We consider difference schemes for dynamical systems \( \dot{x}=f(x) \) with quadratic right-hand side that have t-symmetry and are reversible. Reversibility is interpreted in the sense that a Cremona transformation is performed at each step of the computations using the difference scheme. The inheritance of periodicity and the Painlevé property by the approximate solution is investigated. In the computer algebra system Sage, values are found for the step Δt for which the approximate solution is a sequence of points with period n ∈ ℕ. Examples are given, and conjectures about the structure of the sets of initial data generating sequences with period n are formulated.
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References
G. J. Cooper, “Stability of Runge–Kutta methods for trajectory problems,” IMA J. Numer. Anal., 7, 1–13 (1987).
Y. B. Suris, “Preservation of symplectic structure in the numerical solution of Hamiltonian systems,” in: S. S. Filippov (ed.), Numerical Solution of Differential Equations [in Russian], Akad. Nauk. SSSR, Inst. Prikl. Mat., Moscow (1988), pp. 138–144.
Yu. B. Suris, “Hamiltonian methods of Runge–Kutta type and their variational interpretation,” Math. Model., 2, 78–87 (1990).
E. Hairer, G. Wanner, and Ch. Lubich, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer, Berlin–Heidelberg–New York (2000).
D. Greenspan, “Completely conservative and covariant numerical methodology for N-body problems with distance-dependent potentials,” technical report No. 285 at http://hdl.handle.net/10106/2267 (1992).
D. Greenspan, “Completely conservative, covariant numerical methodology,” Comp. Math. Appl., 29, No. 4, 37–43 (1995).
D. Greenspan, “Completely conservative, covariant numerical solution of systems of ordinary differential equations with applications,” Rend. Sem. Mat. Fis. Milano, 65, 63–87 (1995).
D. Greenspan, N-Body Problems and Models, World Scientific (2004).
J. C. Simo and O. González, “Assessment of energy-momentum and symplectic schemes for stiff dynamical systems,” presented at the ASME Annual Meeting (1993).
E. Graham, G. Jeleni, and M. A. Crisfield, “A note on the equivalence of two recent time-integration schemes for N-body problems,” Comm. Numer. Methods Engrg., 18, 615–620 (2002).
E. A. Ayryan, M. D. Malykh, L. A. Sevastianov, and Yu Ying, “On periodic approximate solutions of the three-body problem found by conservative difference schemes,” Lect. Notes Comput. Sci., 12291, 77–90 (2020).
X. Yang and L. Ju, “Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model,” Comput. Methods Appl. Mech. Engrg., 315, 691–711 (2016).
X. Yang and L. Ju, “Linear and unconditionally energy stable schemes for the binary fluid-surfactant phase field model,” Comput. Methods Appl. Mech. Engrg., 318, 1005–1029 (2017).
H. Zhang, X. Qian, and S. Song, “Novel high-order energy-preserving diagonally implicit Runge–Kutta schemes for nonlinear Hamiltonian ODEs,” Appl. Math. Lett., 102, 106091 (2020).
J. Shen, J. Xu, and J. Yang, “The scalar auxiliary variable (SAV) approach for gradient flows,” J. Comput. Phys., 353, 407–416 (2018).
P. Painlevé, “Le, cons sur la theorie analytique des equations differentielles,” in: OEuvres de Paul Painlevé, Vol. 1 (1971).
H. Umemura, “Birational automorphism groups and differential equations,” Nagoya Math. J., 119, 1–80 (1990).
M. D. Malykh, “On transcendental functions arising from integrating differential equations in finite terms,” J. Math. Sci., 209, 935–952 (2015).
L. Cremona, “Sulle trasformazioni geometriche delle figure piane,” in: Opere matematiche di Luigi Cremona, Vol. 2, U. Hoepli, Milano (1915), pp. 54–61, 193–218.
M. Hénon, “A two-dimensional mapping with a strange attractor,” Comm. Math. Phys., 50, 69–77 (1976).
M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley (1989).
E. A. Ayryan, M. D. Malykh, L. A. Sevastianov, and Yu Ying, “On explicit difference schemes for autonomous systems of differential equations on manifolds,” Lect. Notes Comput. Sci., 11661, 343–361 (2019).
A. Goriely, Integrability and Nonintegrability of Dynamical Systems, World Scientific (2001).
W. W. Golubev, Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point, Jerusalem (1960).
F. Klein, Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert, Vol. 1, Springer, Berlin–Heidelberg (1979).
E. A. Ayryan, M. D. Malykh, and L. A. Sevastianov, “On difference schemes approximating first-order differential equations and defining a projective correspondence between layers,” J. Math. Sci., 240, 634–645 (2019).
M. N. Lagutinski, “Sur la forme la plus simple du système d’equations différentielles ordinaires,” Mat. Sb., 27, No. 4, 420–423 (1911).
A. Bychkov and G. Pogudin, “Optimal monomial quadratization for ODE systems,” arXiv:2103.08013 (2021).
B. Grammaticos, F. W. Nijhoff, and A. Ramani, “Discrete Painlevé equations,” in: The Painlevé Property, One Century Later, Springer, Berlin–Heidelberg (1999), pp. 413–516.
P. A. Clarkson, E. L. Mansfield, and H. N. Webster, “On the relation between the continuous and discrete Painlevé equations,” Theor. Math. Phys., 122, 1–16 (2000).
K. Ishizaki and R. Korhonen, “Meromorphic solutions of algebraic difference equations,” Constr. Approx., 48, 371–384 (2018).
V. P. Gerdt, M. D. Malykh, L. A. Sevastianov, and Yu Ying, “On the properties of numerical solutions of dynamical systems obtained using the midpoint method,” Discrete Contin. Models Appl. Comput. Sci., 27, No. 3, 242–262 (2019).
SageMath, the Sage Mathematics Software System, Version 7.4 (2016); https://www.sagemath.org.
R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, Dover, Mineola (2010).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 507, 2021, pp. 157–172.
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Baddour, A., Malykh, M. & Sevastianov, L. On Periodic Approximate Solutions of Dynamical Systems with Quadratic Right-Hand Side. J Math Sci 261, 698–708 (2022). https://doi.org/10.1007/s10958-022-05781-4
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DOI: https://doi.org/10.1007/s10958-022-05781-4