Abstract
The normal form method is widely used in the theory of nonlinear ordinary differential equations (ODEs). But in practice it is impossible to evaluate the corresponding transformations without computer algebra packages. Here we describe an algorithm for normalization of nonlinear autonomous ODEs. Some implementations of these algorithms are also discussed.
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Poincaré, H.: Les Méthodes Nouvelles de la Méchanique Celeste. vol. 1,2,3. Gauthier-Villars, Paris (1872–1879) Reprint: Dover, New York, 1957. National Aeronautics and Space Administration, Washington (1967)
Dulac, H.: Solution d’une système d’équations différentielles dans le voisinage des valeurs singulieres. Bull. Soc. Math. France 40, 324–384 (1912)
Bruno(Brjuno), A.D.: The normal form of differential equations. Dokl. Akad. Nauk SSSR 157, 1276–1279 (1964) [Russian]. Sov. Math. Dokl. 5, 1105–1108 (1964) [English]
Arnold, V.I., Anosov, D.V. (eds.): Dynamical Systems, I. Encyclopaedia of Mathematical Sciences. Springer, New York (1988)
Rom, A.: Mechanized algebraic operations (MAO). Celestial Mechanics 1, 301–319 (1970)
Bruno(Brjuno), A.D.: Analytical form of differential equations. Trans. Mosc. Mat. Soc. 25, 131–288 (1971), 26, 199–239 (1972)
Bruno, A.D.: Local method in nonlinear differential equations. Springer Series in Soviet Mathematics, pages 370 (1989), ISBN 3-540-18926-2
Bruno, A.D.: Normal forms. J. Mathematics and Computers in Simulation 45, 413–427 (1998)
Bruno, A.D.: The power geometry in algebraic and differential equations, pages 385. Elsevier, Amsterdam (2000)
Edneral, V.F., Khrustalev, O.A.: The normalizing transformation for nonlinear systems of ODEs. The realization of the algorithm (Russian). In: Proc. Int. Conf. on Computer Algebra and its Application in Theoretical Physics, USSR, Dubna, pp. 219–224. JINR publ., Dubna (1985)
Edneral, V.F., Khrustalev, O.A.: Program for recasting ODE systems in normal form (Russian). Sov. J. Programmirovanie (5), 73–80 (1992)
Edneral, V.F., Khanin, R.: Multivariate power series and normal form calculation in MATHEMATICA. In: Ganzha, V., et al. (eds.) CASC 2002. Proc. Fifth Workshop on Computer Algebra in Scientific Computing, Big Yalta, Ukraine, pp. 63–70. Tech. Univ. München, Munich (2002)
Vallier, L.: An Algorithm for the computation of normal forms and invariant manifolds. In: Bronstein, M. (ed.) Proc. 1993 Int. Symp. on Symbolic and Algebraic Computation, Kiev, Ukraine, pp. 225–233. ACM Press, New York (1993)
Mikram, J., Zinoun, F.: Normal form methods for symbolic creation of approximate solutions of nonlinear dynamical systems. J. Mathematics and Computers in Simulation 57, 253–290 (2001)
Edneral, V.F.: Computer generation of normalizing transformation for systems of nonlinear ODE. In: Bronstein, M. (ed.) Proc. 1993 Int. Symp. on Symbolic and Algebraic Computation, Kiev, Ukraine, pp. 14–19. ACM Press, New York (1993)
Edneral, V.F.: A symbolic approximation of periodic solutions of the Henon–Heiles system by the normal form method. J. Mathematics and Computers in Simulation 45, 445–463 (1998)
Edneral, V.F.: Bifurcation analysis of low resonant case of the generalized Henon - Heiles system. In: Ganzha, et al. (eds.) CASC 2001. Proc. Fourth Workshop on Computer Algebra in Scientific Computing, Konstanz, Germany, pp. 167–176. Springer, Heidelberg (2001)
Edneral, V.F.: Periodic solutions of a cubic ODE system. In: Ganzha, et al. (eds.) CASC 2003. Proc. Fifth Workshop on Computer Algebra in Scientific Computing, Passau, Germany, pp. 77–80. Tech. Univ. München, Munich (2003)
Edneral, V.F., Khanin, R.: Application of the resonant normal form to high order nonlinear ODEs using MATHEMATICA. Nuclear Inst. and Methods in Physics Research A 502, 643–645 (2003)
Edneral, V.F., Khanin, R.: Investigation of the double pendulum system by the normal form method in MATHEMATICA. Programming and Computer Software 30, 115–117 (2004)
Edneral, V.F.: Looking for periodic solutions of ode systems by the normal form method. In: Wang, D., Zheng, Z. (eds.) Differential Equations with Symbolic Computation, Birkhauzer, Basel, Boston, Berlin, pp. 173–200 (2005)
Verhulst, F.: Nonlinear differential equations and dynamical systems, p. 227. Springer, Heidelberg (1990)
Edneral, V.F.: Computer evaluation of cyclicity in planar cubic system. In: Küchlin, W. (ed.) Proc. ISSAC 1997, Hawaii, USA, pp. 305–309. ACM, New York (1997)
Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation, p. 280. Cambridge Univ. Press, Cambridge (1981)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, New York (1986)
Rand, R., Armbruster, D.: Perturbation Methods, Bifurcation Theory and Computer Algebra. Springer, New York (1987)
Bruno, A.D.: Bifurcation of the periodic solutions in the case of a multiple pair of imaginary eigenvalues. Selecta Mathematica formerly Sovietica 12, 1–12 (1993)
Bruno, A.D.: Local integrability of the Euler–Poisson equations. Doklady Math. 74, 512–516 (2006)
Bruno, A.D., Edneral, V.F.: Normal forms and integrability of ODE systems. Programming and Computer Software 32, 139–144 (2006)
Bruno, A.D., Edneral, V.F.: On integrability of the Euler–Poisson equations. Fundamental and Applied Mathematics 13, 45–59 (2007)
Bruno, A.D., Edneral, V.F.: Computation of normal forms of the Euler–Poisson equations (Russian). Preprint No. 1 of the Keldysh Institute of Applied Mathematics of RAS. Moscow, pages 17 (2007)
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Edneral, V.F. (2007). An Algorithm for Construction of Normal Forms. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_10
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DOI: https://doi.org/10.1007/978-3-540-75187-8_10
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