Abstract
We describe usage of the normal form method and corresponding computer algebra packages for building an approximation of local periodic solutions of nonlinear autonomous ordinary differential equations (ODEs). For illustration a number of systems are treated.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Andersen, G. M. and Geer, J. F., Power series expansions for the frequency and period of the limit cycle of the van der Pol equation. SIAM J. Appl. Math. 42 (1983), 678–693.
Arnold, V. I. and Anosov, D. V. (eds.), Dynamical Systems I. Encyclopaedia of Mathematical Sciences. Springer-Verlag, New York, 1988.
Bibikov, Yu. N., Local Theory of Nonlinear Analytic Ordinary Differential Equations. LNM 702, Springer-Verlag, New York, 1979.
Boege, W., Gebauer, R. and Kredel, H., Some examples for solving systems of algebraic equations by calculating Groebner bases. J. Symb. Comput. 1 (1986), 83–98.
Bruno (Brjuno), A. D., Analytical form of differential equations I. Trans. Mosc. Mat. Soc. 25 (1971), 131–288.
Bruno (Brjuno), A. D., Analytical form of differential equations II. Trans. Mosc. Mat. Soc. 26 (1972), 199–239.
Bruno, A. D., Local Method in Nonlinear Differential Equations. Part I — The Local Method of Nonlinear Analyses of Differential Equations, Part II — The Sets of Analyticity of a Normalizing Transformation. Springer Series in Soviet Mathematics, Springer-Verlag, Berlin New York, 1989.
Bruno, A. D., Bifurcation of the periodic solutions in the case of a multiple pair of imaginary eigenvalues. Selecta Mathematica (formerly Sovietica) 12(1) (1993), 1–12.
Bruno, A. D., The Restricted 3-Body Problem: Plane Periodic Orbits. De Gruyter Expositions in Mathematics, v. 17, De Gruyter, Berlin New York, 1994.
Bruno, A. D., Normal forms. J. Math. Comput. Simul. 45 (1998), 413–427.
Bruno, A. D., The Power Geometry in Algebraic and Differential Equations. Elsevier, North-Holland, 2000.
Deprit, A., Canonical transformation depending on a small parameter. Celestial Mechanics 1(1) (1969), 12–30.
Edneral, V. F., Computer generation of normalizing transformation for systems of nonlinear ODE. Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation (Kiev, Ukraine, July 1993) (M. Bronstein, ed.), pp. 14–19. ACM Press, New York, 1993.
Edneral, V. F., Complex periodic solutions of autonomous ODE systems with analytical right sides near an equilibrium point. Fundamentalnaya i Prikladnaya Matematika 1(2) (1995), 393–398 (in Russian).
Edneral, V. F., Computer evaluation of cyclicity in planar cubic system. Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation (Hawaii, USA, July 1997) (W. Küchlin, ed.), pp. 305–309. ACM Press, New York, 1997.
Edneral, V. F., A symbolic approximation of periodic solutions of the Henon-Heiles system by the normal form method. J. Math. Comput. Simul. 45 (1998), 445–463.
Edneral, V. F., About normal form method. Proceedings of the Second Workshop on Computer Algebra in Scientific Computing (CASC’ 99, Munich, Germany, 1999) (Ganzha et al., eds.), pp. 51–66. Springer-Verlag, Berlin Heidelberg, 1999.
Edneral, V. F., Bifurcation analysis of low resonant case of the generalized Henon-Heiles system. Proceedengs of the Fourth Workshop on Computer Algebra in Scientific Computing (CASC 2001, Konstanz, Germany, 2001) (Ganzha et al., eds.), pp. 167–176. Springer-Verlag, Berlin Heidelberg, 2001.
Edneral, V. F., Periodic solutions of a cubic ODE system. Proceedings of the Fifth Workshop on Computer Algebra in Scientific Computing (CASC 2003, Passau, Germany, September 20–26, 2003) (Ganzha et al., eds.), pp. 77–80. Tech. Univ. München, Munich, 2003.
Edneral, V. F. and Khanin, R., Multivariate power series and normal form calculation in Mathematica. Proceedings of the Fifth Workshop on Computer Algebra in Scientific Computing (CASC 2002, Big Yalta, Ukraine, September 2002) (Ganzha et al., eds.), pp. 63–70. Tech. Univ. München, Munich, 2002.
Edneral, V. F. and Khanin, R., Application of the resonant normal form to high order nonlinear ODEs using MATHEMATICA. Nuclear Inst. and Methods in Physics Research A 502(2-3) (2003), 643–645.
Edneral, V. F. and Khanin, R., Investigation of the double pendulum system by the normal form method in MATHEMATICA. Programming and Computer Software 30(2) (2004), 115–117. Translated from Programmirovanie 30(2).
Edneral, V. F. and Khrustalev, O. A., The normalizing transformation for nonlinear systems of ODEs. The realization of the algorithm. Proceedings of International Conference on Computer Algebra and its Application in Theoretical Physics (USSR, Dubna, September 1985), pp. 219–224. JINR Publ., Dubna, 1985 (in Russian).
Edneral, V. F. and Khrustalev, O. A., Program for recasting ODE systems in normal form. Sov. J. Programmirovanie 5 (1992), 73–80 (in Russian).
Godziewski, K. and Maciejewski, A. J., Celest. Mech. Dyn. Astron. 49 (1990), 1.
Guckenheimer, J. and Holmes P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer-Verlag, New York, 1986.
Gustavson, A. G., On constructing formal integrals of a Hamiltonian system near an equilibrium point. Astronomical J. 71 (1966), 670–686.
Hassard, B. D., Kazarinoff, N. D. and Wan, Y. H., Theory and Applications of Hopf Bifurcation. Cambridge Univ. Press, Cambridge, 1981.
Hearn, A. C., REDUCE User’s Manual. Rand Publication, CP78, 1987.
Henon, M. and Heiles, C., The applicability of the third integral of motion: Some numerical experiments. Astronomical J. 69 (1964), 73–79.
Hori, G. I., Theory of general perturbations with unspecified canonical variables. J. Japan Astron. Soc. 18(4) (1966), 287–296.
Ito, H., Convergence of Birkhoff normal forms for integrable systems. Comment. Math. Helv. 64 (1989), 412–461.
Ito, H., Integrability of Hamiltonian systems and Birkhoff normal forms in the simple resonance case. Math. Ann. 292 (1992), 411–444.
Marsden, J. E., McCracken, M., The Hopf Bifurcation and Its Applications. Springer Applied Math. Series, v. 19, Springer-Verlag, New York, 1976.
Mersman, W. A., A new algorithm for Lie transformation. Celestial Mechanics 3(1) (1970), 81–89.
Rand, R. and Armbruster, D., Perturbation Methods, Bifurcation Theory and Computer Algebra. Springer-Verlag, New York, 1987.
Shevchenko, I. I. and Sokolsky, A. G., Algorithms for normalization of Hamiltonian systems by means of computer algebra. Comp. Phys. Comm. 77 (1993), 11–18.
Vallier, L., An algorithm for the computation of normal forms and invariant manifolds. Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation (Kiev, Ukraine, July 1993) (M. Bronstein, ed.), pp. 225–233. ACM Press, New York, 1993.
Walcher, S., On differential equations in normal form. Math. Ann. 291 (1991), 293–314.
Walcher, S., On transformations into normal form. J. Math. Anal. Appl. 180 (1993), 617–632.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Edneral, V.F. (2005). Looking for Periodic Solutions of ODE Systems by the Normal Form Method. In: Wang, D., Zheng, Z. (eds) Differential Equations with Symbolic Computation. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7429-2_11
Download citation
DOI: https://doi.org/10.1007/3-7643-7429-2_11
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7368-9
Online ISBN: 978-3-7643-7429-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)