Abstract
We consider analytical systems of ODE with a real or complex time. Integration of such ODE is equivalent to an analytical continuation of a solution along some path, which usually belongs to the real axis. The problems that may appear along this path are often caused by singularities of the solution that lie outside the real axis. It is possible to circumvent problematic parts of the path (including singularities) by going on the Riemann surface of the solution (i.e., in the complex domain). A natural way to realize this program is to use the method of Taylor expansions, which does not require a formal complexification of the system (i.e., a change of variables). We use two classical problems, i.e., the Restricted Three-Body problem, and Van der Pol equation, for demonstration of how Taylor expansions can be used for integration of ODE with an arbitrary precision. We obtained some new results in these problems.
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REFERENCES
W. H. Press, et al., Numerical Recipes: The Art of Scientific Computing (Cambridge Univ. Press, Cambridge, 1986).
N. N. Kalitkin, Numerical Methods (Nauka, Moscow, 1978) [in Russian].
K. Atkinson and W. Han, Elementary Numerical Analysis (Wiley, New York, 2004).
R. E. Moore, Methods and Applications of Interval Analysis (SIAM, Philadelphia, 1979).
L. Euler, Introduction to the Analysis of the Infinite (Springer, New York, 1988).
R. Broucke, “Solution of the n-body problem with recurrent power series,” Cel. Mech., No. 4, 110–115 (1971).
M. R. Broucke, “Periodic orbits in the restricted three-body problem with Earth–Moon masses,” NASA Tech. Rep., 32–1168 (Pasadena, 1968).
K. I. Babenko, Principles of Numerical Analysis (Nauka, Moscow, 1986) [in Russian].
O. Aberth, Introduction to Precise Numerical Methods (Elsevier, Amsterdam, 2007).
G. Dahlquist and A. Bjorck, Numerical Methods (Dover, New York, 1974).
V. Szebehely, Theory of Orbits: The Restricted Problem of Three Bodies (Academic, New York, 1967).
V. P. Varin, “Closed families of periodic solutions of the restricted three-body problem,” Preprint No. 16, IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, Moscow, 2008) [in Russian].
A. D. Bruno and V. P. Varin, “Closed families of periodic solutions of the restricted three-body problem,” Astron. Vestn. 43 (3), 265–288 (2009).
A. N. Sharkovskii, “Coexistence of cycles of a continuous mapping of the line into itself,” Ukr. Mat. Zh. 16 (1), 61–71 (1964).
J.-P. Eckmann, “Roads to turbulence in dissipative dynamical systems,” Rev. Mod. Phys. 53 (4) Part I (1981).
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Varin, V.P. Integration of Ordinary Differential Equations on Riemann Surfaces with Unbounded Precision. Comput. Math. and Math. Phys. 59, 1105–1120 (2019). https://doi.org/10.1134/S0965542519070121
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DOI: https://doi.org/10.1134/S0965542519070121