Numerical integration of differential equations in the presence of first integrals: observer method

Busvelle, Eric ; Kharab, Rachid ; Maciejewski, A. ; Strelcyn, Jean-Marie

Applicationes Mathematicae, Tome 22 (1994), p. 373-418 / Harvested from The Polish Digital Mathematics Library

Résumé

We introduce a simple and powerful procedure-the observer method-in order to obtain a reliable method of numerical integration over an arbitrary long interval of time for systems of ordinary differential equations having first integrals. This aim is achieved by a modification of the original system such that the level manifold of the first integrals becomes a local attractor. We provide a theoretical justification of this procedure. We report many tests and examples dealing with a large spectrum of systems with different dynamical behaviour. The comparison with standard and symplectic methods of integration is also provided.

Détail

Publié le : 1994-01-01

Zbl 0819.34008

EUDML-ID : urn:eudml:doc:219103

Bibliographie

[000] [1] M. Adler and P. van Moerbeke, The algebraic integrability of geodesic flow on SO(4), Invent. Math., 67 (1982) 297-331. | Zbl 0539.58012

[001] [2] V. I. Arnold, Mathematical Methods of Classical Mechanics, Graduate Texts in Math. 60, Springer, Berlin (1989).

[002] [3] V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical aspects of classical and celestial mechanics, V. I. Arnold (ed.) Dynamical Systems III, Encyclopaedia Math. Sci. 3, Springer, Berlin (1988). | Zbl 0885.70001

[003] [4] M. Artigue, V. Gautheron et E. Isambert, Ensemble de bifurcation et topologie des variétés intégrales dans le problème du solide pesant, J. Méc. Théor. Appl., 5 (1986) 429-469. | Zbl 0633.70005

[004] [5] M. Audin et R. Shilol, Variétés abéliennes réelles et toupie de Kovalevski, Publ. IRMA (preprint), Strasbourg (1992).

[005] [6] J. Baumgarte, Stabilization of the differential equations of Keplerian motion, in: B. D. Tapley and V. Szebehely (eds.), Recent Advances in Dynamical Astronomy, Reidel, Dordrecht, (1973) 38-44. | Zbl 0272.70004

[006] [7] J. Baumgarte, Stabilization, manipulation and analytic step adaptation, in: V. Szebehely and B. D. Tapley (eds.), Long Time Predictions in Dynamics, Reidel, Dordrecht, (1976) 153-163.

[007] [8] G. Benettin, M. Casartelli, L. Galgani, A. Giorgilli and J.-M. Strelcyn, On the reliability of numerical study of stochasticity. Part I: Existence of time averages, Nuovo Cimento, 44B (1978) 183-195.

[008] [9] G. Benettin, M. Casartelli, L. Galgani, A. Giorgilli and J.-M. Strelcyn, On the reliability of numerical study of stochasticity. Part II: Identification of time averages, ibid., 50B (1979) 211-232.

[009] [10] D. G. Bettis (ed.), Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations, 19- 20 October 1972, the University of Texas at Austin, Lecture Notes in Math. 362, Springer, Berlin, 1974.

[010] [11] O. I. Bogoyavlensky, Integrable Euler equations on Lie algebras arising in problems of mathematical physics, Math. USSR-Izv., 25 (1984) 207-257.

[011] [12] O. I. Bogoyavlensky, Integrable Euler equations on SO(4) and their physical applications, Comm. Math. Phys., 93 (1984) 417-436. | Zbl 0567.58012

[012] [13] O. I. Bogoyavlensky, New integrable problems of classical mechanics, ibid., 94 (1984) 225-269. | Zbl 0576.70004

[013] [14] O. I. Bogoyavlensky, Euler equations on finite dimensional Lie algebras arising in physical problems, ibid., 95 (1984) 307-315.

[014] [15] O. I. Bogoyavlensky, Periodic solutions in a model of pulsar rotation, ibid., 102 (1985) 349-359.

[015] [16] O. I. Bogoyavlensky, Integrable cases of a rigid body dynamics and integrable systems on the ellipsoids, ibid., 103 (1986) 305-322.

[016] [17] O. I. Bogoyavlensky, Some integrable cases of Euler equations I, Dokl. Akad. Nauk SSSR, 287 (1986) 1105-1108 (in Russian).

[017] [18] O. I. Bogoyavlensky, Some integrable cases of Euler equations II, ibid., 292 (1987) 318-322 (in Russian).

[018] [19] O. I. Bogoyavlensky, Boundary value problems of mathematical physics, Proc. Steklov Inst. Math., 95 (1988).

[019] [20] O. I. Bogoyavlensky, Euler equations on finite dimensional Lie coalgebras, Uspekhi Mat. Nauk, 47 (1) (1992) 107-146 (in Russian); English transl.: Russian Math. Surveys 47 (1).

[020] [21] J. C. Butcher, Implicit Runge-Kutta processes, Math. Comput., 18 (1964) 50-64. | Zbl 0123.11701

[021] [22] P. J. Channell and J. C. Scovel, Symplectic integrators of Hamiltonian systems, Nonlinearity, 3 (1990) 231-259. | Zbl 0704.65052

[022] [23] P. J. Channell and J. C. Scovel, Integrators for Lie-Poisson dynamical systems, Phys. D, 50 (1991) 80-88. | Zbl 0735.58017

[023] [24] I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai, Ergodic Theory, Springer, Berlin (1982).

[024] [25] R. L. Devaney, Homoclinic orbits in Hamiltonian systems, J. Differential Equations, 21 (1976) 431-438. | Zbl 0343.58005

[025] [26] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry-Methods and Applications. Part I: The Geometry of Surfaces. Transformation Groups and Fields, Graduate Texts in Math. 93, Springer, Berlin (1984). | Zbl 0529.53002

[026] [27] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry - Methods and Applications. Part II: The Geometry and Topology of Manifolds, Graduate Texts in Math. 104, Springer, Berlin (1985). | Zbl 0565.57001

[027] [28] T. Eirola and J. M. Sanz-Serna, Conservation of integrals and symplectic structure in the integration of differential equations by multistep methods, Numer. Math., 61 (1992) 281-290. | Zbl 0741.65056

[028] [29] El Hamidi, Propriétés stochastiques d'un système non-linéaire en dimension finie, Thèse de physique théorique, Université de Pau (1989).

[029] [30] K. Feng and M.-Z. Qin, The symplectic methods for the computation of hamiltonian equations, in: Y.-W. Zhu and B.-Y. Guo (eds.), Numerical Methods for Partial Differential Equations, Lecture Notes in Math. 1297, Springer, Berlin, (1987), 1-37.

[030] [31] A. F. Filippov, Differential Equations with Discontinuous Right Hand Side, Nauka, Moscow (1985) (in Russian). | Zbl 0571.34001

[031] [32] A. T. Fomenko, Integrability and Nonintegrability in Geometry and Mechanics, Kluwer, Dordrecht (1988).

[032] [33] A. T. Fomenko and V. V. Trofimov, Integrable Systems on Lie Algebras and Symmetric Spaces, Gordon and Breach, New York (1988).

[033] [34] N. K. Gavrilov and L. P. Shil'nikov, On bifurcations of the equilibrium states of a hamiltonian system with two degrees of freedom, Selecta Math. Sov., 10 (1) (1991) 61-68.

[034] [35] Z. G. Ge and J. E. Marsden, Lie-Poisson-Hamilton-Jacobi theory and Lie- Poisson integrators, Phys. Lett. A, 133 (3) (1988) 134-139.

[035] [36] C. W. Gear, Invariants and numerical methods for ODEs, Phys. D, 60 (1992) 303-310. | Zbl 0779.34012

[036] [37] B. Gladman and M. Dunca, Symplectic integrators for long-term integrations in celestial mechanics, Celestial Mech., 52 (1991) 221-240. | Zbl 0744.70016

[037] [38] D. Greenspan, Arithmetic Applied Mathematics, Pergamon Press, Oxford (1980).

[038] [39] J. W. Grizzle and P. E. Moraal, Newton, observers and nonlinear discrete-time control, in: Proc. 29th Conf. on Decision and Control, Honolulu, Hawaii, 1990, 760-767.

[039] [40] L. Haine, Geodesic flow on SO(4) and abelian surfaces, Math. Ann., 263 (1983) 435-472. | Zbl 0521.58042

[040] [41] L. Haine, The algebraic complete integrability of geodesic flow on SO(N), Comm. Math. Phys., 94 (1984) 271-287. | Zbl 0584.58023

[041] [42] E. Hairer, S. P. Norsett and G. Wanner, Solving Ordinary Differential Equa- tions I. Nonstiff Problems, Springer, Berlin (1987). | Zbl 0638.65058

[042] [43] P. Hartman, Ordinary Differential Equations, Wiley, New York (1964).

[043] [44] K. Hockett, Chaotic numerics from an integrable hamiltonian system, Proc. Amer. Math. Soc., 108 (1990) 271-281. | Zbl 0678.58021

[044] [45] V. D. Irtegov, Invariant Manifolds of Stationary Motions and Their Stability, Nauka, Siberian branch, Novosibirsk (1985 (in Russian). | Zbl 0602.70023

[045] [46] H. Kinoshita, H. Yoshida and H. Nakai, Symplectic integrators and their application to dynamical astronomy, Celestial Mech., 50 (1991) 59-71. | Zbl 0724.70019

[046] [47] A. M. Kovalev and A. N. Chudnenko, On stability of the equilibrium point of a Hamiltonian system with two degrees of freedom in the case of equal frequencies, Dokl. Akad. Nauk Ukrain. SSSR Ser. A, 11 (1977) 1010-1013 (in Russian). | Zbl 0376.70020

[047] [48] V. V. Kozlov, Integrability and nonintegrability in hamiltonian mechanics, Uspekhi Mat. Nauk, 38 (1) (1983) 3-67 (in Russian).

[048] [49] V. V. Kozlov and D. A. Onishenko, Nonintegrability of Kirchhoff equations, Dokl. Akad. Nauk SSSR, 266 (1982) 271-281 (in Russian).

[049] [50] J. D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, New York (1973). | Zbl 0258.65069

[050] [51] M. Lecar, A comparison of eleven numerical integrations of the same gravitational 25-body problem, Bull. Astronom., Sér. 3 vol. III fasc. 1 (1968) 91-104.

[051] [52] E. Leimanis, The General Problem of the Motion of Coupled Rigid Bodies about a Fixed Point, Springer, Berlin (1965). | Zbl 0128.41606

[052] [53] L. M. Lerman and Ya. L. Umanskiĭ, On the existence of separatrix loops in four-dimensional systems similar to the integrable hamiltonian systems, J. Appl. Math. Mech., 47 (3) (1983) 335-340.

[053] [54] T. Levi-Civita and G. Amaldi, Lezioni di meccanica razionale, Vol. II, Part 2, Zanichelli, Bologna (1927). | Zbl 49.0570.01

[054] [55] A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Appl. Math. Sci. 38, Springer, Berlin (1983). | Zbl 0506.70016

[055] [56] A. J. Maciejewski and J.-M. Strelcyn, Numerical integration of differential equations in presence of first integrals: partial first integrals, to appear. | Zbl 0819.34008

[056] [57] A. Marciniak, Numerical Solutions of the n-Body Problem, Reidel, Dordrecht (1985).

[057] [58] A. Marciniak, The Selected Numerical Methods for Solving the n-Body Problem, Poznań Technical University, (1989).

[058] [59] A. Marciniak and D. Greenspan, Energy conserving numerical solutions of simplified turbulence equations, Dept. Math., the University of Texas at Arlington, Technical Report 269 (1990).

[059] [60] A. Marciniak and D. Greenspan, Arbitrary order Hamiltonian conserving numerical solutions of Calogero and Toda systems, Computers Math. Appl., 22 (7) (1991) 11-35. | Zbl 0734.65059

[060] [61] P. E. Nacozy, The use of integrals in numerical integrations of the n-body problem, Astrophys. and Space Sci., 14 (1971) 40-51.

[061] [62] O. Neron de Surgy, Sur les intégrateurs symplectiques pour les systèmes Hamiltoniens standards et les systèmes de Lie-Poisson; une application aux équations d'Euler sur SO(4) Rapport de stage fait à Electricité de France, 1991.

[062] [63] P. J. Olver, Applications of Lie Groups to Differential Equations, Graduate Texts in Math. 107, Springer, Berlin (1986).

[063] [64] S. A. Orszag and J. B. McLaughlin, Evidence that random behavior is generic for nonlinear differential equations, Phys. D 1, (1980)) 68-79. | Zbl 1194.37060

[064] [65] A. M. Perelomov, Dynamical systems of classical mechanics with hidden symmetry, Preprints Inst. Theor. Exper. Phys. Moscow 82, 1981 (in Russian)..

[065] [66] A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. The simplest systems, Preprints Inst. Theor. Exper. Phys. Moscow 149, 1981 (in Russian)..

[066] [67] A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. Constrained systems, Preprints Inst. Theor. Exper. Phys. Moscow 116, 1983 (in Russian)..

[067] [68] A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. The motion of a rigid body with fixed point, Preprints Inst. Theor. Exper. Phys. Moscow 147, 1983 (in Russian)..

[068] [69] A. M. Perelomov, Integrable systems of classical mechanics and Lie algebras. The motion of a rigid body in the ideal fluid, Preprints Inst. Theor. Exper. Phys. Moscow 151, 1984 (in Russian)..

[069] [70] A. M. Perelomov, Integrable Systems of Classical Mechanics and Lie Algebras, Vol. I, Birkhäuser, Basel (1990). | Zbl 0717.70003

[070] [71] H. Pollard, Celestial Mechanics, Carus Math. Monogr. 18, Math. Assoc. Amer. (1976).

[071] [72] J. M. Sanz-Serna, Symplectic integrators for Hamiltonian problems: an overview, Acta Numerica 1992, 243-286. | Zbl 0762.65043

[072] [73] Ya. G. Sinai et al., Ergodic theory with applications to dynamical systems and statistical mechanics, in: Ya. G. Sinai (ed.), Dynamical Systems II, Encyclopaedia Math. Sci. 2, Springer, Berlin (1989).

[073] [74] S. Smale, Topology and mechanics. I, Invent. Math., 10 (1970) 305-331. | Zbl 0202.23201

[074] [75] S. Smale, Topology and mechanics. II, ibid., 11 (1970) 45-64. | Zbl 0203.26102

[075] [76] A. G. Sokol'skiĭ, On the stability of an autonomous hamiltonian system with two degres of freedom in the case of equal frequencies, J. Appl. Math. Mech., 38 (1974) 741-749.

[076] [77] E. Stiefel and G. Schiefele, Linear and Regular Celestial Mechanics, Springer, Berlin (1971). | Zbl 0226.70005

[077] [78] J.-M. Strelcyn, The 'coexistence problem' for conservative dynamical systems: a review, Colloq. Math., 62 (1991) 331-345. | Zbl 0791.58062

[078] [79] V. V. Strygin and V. A. Sobolev, Separation of Motions by the Method of Integral Manifolds, Nauka, Moscow (1988 (in Russian). | Zbl 0657.70002

[079] [80] V. V. Trofimov, Introduction to the Geometry of Manifolds with Symmetries, Moscow University Press, Moscow (1989). | Zbl 0705.53002

[080] [81] S. Ushiki, Central difference schema and chaos, Phys., D 4 (1982) 407-424. | Zbl 1194.65097

[081] [82] A. P. Veselov, On conditions of integrability of Euler equations on SO(4) Dokl. Akad. Nauk SSSR, 270 (1983) 1298-1300 (in Russian).

[082] [83] H. Yoshida, Construction of higher order symplectic integrators, Phys. Lett. A, 150 (1990) 262-268.