American Institute of Mathematical Sciences

September  2012, 32(9): 3009-3027. doi: 10.3934/dcds.2012.32.3009

A formal series approach to averaging: Exponentially small error estimates

 1 INRIA Rennes and ENS Cachan Bretagne, Campus Ker-Lann, av. Robert Schumann, F-35170 Bruz, France 2 Konputazio Zientziak eta A. A. Saila, Informatika Fakultatea, UPV/EHU, E-20018 Donostia-San Sebastián, Spain 3 Departamento de Matemática Aplicada e IMUVA, Facultad de Ciencias, Universidad de Valladolid, Valladolid, Spain

Received  December 2011 Revised  March 2012 Published  April 2012

The techniques, based on formal series and combinatorics, used nowadays to analyze numerical integrators may be applied to perform high-order averaging in oscillatory periodic or quasi-periodic dynamical systems. When this approach is employed, the averaged system may be written in terms of (i) scalar coefficients that are universal, i.e. independent of the system under consideration and (ii) basis functions that may be written in an explicit, systematic way in terms of the derivatives of the Fourier coefficients of the vector field being averaged. The coefficients may be recursively computed in a simple fashion. We show that this approach may be used to obtain exponentially small error estimates, as those first derived by Neishtadt. All the constants that feature in the estimates have a simple explicit expression.
Citation: Philippe Chartier, Ander Murua, Jesús María Sanz-Serna. A formal series approach to averaging: Exponentially small error estimates. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3009-3027. doi: 10.3934/dcds.2012.32.3009
References:
 [1] V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 250, Springer-Verlag, New York, 1988. [2] V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. [3] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001. [4] M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, A stroboscopic method for highly oscillatory problems, in "Numerical Analysis of Multiscale Computations" (eds. B. Engquist, O. Runborg and R. Tsai), Springer-Verlag, (2011), 73-87. [5] M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math., 61 (2011), 1077-1095. doi: 10.1016/j.apnum.2011.06.007. [6] F. Casas, J. A. Oteo and J. Ros, Floquet theory: Exponential perturbative treatment, J. Phys. A, 34 (2001), 3379-3388. doi: 10.1088/0305-4470/34/16/305. [7] P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series, Found. Comput. Math., 10 (2010), 695-727. doi: 10.1007/s10208-010-9074-0. [8] P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration II: The quasi-periodic case,, Found. Comput. Math., (). [9] K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Haussdorff formula, Annals Math. (2), 65 (1957), 163-178. doi: 10.2307/1969671. [10] M. Fliess, Fonctionelles causales nonlinéaires et indeterminées non commutatives, Bull. Soc. Math. France, 109 (1981), 3-40. [11] E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," 2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. [12] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1993. [13] P. Lochak and C. Meunier, "Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems," Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988. [14] A. Murua, Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251. doi: 10.1016/S0168-9274(98)00064-6. [15] A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6 (2006), 387-426. doi: 10.1007/s10208-003-0111-0. [16] A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9. [17] J.-P. Ramis and R. Schäfke, Gevrey separation of fast and slow variables, Nonlinearity, 9 (1996), 353-384. doi: 10.1088/0951-7715/9/2/004. [18] J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Applied Mathematical Sciences, 59, Springer-Verlag, New York, 2007. [19] J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. [20] C. Simó, Averaging under fast quasiperiodic forcing, in "Hamiltonian Mechanics" (Toruń, 1993) (ed. I. Seimenis), NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, (1994), 13-34. [21] H. Sussman, A product expansion of the Chen series, in "Theory and Applications of Nonlinear Control Systems" (Stockholm, 1985) (eds. C. Byrnes and A. Linquist), North Holland, Amsterdam, (1986), 325-335.

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References:
 [1] V. I. Arnol'd, "Geometrical Methods in the Theory of Ordinary Differential Equations," 2nd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 250, Springer-Verlag, New York, 1988. [2] V. I. Arnol'd, "Mathematical Methods of Classical Mechanics," 2nd edition, Graduate Texts in Mathematics, 60, Springer-Verlag, New York, 1989. [3] S. Blanes, F. Casas, J. A. Oteo and J. Ros, The Magnus expansion and some of its applications, Phys. Rep., 470 (2009), 151-238. doi: 10.1016/j.physrep.2008.11.001. [4] M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, A stroboscopic method for highly oscillatory problems, in "Numerical Analysis of Multiscale Computations" (eds. B. Engquist, O. Runborg and R. Tsai), Springer-Verlag, (2011), 73-87. [5] M. P. Calvo, Ph. Chartier, A. Murua and J. M. Sanz-Serna, Numerical stroboscopic averaging for ODEs and DAEs, Appl. Numer. Math., 61 (2011), 1077-1095. doi: 10.1016/j.apnum.2011.06.007. [6] F. Casas, J. A. Oteo and J. Ros, Floquet theory: Exponential perturbative treatment, J. Phys. A, 34 (2001), 3379-3388. doi: 10.1088/0305-4470/34/16/305. [7] P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration I: B-series, Found. Comput. Math., 10 (2010), 695-727. doi: 10.1007/s10208-010-9074-0. [8] P. Chartier, A. Murua and J. M. Sanz-Serna, Higher-order averaging, formal series and numerical integration II: The quasi-periodic case,, Found. Comput. Math., (). [9] K.-T. Chen, Integration of paths, geometric invariants and a generalized Baker-Haussdorff formula, Annals Math. (2), 65 (1957), 163-178. doi: 10.2307/1969671. [10] M. Fliess, Fonctionelles causales nonlinéaires et indeterminées non commutatives, Bull. Soc. Math. France, 109 (1981), 3-40. [11] E. Hairer, Ch. Lubich and G. Wanner, "Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations," 2nd edition, Springer Series in Computational Mathematics, 31, Springer-Verlag, Berlin, 2006. [12] E. Hairer, S. P. Nørsett and G. Wanner, "Solving Ordinary Differential Equations. I. Nonstiff Problems," 2nd edition, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1993. [13] P. Lochak and C. Meunier, "Multiphase Averaging for Classical Systems. With Applications to Adiabatic Theorems," Applied Mathematical Sciences, 72, Springer-Verlag, New York, 1988. [14] A. Murua, Formal series and numerical integrators. I. Systems of ODEs and symplectic integrators, Appl. Numer. Math., 29 (1999), 221-251. doi: 10.1016/S0168-9274(98)00064-6. [15] A. Murua, The Hopf algebra of rooted trees, free Lie algebras, and Lie series, Found. Comput. Math., 6 (2006), 387-426. doi: 10.1007/s10208-003-0111-0. [16] A. I. Neishtadt, The separation of motions in systems with rapidly rotating phase, J. Appl. Math. Mech., 48 (1984), 133-139. doi: 10.1016/0021-8928(84)90078-9. [17] J.-P. Ramis and R. Schäfke, Gevrey separation of fast and slow variables, Nonlinearity, 9 (1996), 353-384. doi: 10.1088/0951-7715/9/2/004. [18] J. A. Sanders, F. Verhulst and J. Murdock, "Averaging Methods in Nonlinear Dynamical Systems," 2nd edition, Applied Mathematical Sciences, 59, Springer-Verlag, New York, 2007. [19] J. M. Sanz-Serna and M. P. Calvo, "Numerical Hamiltonian Problems," Applied Mathematics and Mathematical Computation, 7, Chapman & Hall, London, 1994. [20] C. Simó, Averaging under fast quasiperiodic forcing, in "Hamiltonian Mechanics" (Toruń, 1993) (ed. I. Seimenis), NATO Adv. Sci. Inst. Ser. B Phys., 331, Plenum, New York, (1994), 13-34. [21] H. Sussman, A product expansion of the Chen series, in "Theory and Applications of Nonlinear Control Systems" (Stockholm, 1985) (eds. C. Byrnes and A. Linquist), North Holland, Amsterdam, (1986), 325-335.
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