×

Computation of periodic solutions to pendulum type systems with a small parameter. (English. Russian original) Zbl 1459.34107

Comput. Math. Math. Phys. 60, No. 12, 1990-2006 (2020); translation from Zh. Vychisl. Mat. Mat. Fiz. 60, No. 12, 2055-2072 (2020).
Summary: Periodic solutions of pendulum-type ODE systems are considered. Finding such solutions is a classical problem in mechanics. Numerous methods are available for computing periodic solutions, and these methods have existed as long as the problems themselves. However, they were designed for manual calculation, and attempts to program them in computer algebra systems (CAS) are sometimes ineffective. For computing such solutions, we propose a method intended for CAS. The method is based on the use of high-order variational equations and symbolic differentiation. It is shown on a number of examples that all computations are reduced to operations with polynomials.

MSC:

34C25 Periodic solutions to ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
37C60 Nonautonomous smooth dynamical systems
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
68W30 Symbolic computation and algebraic computation
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Chernousko, F. L., Resonance phenomena in the motion of a satellite relative to its mass center, USSR Comput. Math. Math. Phys., 3, 699-713 (1963)
[2] A. D. Bruno, Local Methods in Nonlinear Differential Equations (Nauka, Moscow, 1979; Springer-Verlag, Berlin, 1989).
[3] Varin, V. P., Degeneracies of periodic solutions to the Beletsky equation, Regular Chaotic Dyn., 5, 313-328 (2000) · Zbl 0984.34030
[4] Varin, V. P., Isolated generating periodic solutions to the Beletsky equation, Cosmic Res., 45, 78-84 (2007)
[5] Varin, V. P., Integration of ordinary differential equations on Riemann surfaces with unbounded precision, Comput. Math. Math. Phys., 59, 1105-1120 (2019) · Zbl 1430.34015
[6] Grebenikov, E. A., Averaging Method in Applications (1986), Moscow: Nauka, Moscow
[7] V. F. Edneral and O. D. Timofeevskaya, “Computer algebra in scientific computations: Search for families of periodic solutions of ordinary differential equations by the method of normal forms 1,” Vestn. RUDN Ser. Mat. Inf. Fiz., No. 3, 28-45 (2014).
[8] Hénon, M.; Heiles, C., The applicability of the third integral of motion: Some numerical experiments, Astron. J., 69, 73-79 (1964)
[9] V. F. Edneral and O. D. Timofeevskaya, “Search for periodic solutions of ordinary differential equations by the method of normal forms: The case of fourth-order equations,” Inf.-Upr. Sist., No. 6, 24-34 (2018).
[10] Mikram, J.; Zinoun, F., Normal form methods for symbolic creation of approximate solutions of nonlinear dynamical systems, Math. Comput. Simul., 57, 253-289 (2001) · Zbl 0986.65123
[11] Poincaré, H., New Methods of Celestial Mechanics (1993) · Zbl 0776.01009
[12] Benbachir, S., Research of the periodic solutions of the Hénon-Heiles nonintegrable Hamiltonian system by the Lindstedt-Poincaré method, J. Phys. A: Math. Gen., 31, 5083-5103 (1998) · Zbl 0953.70012
[13] Meyer, K. R., Periodic Solutions of the N-Body Problem (1999), Berlin: Springer, Berlin · Zbl 0958.70001
[14] Tkhai, V. N., Periodic motions of a reversible second-order mechanical system: Application to the Sitnikov problem, J. Appl. Math. Mech., 70, 734-753 (2006)
[15] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series, and Products (2007), New York: Academic, New York · Zbl 1208.65001
[16] Petrov, A. G., Asymptotic methods for solving the Hamilton equations with the use of a parametrization of canonical transformations, Differ. Equations, 40, 672-685 (2004) · Zbl 1077.37041
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.