, Error polytope for the toy example

, Tightness cones for the toy example

. .. , Almost-banded structure of vector-valued integral operators

, Solution of the highly oscillating example

, Poincaré return map and displacement function

, Level curves of the potential function

, Abelian integral on small and big ovals

.. .. Inertial,

, Parameters for LODE validation in function of eccentricity and total time, p.239

, Steering into the hovering region within N velocity corrections, p.242

, Illustration of model predictive control strategy and receding horizon, p.243

, Obtained relative trajectory without certification

, X-coordinate in function of true anomaly for RPAs of degrees 5 and 7, p.245

, Y-coordinate in function of true anomaly for RPAs of degrees 5 and 7, p.246

. .. Error, 249 7.10 Error between numerical integration and the semi-analytical solution, p.251

, Approximation of Ai over, vol.2

, Approximation of Ai over, vol.2

, Approximation of Ai over, vol.2

. .. , Error plots for different polynomial approximations of arcsin, p.274

. .. , Error plots for different polynomial approximations of shifted arcsin, vol.294

, Dulac's result was given new (extremely complicated) proofs by Ilyashenko [118] and Écalle, vol.77

, Ilyashenko commented: Thus, after eighty years of development, our knowledge on Hilbert's 16th problem was almost the same as at the time when the problem was stated

S. Both, L. Shi-;, M. Chen, and . Wang,

L. Li and Q. Huang, , vol.155

R. Roussarie, 216] reduced the question of uniform finiteness in the quadratic case to proving finite cyclicity of 121 graphics. Today ca 85 of these have been successfully dealt with

J. Li, , vol.156

C. Christopher,

C. Li, C. Liu, and J. Yang, , p.13

?. and L. Ann, 16 Let G and g be as in Assumption 9, vol.9

. Then-g-r-l-?-ann,

. Proof, Using Proposition 9.15, one needs to prove that G g r ?(L · f )dx and ?G L L (f, g r ?) · n dS are zero. The first one is trivial since L ? Ann(f ). For the second, L L (f, g r ?) involves derivatives ? ? x (g r ?) with |?| < r

;. .. Hence and A. Of, and g ? |[x] vanishing over ?G, each operator g r i L i (with r i the order of L i ) annihilates f ? G as a distribution. Therefore, each operator R i := (g r i L i ) M gives a valid recurrence for the sequence of moments

, g r k L k } is holonomic. Similarly, we are not able to prove (or refute) that {R 1 , . . . , R k } is holonomic in general. Nevertheless, one can apply a Gröbner basis algorithm, which will possibly return a basis of a holonomic ideal. This heuristic is given in Algorithm RecurrencesMoments, However, from the fact that f is holonomic one can not directly guarantee that the ideal generated by {g r 1 L 1

. Bibliography,

S. A. Abramov, M. A. Barkatou, and M. Van-hoeij, Apparent singularities of linear difference equations with polynomial coefficients, Appl. Algebra Eng. Commun. Comput, vol.17, issue.2, pp.117-133, 2006.
URL : https://hal.archives-ouvertes.fr/hal-00116031

M. Abramowitz and I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series. Courier Corporation, vol.55, 1964.

R. P. Agarwal, Contraction and approximate contraction with an application to multipoint boundary value problems, J. Comput. Appl. Math, vol.9, issue.4, pp.315-325, 1983.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Dongarra et al., Sven Hammarling, Alan McKenney, and Danny Sorensen. LAPACK Users' guide, vol.9, 1999.

P. Gilz, Embedded and validated control algorithms for the spacecraft rendezvous, 2018.
URL : https://hal.archives-ouvertes.fr/tel-01922288

P. R. , A. Gilz, F. Bréhard, and C. Gazzino, Validated Semi-Analytical Transition Matrix for Linearized Relative Spacecraft Dynamics via Chebyshev Polynomials, Space Flight Mechanics Meeting, p.24, 2018.

P. R. , A. Gilz, and M. Joldes, Model predictive control for rendezvous hovering phases based on a novel description of constrained trajectories, Proceedings of the 20th IFAC World Congress, pp.7490-7495, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01484764

P. R. , A. Gilz, and C. Louembet, Predictive control algorithm for spacecraft rendezvous hovering phases, Control Conference (ECC), pp.2085-2090, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01078506

G. Arioli and H. Koch, Integration of dissipative partial differential equations: a case study, SIAM Journal on Applied Dynamical Systems, vol.9, issue.3, pp.1119-1133, 2010.

V. I. , Loss of stability of self-induced oscillations near resonance, and versal deformations of equivariant vector fields, Funkcional. Anal. i Prilo?en, vol.11, issue.2, pp.85-92, 1977.

V. I. , Arnol'd. Ten problems, Adv. Soviet Math. Amer. Math. Soc, vol.1, 1990.

D. Arzelier, F. Bréhard, N. Deak, M. Joldes, C. Louembet et al., Linearized impulsive fixed-time fuel-optimal space rendezvous: A new numerical approach, IFAC-PapersOnLine, vol.49, issue.17, pp.373-378, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01275427

D. Arzelier, F. Bréhard, and M. Joldes, Exchange algorithm for evaluation and approximation error-optimized polynomials, 26th IEEE Symposium on Computer Arithmetic (ARITH-26), 2019.
URL : https://hal.archives-ouvertes.fr/hal-02006606

R. Barrio, H. Jiang, and S. Serrano, A general condition number for polynomials, SIAM J. Numer. Anal, vol.51, issue.2, pp.1280-1294, 2013.

D. Batenkov, Moment inversion problem for piecewise D-finite functions, Inverse Problems, vol.25, issue.10, p.24, 2009.

R. H. Battin, An introduction to the mathematics and methods of astrodynamics, AIAA Education Series. American Institute of Aeronautics and Astronautics, 1999.

B. Beckermann, The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numer. Math, vol.85, issue.4, pp.553-577, 2000.

A. Benoit, Algorithmique semi-numérique rapide des séries de Tchebychev, 2012.

A. Benoit, M. Joldes, and M. Mezzarobba, Rigorous uniform approximation of D-finite functions using Chebyshev expansions, Math. Comp, vol.86, issue.305, pp.1303-1341, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01022420

V. Berinde, Iterative approximation of fixed points, Lecture Notes in Mathematics, vol.1912, 2007.

I. N. Bern?te?n, Modules over a ring of differential operators. An investigation of the fundamental solutions of equations with constant coefficients, Funkcional. Anal. i Prilo?en, vol.5, issue.2, pp.1-16, 1971.

Y. Bertot and P. Castéran, Interactive theorem proving and program development: Coq'Art: the calculus of inductive constructions, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00344237

M. Berz and K. Makino, Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by shrink wrapping, Int. J. Diff. Eq. Appl, vol.10, pp.385-403, 2005.

M. Berz and K. Makino, Rigorous global search using Taylor models, Proceedings of the 2009 conference on Symbolic numeric computation, pp.11-20, 2009.

J. T. Betts, Survey of numerical methods for trajectory optimization, J. Guidance Control Dynam, vol.21, issue.2, pp.193-207, 1998.

G. Binyamini, D. Novikov, and S. Yakovenko, On the number of zeros of Abelian integrals, Invent. Math, vol.181, issue.2, pp.227-289, 2010.

S. Boldo, F. Clément, F. Faissole, V. Martin, and M. Mayero, A Coq formal proof of the Lax-Milgram theorem, 6th ACM SIGPLAN Conference on Certified Programs and Proofs, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01391578

S. Boldo, F. Faissole, and A. Chapoutot, Round-off error analysis of explicit one-step numerical integration methods, 2017 IEEE 24th Symposium on Computer Arithmetic (ARITH), pp.82-89, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01581794

S. Boldo, C. Lelay, and G. Melquiond, Coquelicot: a user-friendly library of real analysis for Coq, Math. Comput. Sci, vol.9, issue.1, pp.41-62, 2015.
URL : https://hal.archives-ouvertes.fr/hal-00860648

S. Boldo and G. Melquiond, Flocq: A unified library for proving floatingpoint algorithms in Coq, 2011 IEEE 20th Symposium on Computer Arithmetic, pp.243-252, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00534854

A. Bose, Did You Know : The History of Egyptian Mathematics (Part II) -Egyptian Numerals, 2015.

A. Bostan, F. Chyzak, P. Lairez, and B. Salvy, Generalized Hermite reduction, creative telescoping and definite integration of D-finite functions, Proceedings of International Symposium on Symbolic and Algebraic Computation, pp.95-102, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01788619

N. Bourbaki and . Topology, , 1971.

J. P. Boyd, Chebyshev and Fourier spectral methods, 2001.

S. Boyd and L. Vandenberghe, Convex Optimization, 2004.

T. Braconnier and P. Langlois, From rounding error estimation to automatic correction with automatic differentiation, Automatic differentiation of algorithms, pp.351-357, 2002.

F. Bréhard, A Newton-like validation method for chebyshev approximate solutions of linear ordinary differential systems, ISSAC 2018-43rd International Symposium on Symbolic and Algebraic Computation, pp.103-110, 2018.

F. Bréhard, N. Brisebarre, and M. Joldes, Validated and numerically efficient chebyshev spectral methods for linear ordinary differential equations, ACM Trans. Math. Software, vol.44, issue.4, 2018.

F. Bréhard, M. Joldes, and J. Lasserre, On a moment problem with holonomic functions, 44th International Symposium on Symbolic and Algebraic Computation, 2019.

F. Bréhard, A. Mahboubi, and D. Pous, A certificate-based approach to formally verified approximations, 2019.

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, 2010.

N. Brisebarre, J. Muller, and A. Tisserand, Computing machine-efficient polynomial approximations, ACM Trans. Math. Software, vol.32, issue.2, pp.236-256, 2006.
URL : https://hal.archives-ouvertes.fr/ensl-00086826

N. Brisebarre and S. Chevillard, Efficient polynomial L ? approximations, 18th IEEE Symposium on Computer Arithmetic (ARITH-18), pp.169-176, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00119513

N. Brisebarre and M. Joldes, Chebyshev interpolation polynomial-based tools for rigorous computing, Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, pp.147-154, 2010.
URL : https://hal.archives-ouvertes.fr/ensl-00472509

N. Brisebarre, M. Joldes, J. Muller, A. Nanes, ,. et al., Error analysis of some operations involved in the fast Fourier transform, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01949458

G. Campan and P. Brousse, ORANGE: Orbital analytical model for geosynchronous satellite, Revista Brasileira de Ciencias Mecanicas, vol.16, pp.561-572, 1994.

E. Capello, F. Dabbene, G. Guglieri, and E. Punta, Flyable" Guidance and Control Algorithms for Orbital Rendezvous Maneuver, SICE Journal of Control, Measurement, and System Integration, vol.11, issue.1, pp.14-24, 2018.

C. Carasso and P. J. Laurent, Un algorithme général pour l'approximation au sens de Tchebycheff de fonctions bornées sur un ensemble quelconque, Approximation Theory, number 556 in Lecture Notes in Mathematics, pp.99-121, 1976.

C. Carasso, L'algorithme d'échange en optimisation convexe, 1973.

B. L. Chalmers, The Remez exchange algorithm for approximation with linear restrictions, Trans. Amer. Math. Soc, vol.223, pp.103-131, 1976.

F. Chen, C. Li, J. Llibre, and Z. Zhang, A unified proof on the weak Hilbert 16th problem for n = 2, J. Differential Equations, vol.221, issue.2, pp.309-342, 2006.

L. Sun-chen and M. S. Wang, The relative position, and the number, of limit cycles of a quadratic differential system, Acta Math. Sinica, vol.22, issue.6, pp.751-758, 1979.

E. W. Cheney, Introduction to approximation theory, 1998.

S. Chevillard, J. Harrison, M. Joldes, and C. Lauter, Efficient and accurate computation of upper bounds of approximation errors, Theoret. Comput. Sci, vol.412, issue.16, pp.1523-1543, 2011.
URL : https://hal.archives-ouvertes.fr/ensl-00445343

M. Sylvain-chevillard, C. Joldes, and . Lauter, Sollya: An environment for the development of numerical codes, International Congress on Mathematical Software, pp.28-31, 2010.

C. Christopher, Estimating limit cycle bifurcations from centers. In Differential equations with symbolic computation, Trends Math, pp.23-35, 2005.

C. Christopher and C. Li, Limit cycles of differential equations, Advanced Courses in Mathematics. CRM Barcelona, 2007.

F. Chyzak, Fonctions holonomes en calcul formel, 1998.
URL : https://hal.archives-ouvertes.fr/tel-00991717

F. Chyzak, The ABC of Creative Telescoping: Algorithms, Bounds, Complexity. Memoir of accreditation to supervise research (HDR), 2014.
URL : https://hal.archives-ouvertes.fr/tel-01069831

C. W. Clenshaw, A note on the summation of Chebyshev series, Math. Tables Aids Comput, vol.9, pp.118-120, 1955.

C. W. Clenshaw, The numerical solution of linear differential equations in Chebyshev series, Proc. Cambridge Philos. Soc, vol.53, pp.134-149, 1957.

A. Earl, N. Coddington, and . Levinson, Theory of ordinary differential equations, 1955.

L. Cruz-filipe, H. Geuvers, and F. Wiedijk, C-corn, the constructive coq repository at nijmegen, International Conference on Mathematical Knowledge Management, pp.88-103, 2004.

C. Daramy-loirat, D. Defour, F. De-dinechin, M. Gallet, N. Gast et al., CR-LIBM, A library of correctly-rounded elementary functions in doubleprecision, 2006.
URL : https://hal.archives-ouvertes.fr/ensl-01529804

E. Darulova and V. Kuncak, Towards a compiler for reals, ACM Trans. Program. Lang. Syst, vol.39, issue.2, p.28, 2017.

M. Daumas and G. Melquiond, Certification of bounds on expressions involving rounded operators, ACM Trans. Math. Software, vol.37, issue.1, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00127769

C. Florent-de-dinechin, G. Lauter, and . Melquiond, Certifying the floatingpoint implementation of an elementary function using Gappa, IEEE Trans. Comput, vol.60, issue.2, pp.242-253, 2011.

C. Q. Florent-de-dinechin, J. Lauter, and . Muller, Fast and correctly rounded logarithms in double-precision, Theor. Inform. Appl, vol.41, issue.1, pp.85-102, 2007.

G. Deaconu, On the trajectory design, guidance and control for spacecraft rendezvous and proximity operations, 2013.
URL : https://hal.archives-ouvertes.fr/tel-00919883

G. Deaconu, C. Louembet, and A. Théron, Constrained periodic spacecraft relative motion using non-negative polynomials, American Control Conference (ACC), pp.6715-6720, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00639905

G. D. Mauro, M. Schlotterer, S. Theil, and M. Lavagna, Nonlinear control for proximity operations based on differential algebra, J. Guidance Control Dynam, vol.38, issue.11, pp.2173-2187, 2015.

T. Driscoll, N. Hale, and L. N. Trefethen, , 2014.

K. Du, On well-conditioned spectral collocation and spectral methods by the integral reformulation, SIAM J. Sci. Comput, vol.38, issue.5, pp.3247-3263, 2016.

H. Dulac, Sur les cycles limites, Bull. Soc. Math. France, vol.51, pp.45-188, 1923.

T. Dzetkuli?, Rigorous integration of non-linear ordinary differential equations in Chebyshev basis, Numer. Algorithms, vol.69, pp.183-205, 2015.

, of Operator Theory: Advances and Applications, vol.156, 2005.

J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, 1992.

D. Elliott, D. F. Paget, G. M. Phillips, and P. J. Taylor, Error of truncated Chebyshev series and other near minimax polynomial approximations, J. Approx. Theory, vol.50, issue.1, pp.49-57, 1987.

C. Epstein, W. L. Miranker, and T. J. Rivlin, Ultra-arithmetic. I. Function data types, Math. Comput. Simulation, vol.24, issue.1, pp.1-18, 1982.

C. Epstein, W. L. Miranker, and T. J. Rivlin, Ultra-arithmetic. II. Intervals of polynomials, Math. Comput. Simulation, vol.24, issue.1, pp.19-29, 1982.

W. Fehse, Automated rendezvous and docking of spacecraft, vol.16, 2003.

. Silviu-ioan and . Filip, Robust tools for weighted Chebyshev approximation and applications to digital filter design, 2016.

L. Fousse, G. Hanrot, V. Lefèvre, P. Pélissier, and P. Zimmermann, MPFR: A Multiple-Precision Binary Floating-Point Library with Correct Rounding, ACM Trans. Math. Software, vol.33, issue.2, 2007.
URL : https://hal.archives-ouvertes.fr/inria-00070266

L. Fox and I. B. Parker, Chebyshev polynomials in numerical analysis, 1968.

A. Galligo, Some algorithmic questions on ideals of differential operators, European Conference on Computer Algebra, pp.413-421, 1985.

C. Gazzino, Dynamics of a Geostationary Satellite, LAAS-CNRS, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01644934

C. Gazzino, D. Arzelier, L. Cerri, D. Losa, C. Louembet et al., Solving the Minimum-Fuel Low-Thrust Geostationary Station Keeping Problem via the Switching Systems Theory, European Conference for Aeronautics and AeroSpace Sciences, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01562696

C. Gazzino, C. Louembet, D. Arzelier, N. Jozefowiez, D. Losa et al., Integer Programming for Optimal Control of Geostationary Station Keeping of Low-Thrust Satellites, IFAC 2017 World Congress, pp.8169-8174, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01416694

C. Gazzino, Stratégies de maintien à poste pour un satellite géostationnaire à propulsion tout électrique, 2018.

C. Gazzino, D. Arzelier, C. Louembet, L. Cerri, C. Pittet et al., Long-term electric-propulsion geostationary station-keeping via integer programming, J. Guidance Control Dynam, vol.42, issue.5, pp.976-991, 2019.
URL : https://hal.archives-ouvertes.fr/hal-02053120

D. Gim and K. , Satellite relative motion using differential equinoctial elements, Celestial Mechanics and Dynamical Astronomy, vol.92, issue.4, pp.295-336, 2005.

J. Girard, P. Taylor, and Y. Lafont, Proofs and types, Cambridge Tracts in Theoretical Computer Science, vol.7, 1989.

I. Gohberg, S. Goldberg, and M. A. Kaashoek, Basic classes of linear operators, 2003.

G. H. Golub, P. Milanfar, and J. Varah, A stable numerical method for inverting shape from moments, SIAM J. Sci. Comput, vol.21, issue.4, p.0, 1999.

G. Gonthier, Formal proof-the four-color theorem, Notices Amer. Math. Soc, vol.55, issue.11, pp.1382-1393, 2008.

D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, vol.26, 1977.

T. Granlund, The GNU multiple precision arithmetic library, 1996.

N. Gravin, J. Lasserre, V. Dmitrii, S. Pasechnik, and . Robins, The inverse moment problem for convex polytopes, Discrete & Comp. Geometry, vol.48, issue.3, pp.596-621, 2012.

L. Greengard, Spectral integration and two-point boundary value problems, SIAM J. Numer. Anal, vol.28, issue.4, pp.1071-1080, 1991.

M. Grimmer, K. Petras, and N. Revol, Multiple precision interval packages: Comparing different approaches, Numerical Software with Result Verification, pp.64-90, 2004.
URL : https://hal.archives-ouvertes.fr/hal-02101997

T. Hales, M. Adams, G. Bauer, T. Dang, J. Harrison et al., A formal proof of the kepler conjecture, In Forum of Mathematics, Pi, vol.5, 2017.

C. Thomas and . Hales, A proof of the Kepler conjecture, Ann. of Math, vol.162, issue.2, pp.1065-1185, 2005.

R. Halsey and F. Patrick, Real Analysis, 2010.

E. N. Hartley, A tutorial on model predictive control for spacecraft rendezvous, 2015 European Control Conference (ECC), pp.1355-1361, 2015.

E. Hecke, ;. George, U. Brauer, J. R. Goldman, and R. Kotzen, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol.77, 1981.

N. J. Higham, Accuracy and stability of numerical algorithms, 2002.

D. Hilbert, Mathematische Probleme. Vortrag, gehalten auf dem internationalen Mathematiker-Congress zu Paris, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl, pp.253-297, 1900.

J. Hölzl, F. Immler, and B. Huffman, Type classes and filters for mathematical analysis in Isabelle/HOL, Interactive Theorem Proving, pp.279-294, 2013.

K. Hrbacek and T. Jech, Introduction to set theory, revised and expanded, 1999.

D. G. Hull, Conversion of optimal control problems into parameter optimization problems, J. Guidance Control Dynam, vol.20, issue.1, pp.57-60, 1997.

A. Hungria, J. Lessard, and J. D. James, Rigorous numerics for analytic solutions of differential equations: the radii polynomial approach, Math. Comp, vol.85, issue.299, pp.1427-1459, 2016.

, IEEE Computer Society. IEEE Standard for Floating-Point Arithmetic. IEEE Standard, 1985.

, IEEE Computer Society. IEEE Standard for Floating-Point Arithmetic. IEEE Stan, pp.754-2008, 2008.

G. Ifrah, The Universal History of Computing: From the Abacus to the Quantum Computer, 2001.

G. Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, 2001.

. Yu and . Ilyashenko, Centennial history of Hilbert's 16th problem, Bull. Amer. Math. Soc. (N.S.), vol.39, issue.3, pp.301-354, 2002.

Y. S. , In the theory of normal forms of analytic differential equations violating the conditions of A. D. Bryuno divergence is the rule and convergence the exception, Vestnik Moskov. Univ. Ser. I Mat. Mekh, vol.86, issue.2, pp.10-16, 1981.

Y. S. , Finiteness theorems for limit cycles, Translations of Mathematical Monographs, vol.94, 1991.

F. Immler, A verified ODE solver and the Lorenz attractor, J. Automat. Reason, vol.61, issue.1-4, pp.73-111, 2018.

F. Immler and J. Hölzl, Numerical analysis of ordinary differential equations in Isabelle/HOL, International Conference on Interactive Theorem Proving, pp.377-392, 2012.

G. Inalhan, M. Tillerson, and J. P. How, Relative dynamics and control of spacecraft formations in eccentric orbits, J. Guidance Control Dynam, vol.25, issue.1, pp.48-59, 2002.

E. L. Ince, Ordinary Differential Equations, 1956.

D. J. Irvin, R. Cobb, and T. A. Lovell, Fuel-optimal maneuvers for constrained relative satellite orbits, Journal of guidance, control, and dynamics, vol.32, issue.3, pp.960-973, 2009.

A. Iserles, A first course in the numerical analysis of differential equations, Cambridge Texts in Applied Mathematics, 2009.

F. Johansson, Arb: efficient arbitrary-precision midpoint-radius interval arithmetic, IEEE Trans. Comput, vol.66, issue.8, pp.1281-1292, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01394258

T. Johnson, A quartic system with twenty-six limit cycles, Exp. Math, vol.20, issue.3, pp.323-328, 2011.

T. Johnson and W. Tucker, An improved lower bound on the number of limit cycles bifurcating from a Hamiltonian planar vector field of degree 7, Internat. J. Bifur. Chaos Appl. Sci. Engrg, vol.20, issue.5, pp.1451-1458, 2010.

T. Johnson and W. Tucker, An improved lower bound on the number of limit cycles bifurcating from a quintic Hamiltonian planar vector field under quintic perturbation, Internat. J. Bifur. Chaos Appl. Sci. Engrg, vol.20, issue.1, pp.63-70, 2010.

M. Joldes, Rigorous Polynomial Approximations and Applications, 2011.
URL : https://hal.archives-ouvertes.fr/tel-00657843

M. Joldes, J. Muller, V. Popescu, and W. Tucker, Campary: cuda multiple precision arithmetic library and applications, International Congress on Mathematical Software, pp.232-240, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01312858

L. V. Kantorovich and G. P. Akilov, Functional analysis, 1982.

L. V. Kantorovich, B. Z. Vulikh, and A. G. Pinsker, Functional analysis in partially ordered spaces, 1950.

S. Karlin and W. J. Studden, Tchebycheff systems: With applications in analysis and statistics, Pure and Applied Mathematics, 1966.

Y. Katznelson, An introduction to harmonic analysis, 2004.

E. Kaucher, Solving function space problems with guaranteed close bounds, Proc. of the symposium on A new approach to scientific computation, pp.139-164, 1983.

W. Edgar, W. L. Kaucher, and . Miranker, Self-validating numerics for function space problems: Computation with guarantees for differential and integral equations, vol.9, 1984.

G. Kedem, A posteriori error bounds for two-point boundary value problems, SIAM Journal on Numerical Analysis, vol.18, issue.3, pp.431-448, 1981.

A. G. Khovanski?, Real analytic manifolds with the property of finiteness, and complex abelian integrals, Funktsional. Anal. i Prilozhen, vol.18, issue.2, pp.119-127, 1984.

R. Klatte, U. Kulisch, A. Wiethoff, and M. Rauch, C-XSC: A C++ class library for extended scientific computing, 2012.

C. Koutschan, Advanced applications of the holonomic systems approach, ACM Comm. Computer Algebra, vol.43, issue.3, p.119, 2009.

C. Koutschan, Advanced applications of the holonomic systems approach, Research Institute for Symbolic Computation (RISC), 2009.

C. Koutschan, Holonomic functions (user's guide), 2010.

U. Kulisch, An axiomatic approach to rounded computations, Numer. Math, vol.18, pp.1-17, 1971.

O. Kupriianova and C. Lauter, Metalibm: A mathematical functions code generator, International Congress on Mathematical Software, pp.713-717, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01513490

T. Lalescu, Introduction à la théorie des équations intégrales (Introduction to the Theory of Integral Equations), 1911.

E. M. Landis and I. G. Petrovski?, A letter to the editors, Mat. Sb. (N.S.), vol.73, issue.115, p.160, 1967.

O. E. Lanford and I. , A computer-assisted proof of the Feigenbaum conjectures, Bull. Amer. Math. Soc. (N.S.), vol.6, issue.3, pp.427-434, 1982.

J. Bernard-lasserre, Moments, positive polynomials and their applications, vol.1, 2010.

J. Bernard-lasserre, Recovering an homogeneous polynomial from moments of its level set, Discrete Comput. Geom, vol.50, issue.3, pp.673-678, 2013.

J. Bernard-lasserre and M. Putinar, Algebraic-exponential data recovery from moments, Discrete Comput. Geom, vol.54, issue.4, pp.993-1012, 2015.

C. Q. Lauter, Arrondi Correct de Fonctions Mathématiques, 2008.

C. Lauter and M. Mezzarobba, Semi-automatic floating-point implementation of special functions, 2015 IEEE 22nd Symposium on Computer Arithmetic, pp.58-65, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01137953

J. , P. Lessard, and C. Reinhardt, Rigorous numerics for nonlinear differential equations using Chebyshev series, SIAM J. Numer. Anal, vol.52, issue.1, pp.1-22, 2014.

C. Li, C. Liu, and J. Yang, A cubic system with thirteen limit cycles, J. Differential Equations, vol.246, issue.9, pp.3609-3619, 2009.

J. Li and Q. Huang, Bifurcations of limit cycles forming compound eyes in the cubic system, Chinese Ann. Math. Ser. B, vol.8, issue.4, p.643, 1987.

J. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg, vol.13, issue.1, pp.47-106, 2003.

G. Liu and V. Kreinovich, Fast convolution and fast Fourier transform under interval and fuzzy uncertainty, J. Comput. System Sci, vol.76, issue.1, pp.63-76, 2010.

P. Di-lizia, R. Armellin, and M. Lavagna, Application of high order expansions of twopoint boundary value problems to astrodynamics, Celestial Mech. Dynam. Astronom, vol.102, issue.4, pp.355-375, 2008.

P. Di-lizia, R. Armellin, A. Morselli, and F. Bernelli-zazzera, High order optimal feedback control of space trajectories with bounded control, Acta Astronautica, vol.94, issue.1, pp.383-394, 2014.

J. Löfberg, YALMIP: A toolbox for modeling and optimization in MATLAB, Computer Aided Control Systems Design, pp.284-289, 2004.

D. Losa, High vs low thrust station keeping maneuver planning for geostationary satellites, 2007.
URL : https://hal.archives-ouvertes.fr/tel-00173537

N. Magaud and Y. Bertot, Changing data structures in type theory: A study of natural numbers, International Workshop on Types for Proofs and Programs, pp.181-196, 2000.

K. Makino and M. Berz, Suppression of the wrapping effect by Taylor model-based verified integrators: Long-term stabilization by preconditioning, Int. J. Diff. Eq. Appl, vol.10, pp.353-384, 2005.

K. Makino, Rigorous analysis of nonlinear motion in particle accelerators, 1998.

K. Makino and M. Berz, Taylor models and other validated functional inclusion methods, Int. J. Pure Appl. Math, vol.4, issue.4, pp.379-456, 2003.

K. Makino and M. Berz, Cosy infinity version 9, Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, vol.558, issue.1, pp.346-350, 2006.

K. Makino and M. Berz, Suppression of the wrapping effect by Taylor modelbased verified integrators: the single step, Int. J. Pure Appl. Math, vol.36, issue.2, pp.175-197, 2007.

É. Martin, -. Dorel, and G. Melquiond, Proving tight bounds on univariate expressions with elementary functions in coq, Journal of Automated Reasoning, vol.57, issue.3, pp.187-217, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01086460

P. Martin-löf and G. Sambin, Intuitionistic type theory, Bibliopolis Naples, vol.9, 1984.

C. John, D. C. Mason, and . Handscomb, Chebyshev polynomials, 2002.

G. Melquiond, Proving bounds on real-valued functions with computations, International Joint Conference on Automated Reasoning, pp.2-17, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00180138

M. Mezzarobba, Autour de l'évaluation numérique des fonctions D-finies, 2011.

R. E. Moore, Interval Analysis, 1966.

R. E. Moore, R. Baker, M. J. Kearfott, and . Cloud, Introduction to interval analysis, vol.110, 2009.

B. Mourrain, Polynomial-exponential decomposition from moments, Found. Comput. Math, vol.18, issue.6, pp.1435-1492, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01367730

J. Muller, Elementary Functions, 2016.
URL : https://hal.archives-ouvertes.fr/ensl-00989001

J. Muller, N. Brunie, C. Florent-de-dinechin, M. Jeannerod, V. Joldes et al., Handbook of Floating-Point Arithmetic, Birkhäuser Boston, 2018.
URL : https://hal.archives-ouvertes.fr/ensl-00379167

T. Mitsuhiro and . Nakao, A numerical approach to the proof of existence of solutions for elliptic problems, Japan Journal of Applied Mathematics, vol.5, issue.2, p.313, 1988.

T. Mitsuhiro and . Nakao, Numerical verification methods for solutions of ordinary and partial differential equations, Numerical Functional Analysis and Optimization, vol.22, issue.3-4, pp.321-356, 2001.

F. Natterer, The mathematics of computerized tomography, Classics in Applied Mathematics. (SIAM), vol.32, 2001.

M. Neher, N. Kenneth-r-jackson, and . Nedialkov, On Taylor model based integration of ODEs, SIAM J. Numer. Anal, vol.45, issue.1, pp.236-262, 2007.

Y. Nesterov, Squared functional systems and optimization problems, High performance optimization, pp.405-440, 2000.

A. Neumaier, Interval methods for systems of equations, 1990.

A. Neumaier, The Wrapping Effect, Ellipsoid Arithmetic, Stability and Confidence Regions, pp.175-190, 1993.

A. Neumaier, Taylor forms-use and limits, Reliab. Comput, vol.9, issue.1, pp.43-79, 2003.

O. Maria-nica-bolojan, Fixed point methods for nonlinear differential systems with nonlocal conditions, 2013.

D. Douglas, J. Novaes, and . Torregrosa, On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl, vol.448, issue.1, pp.171-186, 2017.

T. Oaku, Algorithms for integrals of holonomic functions over domains defined by polynomial inequalities, J. Symbolic Comput, vol.50, pp.1-27, 2013.

. Shin&apos;ichi-oishi, Numerical verification of existence and inclusion of solutions for nonlinear operator equations, Journal of Computational and Applied Mathematics, vol.60, issue.1-2, pp.171-185, 1995.

J. Oliver, Rounding error propagation in polynomial evaluation schemes, J. Comput. Appl. Math, vol.5, issue.2, pp.85-97, 1979.

S. Olver and A. Townsend, A fast and well-conditioned spectral method, SIAM Rev, vol.55, issue.3, pp.462-489, 2013.

J. M. Ortega and W. C. Rheinboldt, Iterative solution of nonlinear equations in several variables, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), vol.30, 2000.

T. W. Parks and J. H. Mcclellan, Chebyshev Approximation for Nonrecursive Digital Filters with Linear Phase, IEEE Transactions on Circuit Theory, vol.19, issue.2, pp.189-194, 1972.

S. Paszkowski, Zastosowania numeryczne wielomianów i szeregów Czebyszewa. Pa?stwowe Wydawn. Naukowe, 1975.

A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pribli?. Metod. Re?en. Differencial'. Uravnen. Vyp, vol.2, pp.115-134, 1964.

I. G. Petrovski? and E. M. Landis, On the number of limit cycles of the equation dy/dx = P (x, y)/Q(x, y), where P and Q are polynomials of 2nd degree, Amer. Math. Soc. Transl, vol.37, issue.79, pp.177-221, 1955.

S. , U. Pillai, and A. Papoulis, Probability, random variables, and stochastic processes, vol.2, 2002.

G. Plonka and M. Tasche, Fast and numerically stable algorithms for discrete cosine transforms, Linear Algebra Appl, vol.394, pp.309-345, 2005.

M. Plum, Computer-assisted existence proofs for two-point boundary value problems, Computing, vol.46, issue.1, pp.19-34, 1991.

M. Plum, Numerical existence proofs and explicit bounds for solutions of nonlinear elliptic boundary value problems, Computing, vol.49, issue.1, pp.25-44, 1992.

V. Popescu, Towards fast and certified multiple-precision librairies, 2017.
URL : https://hal.archives-ouvertes.fr/tel-01534090

M. J. Powell, On the maximum errors of polynomial approximations defined by interpolation and by least squares criteria, Comput. J, vol.9, pp.404-407, 1967.

R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comput. Model, vol.49, issue.3, pp.703-708, 2009.

, Homotopy type theory: Univalent foundations of mathematics, 2013.

L. B. Rall, Computational solution of nonlinear operator equations, 1979.

L. Rebillard, Étude théorique et algorithmique des séries de Chebyshev solutions d'équations différentielles holonomes, 1998.

R. Reemtsen and J. Rückmann, Semi-Infinite Programming, vol.25, 1998.

E. Remes, Sur le calcul effectif des polynômes d'approximation de Tchebichef (in French), Compt. Rend. Acad. Sci, vol.199, pp.337-340, 1934.

E. Remes, Sur un procédé convergent d'approximations successives pour déterminer les polynômes d'approximation (in French), Compt. Rend. Acad. Sci, vol.198, pp.2063-2065, 1934.

N. Revol and F. Rouillier, Motivations for an arbitrary precision interval arithmetic and the MPFI library, Reliab. Comput, vol.11, issue.4, pp.275-290, 2005.
URL : https://hal.archives-ouvertes.fr/inria-00100985

A. Riccardi, C. Tardioli, and M. Vasile, An intrusive approach to uncertainty propagation in orbital mechanics based on Tchebycheff polynomial algebra, AAS/AIAA Astrodynamics Specialist Conference, vol.8, pp.707-722, 2015.

T. J. Rivlin, The Chebyshev Polynomials, 1974.

F. Robert, Étude et utilisation de normes vectorielles en analyse numérique linéaire (in French), 1968.

A. Robertson, G. Inalhan, and J. P. How, Formation control strategies for a separated spacecraft interferometer, Proceedings of the 1999 American Control Conference, vol.6, pp.4142-4147, 1999.

A. Rocca, V. Magron, and T. Dang, Certified Roundoff Error Bounds using Bernstein Expansions and Sparse Krivine-Stengle Representations, 24th IEEE Symposium on Computer Arithmetic. IEEE, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01956817

R. Roussarie, Bifurcation of planar vector fields and Hilbert's sixteenth problem, Progress in Mathematics, vol.164, 1998.

W. Rudin, Real and complex analysis (3rd), 1986.

W. Rudin, Functional analysis, 1991.

S. M. Rump, Fast and parallel interval arithmetic, BIT, vol.39, issue.3, pp.534-554, 1999.

M. Siegfried and . Rump, Intlab-interval laboratory, Developments in reliable computing, pp.77-104, 1999.

S. M. Rump, Verification methods: rigorous results using floating-point arithmetic, Acta Numer, vol.19, pp.287-449, 2010.

B. Salvy, D-finiteness: Algorithms and applications, Proceedings of the 18th International Symposium on Symbolic and Algebraic Computation, pp.2-3, 2005.

B. Salvy, Linear differential equations as a data-structure, Found. Comput. Math, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01940078

B. Salvy and P. Zimmermann, Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable, ACM Trans. Math. Software, vol.20, issue.2, pp.163-177, 1994.
URL : https://hal.archives-ouvertes.fr/inria-00070025

L. Schwartz, Théorie des distributions, 1966.

R. Serra, D. Arzelier, F. Bréhard, and M. Joldes, Fuel-optimal impulsive fixed-time trajectories in the linearized circular restricted 3-body-problem, IAF Astrodynamics Symposium in 69TH international astronautical congress (IAC 2018), pp.1-9, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01830253

R. Serra, D. Arzelier, M. Joldes, J. Lasserre, A. Rondepierre et al., Fast and accurate computation of orbital collision probability for shortterm encounters, J. Guidance Control Dynam, vol.39, issue.5, pp.1009-1021, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01132149

A. Shapiro, Semi-infinite programming, duality, discretization and optimality conditions, Optimization, vol.58, issue.2, pp.133-161, 2009.

L. Song and . Shi, A concrete example of the existence of four limit cycles for plane quadratic systems, Sci. Sinica, vol.23, issue.2, pp.153-158, 1980.

S. K. Shrivastava, Orbital Perturbations and Stationkeeping of Communication Satellites, Journal of Spacecraft, vol.15, issue.2, 1978.

M. J. Sidi, Spacecraft Dynamics and Control, 1997.

A. Solovyev, M. S. Baranowski, I. Briggs, and C. Jacobsen, Rigorous estimation of floating-point round-off errors with symbolic Taylor expansions, ACM Trans. Program. Lang. Syst, vol.41, issue.1, 2018.

E. M. Soop, Handbook of Geostationary Orbits, 1994.

R. P. Stanley, Differentiably finite power series, European J. Combin, vol.1, issue.2, pp.175-188, 1980.

G. Strang, The discrete cosine transform, SIAM Rev, vol.41, issue.1, pp.135-147, 1999.

A. W. Strzebo?ski, Computing in the field of complex algebraic numbers, J. Symbolic Comput, vol.24, issue.6, pp.647-656, 1997.

J. Sullivan, S. Grimberg, and S. Amico, Comprehensive survey and assessment of spacecraft relative motion dynamics models, Journal of Guidance, Control, and Dynamics, vol.40, issue.8, pp.1837-1859, 2017.

T. Sunaga, Theory of interval algebra and its application to numerical analysis. RAAG memoirs, vol.2, p.209, 1958.

G. Szeg?-;-providence and R. I. , Orthogonal polynomials, 1975.

N. Takayama, An algorithm of constructing the integral of a module -an infinite dimensional analog of gröbner basis, Proceedings of the International Symposium on Symbolic and Algebraic Computation, pp.206-211, 1990.

M. Tillerson, G. Inalhan, and J. P. How, Co-ordination and control of distributed spacecraft systems using convex optimization techniques, International Journal of Robust and Nonlinear Control, vol.12, issue.2-3, pp.207-242, 2002.

A. Tisserand, High-performance hardware operators for polynomial evaluation, International Journal of High Performance Systems Architecture (IJHPSA), vol.1, issue.1, pp.14-23, 2007.
URL : https://hal.archives-ouvertes.fr/lirmm-00140930

L. Nicholas-trefethen, Approximation Theory and Approximation Practice. SIAM, 2013.

J. Tschauner, Elliptic orbit rendezvous, AIAA Journal, vol.5, issue.6, pp.1110-1113, 1967.

J. Tschauner and P. Hempel, Optimale Beschleunigungsprogramme fur das Rendezvous-Manover, Acta Astronautica, vol.10, issue.5-6, pp.296-307, 1964.

W. Tucker, A rigorous ODE solver and Smale's 14th problem, Found. Comput. Math, vol.2, issue.1, pp.53-117, 2002.

W. Tucker, Validated numerics: a short introduction to rigorous computations, 2011.

R. H. Tütüncü, K. C. Toh, and M. J. Todd, Solving semidefinite-quadratic-linear programs using SDPT3, Math. Program, vol.95, issue.2, pp.189-217, 2003.

D. A. Vallado, Fundamentals of astrodynamics and applications, Space Technology Series, 1997.

J. Bouwe-van-den, J. Berg, and . Lessard, Chaotic braided solutions via rigorous numerics: Chaos in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Syst, vol.7, issue.3, pp.988-1031, 2008.

J. Bouwe-van-den, J. Berg, and . Lessard, Rigorous numerics in dynamics, Notices of the AMS, vol.62, issue.9, 2015.

J. Van-der-hoeven, Ball arithmetic, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00432152

J. Van-heijenoort, From Frege to Gödel: a source book in mathematical logic, vol.9, pp.1879-1931, 1967.

A. N. Varchenko, Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles, Funktsional. Anal. i Prilozhen, vol.18, issue.2, pp.98-108, 1984.

M. Vasile, C. Ortega-absil, and A. Riccardi, Set propagation in dynamical systems with generalised polynomial algebra and its computational complexity, Communications in Nonlinear Science and Numerical Simulation, vol.75, pp.22-49, 2019.

J. Neumann, Zur Einführung der transfiniten Zahlen, Acta Litt. Sci. Szeged, vol.1, pp.199-208, 1923.

J. Von-zur-gathen and J. Gerhard, Modern computer algebra, 2013.

S. Wang and P. Yu, Existence of 121 limit cycles in a perturbed planar polynomial Hamiltonian vector field of degree 11, Chaos Solitons Fractals, vol.30, issue.3, pp.606-621, 2006.

G. A. Watson, The calculation of best restricted approximations, SIAM J. Numer. Anal, vol.11, issue.4, pp.693-699, 1974.

A. Wazwaz, Linear and nonlinear integral equations: methods and applications, 2011.

H. Whitney, Geometric integration theory, 1957.

R. Wong, Asymptotic approximations of integrals, vol.34, 2001.

C. E. Woodruff, The evolution of modern numerals from ancient tally marks, The American Mathematical Monthly, vol.16, issue.8/9, pp.125-133, 1909.

K. Xu, The Chebyshev points of the first kind, Appl. Numer. Math, vol.102, pp.17-30, 2016.

N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM J. Numer. Anal, vol.35, issue.5, pp.2004-2013, 1998.

T. Yamamoto, A unified derivation of several error bounds for Newton's process, J. Comput. Appl. Math, vol.12, pp.179-191, 1985.

K. Yamanaka and F. Ankersen, New state transition matrix for relative motion on an arbitrary elliptical orbit, J. Guidance Control Dynam, vol.25, issue.1, pp.60-66, 2002.

O. Zarrouati, Trajectoires spatiales. Cépaduès-e edition, 1987.

D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math, vol.32, issue.3, pp.321-368, 1990.

J. Zemke, b4m: A free interval arithmetic toolbox for MATLAB, 1999.

A. Zygmund, Trigonometric series, vol.I, 2002.