A multiscale method for highly oscillatory ordinary differential equations with resonance, Mathematics of Computation, vol.78, issue.266, pp.929-956, 2009. ,
DOI : 10.1090/S0025-5718-08-02139-X
Geometrical Methods in the Theory of Ordinary Differential Equations, 1988. ,
Mathematical Methods of Classical Mechanics, 1989. ,
Canonical B-series, Numerische Mathematik, vol.67, issue.2, pp.161-175, 1994. ,
DOI : 10.1007/s002110050022
Instabilities and Inaccuracies in the Integration of Highly Oscillatory Problems, SIAM Journal on Scientific Computing, vol.31, issue.3, pp.1653-1677, 2009. ,
DOI : 10.1137/080727658
Carrying an inverted pendulum on a bumpy road, Discrete and Continuous Dynamical Systems - Series B, vol.14, issue.2 ,
DOI : 10.3934/dcdsb.2010.14.429
Heterogeneous multiscale methods for mechanical systems with vibrations (submitted) ,
DOI : 10.1137/080738556
Sanz-Serna, A stroboscopic numerical method for highly oscillatory problems (submitted) ,
Algebraic structures of B-series, Found, Comput. Math ,
The heterogeneous multiscale methods, Commun. Math. Sci, vol.1, pp.87-132, 2003. ,
Vanden-Eijnden, Heterogeneous multiscale methods: a review, Commun. Comput. Phys, vol.2, pp.367-450, 2007. ,
Heterogeneous multiscale methods for stiff ordinary differential equations, Mathematics of Computation, vol.74, issue.252, pp.1707-1742, 2005. ,
DOI : 10.1090/S0025-5718-05-01745-X
Derivation and justification of equations in slow time (the stroboscopic method), USSR Computational Mathematics and Mathematical Physics, vol.14, issue.5, pp.81-118, 1974. ,
DOI : 10.1016/0041-5553(74)90198-0
Backward error analysis of numerical integrators and symplectic methods, Ann. Numer. Math, vol.1, pp.107-132, 1994. ,
Long-Time Energy Conservation of Numerical Methods for Oscillatory Differential Equations, SIAM Journal on Numerical Analysis, vol.38, issue.2, pp.414-441, 2000. ,
DOI : 10.1137/S0036142999353594
??ber die Butchergruppe und allgemeine Multi-Value Methoden, Computing, vol.11, issue.1, pp.287-303, 1974. ,
DOI : 10.1007/BF02268387
Solving Ordinary Differential Equations I, Nonstiff Problems, 1993. ,
DOI : 10.1007/978-3-662-12607-3
Geometric Numerical Integration, 2006. ,
URL : https://hal.archives-ouvertes.fr/hal-01403326
On the global error of discretization methods for highly-oscillatory ordinary differential equations, Bit Numerical Mathematics, vol.42, issue.3, pp.561-599, 2002. ,
DOI : 10.1023/A:1022049814688
Dynamical stability of a pendulum when its point of suspension vibrates, Collected Papers, pp.714-725, 1965. ,
Manifolds of slow solutions for highly oscillatory problems, Math. J, vol.42, pp.1169-1191, 1993. ,
Geometry and physics of averaging with applications, Physica D: Nonlinear Phenomena, vol.132, issue.1-2, pp.150-164, 1999. ,
DOI : 10.1016/S0167-2789(99)00022-6
Deciding the nature of the coarse equation through microscopic simulations: the baby-bathwater scheme, SIAM Rev, pp.469-487, 2007. ,
Formal series and numerical integrators, part I: Systems of ODEs and symplectic integrators, Applied Numerical Mathematics, vol.29, issue.2, pp.221-251, 1999. ,
DOI : 10.1016/S0168-9274(98)00064-6
Higher Order Averaging and Related Methods for Perturbed Periodic and Quasi-Periodic Systems, SIAM Journal on Applied Mathematics, vol.17, issue.4, pp.698-724, 1968. ,
DOI : 10.1137/0117065
Averaging Methods in Nonlinear Dynamical Systems, 2007. ,
DOI : 10.1007/978-1-4757-4575-7
STABILIZING WITH A HAMMER, Stochastics and Dynamics, vol.08, issue.01, pp.45-57, 2008. ,
DOI : 10.1142/S0219493708002263
Modulated Fourier expansions and heterogeneous multiscale methods, IMA Journal of Numerical Analysis, vol.29, issue.3, pp.595-605, 2009. ,
DOI : 10.1093/imanum/drn031
URL : http://imajna.oxfordjournals.org/cgi/content/short/29/3/595
Multiple Time Scale Numerical Methods for the Inverted Pendulum Problem, Engquist, P. Lötsdedt, O. Runborg. Lect. Notes Comput. Sci. Eng, vol.44, pp.241-261, 2005. ,
DOI : 10.1007/3-540-26444-2_13