atwood's machine, Amer. J. Phys., 54 (1986), 142-153. doi: 10.1119/1.14710. [31] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen, 16 (1982), 30-41, 96. [32] H. Zoladek, The Monodromy Group, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series) 67, Birkhäuser Verlag, Basel, 2006.

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##### References:
 [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, A Wiley-Interscience Publication, John Wiley & Sons Inc., New York, 1984, Reprint of the 1972 edition, Selected Government Publications. [2] A. Aparicio-Monforte, Méthodes Effectives Pour L'intégrabilité des Systèmes Dynamiques, Ph.D. thesis, Université de Limoges, December, 2010. [3] A. Aparicio Monforte and J.-A. Weil, A reduction method for higher order variational equations of Hamiltonian systems， Symmetries and related topics in differential and difference equations, Contemp. Math., Amer. Math. Soc., Providence, RI, 549 (2011), 1-15. doi: 10.1090/conm/549/10850. [4] _______ and _______, A reduced form for linear differential systems and its application to integrability of Hamiltonian systems}, J. Symbolic Comput., 47 (2012), 192-213. doi: 10.1016/j.jsc.2011.09.011. [5] A. Aparicio-Monforte, M. Barkatou, S. Simon and J.-A. Weil, Formal first integrals along solutions of differential systems I, ISSAC 2011 - Proceedings of the 36th International Symposium on Symbolic and Algebraic Computation, 19-26, ACM, New York, 2011. doi: 10.1145/1993886.1993896. [6] M. Audin, Les Systèmes Hamiltoniens et Leur Intégrabilité, Cours Spécialisés, vol. 8, Société Mathématique de France, Paris, 2001. [7] M. Barkatou, On rational solutions of systems of linear differential equations, J. Symbolic Comput., 28 (1999), 547-567. doi: 10.1006/jsco.1999.0314. [8] U. Bekbaev, A matrix representation of composition of polynomial maps,, , (). [9] ________, A radius of absolute convergence for power series in many variables,, , (). [10] ________, Matrix representations for symmetric and antisymmetric multi-linear maps,, , (). [11] ________, An inversion formula for multivariate power series,, , (). [12] E. T. Bell, Exponential numbers, Amer. Math. Monthly, 41 (1934), 411-419. doi: 10.2307/2300300. [13] A. Blokhuis and J. J. Seidel, An introduction to multilinear algebra and some applications, Philips J. Res., 39 (1984), 111-120. [14] H. Cartan, Calcul Différentiel, Hermann, Paris, 1967. [15] J. Casasayas, A. Nunes and N. B. Tufillaro, Swinging Atwood's machine: Integrability and dynamics, J. Phys., 51 (1990), 1693-1702. doi: 10.1051/jphys:0199000510160169300. [16] W. Fulton and J. Harris, Representation Theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0979-9. [17] I. M. Gelfand, M. M. Kapranov and A. V. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, 2008. [18] S. Lang, Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. doi: 10.1007/978-1-4613-0041-0. [19] K. Makino and M. Berz, Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by preconditioning, Int. J. Differ. Equ. Appl., 10 (2005), 353-384 (2006). [20] R. Martínez and C. Simó, Non-integrability of the degenerate cases of the swinging Atwood's machine using higher order variational equations, Discrete Contin. Dyn. Syst., 29 (2011), 1-24. doi: 10.3934/dcds.2011.29.1. [21] _______ and _______, Non-integrability of Hamiltonian systems through high order variational equations: summary of results and examples, Regul. Chaotic Dyn., 14 (2009), 323-348. doi: 10.1134/S1560354709030010. [22] J. J. Morales-Ruiz, Differential Galois Theory and Non-integrability of Hamiltonian Systems, Progress in Mathematics, Birkhäuser Verlag, Basel, 1999. [23] J. J. Morales-Ruiz and J.-P. Ramis, Galoisian obstructions to integrability of Hamiltonian systems. I, Methods Appl. Anal., 8 (2001), 33-95. [24] J. J. Morales-Ruiz, J.-P. Ramis and C. Simó, Integrability of Hamiltonian systems and differential Galois groups of higher variational equations, Ann. Sci. École Norm. Sup. (4), 40 (2007), 845-884. doi: 10.1016/j.ansens.2007.09.002. [25] J. J. Morales-Ruiz, C. Simó and S. Simon, Algebraic proof of the non-integrability of Hill's problem, Ergodic Theory Dynam. Systems, 25 (2005), 1237-1256. doi: 10.1017/S0143385704001038. [26] O. Pujol, J.-P. Pérez, J.-P. Ramis, C. Simó, S. Simon and J.-A. Weil, Swinging Atwood machine: Experimental and numerical results, and a theoretical study, Phys. D, 239 (2010), 1067-1081. doi: 10.1016/j.physd.2010.02.017. [27] S. Ramanujan, Notebooks, (2 volumes) Tata Institute of Fundamental Research, Bombay, 1957. [28] S. Simon, Conditions and evidence for non-integrability in the Friedmann-Robertson-Walker Hamiltonian, Journal of Nonlinear Mathematical Physics, 21 (2014), 1-16. doi: 10.1080/14029251.2014.894710. [29] M. van der Put and M. F. Singer, Galois Theory of Linear Differential Equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 328, Springer-Verlag, Berlin, 2003. [30] N. B. Tufillaro, Integrable motion of a swinging Atwood's machine, Amer. J. Phys., 54 (1986), 142-153. doi: 10.1119/1.14710. [31] S. L. Ziglin, Bifurcation of solutions and the nonexistence of first integrals in Hamiltonian mechanics. I, Funktsional. Anal. i Prilozhen, 16 (1982), 30-41, 96. [32] H. Zoladek, The Monodromy Group, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series) 67, Birkhäuser Verlag, Basel, 2006.
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