Abstract
Numerical continuation of solution through certain singular points of the curve of the set of solutions to a system of nonlinear algebraic or transcendental equations with a parameter is considered. Bifurcation points of codimension two and three are investigated. Algorithms and computer programs are developed that implement the procedure of discrete parametric continuation of the solution and find all branches at simple bifurcation points of codimension two and three. Corresponding theorems are proved, and each algorithm is rigorously justified. A novel algorithm for the estimation of errors of tangential vectors at simple bifurcation points of a finite codimension m is proposed. The operation of the computer programs is demonstrated by test examples, which allows one to estimate their efficiency and confirm the theoretical results.
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References
H. B. Keller, Lectures on Numerical Methods in Bifurcation Problems (Springer, Berlin, 1987).
E. I. Grigolyuk and V. I. Shalashilin, Nonlinear Strain Problems (Nauka, Moscow, 1988) [in Russian].
E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Series in Computational Mathematics, Vol. 13 (Springer, Berlin, 1990).
J. D. Crawford, “Introduction to bifurcation theory,” Rev. Modern Phys. 63, 991–1037(1991).
E. B. Kuznetsov, “Criterion-based approach to the analysis of a dynamic panel jumping,” Izv. Ross. Akad. Nauk, Ser. Mekh. Tverd. Tela, No. 1, 150–160 (1996).
V. I. Shalashilin and E. B. Kuznetsov, Parametric Continuation and Optimal Parametrization in Applied Mathematics and Mechanics (Editorial URSS, Moscow, 1999; Kluwer, Dordrecht, 2003).
E. I. Grigolyuk and E. A. Lopanitsyn, Finite Bends, Stability, and Supercritical Behavior of Thin Slightly Sloping Shells (Mosc. Avtomekhanicheskii Institut, Moscow, 2004) [in Russian].
S. D. Krasnikov and E. B. Kuznetsov, “The numerical continuation of solutions at the bifurcation points of mathematical models,” Mat. Model. 21, 47–58 (2009).
S. D. Krasnikov and E. B. Kuznetsov, “Parametrization of a solution at bifurcation points,” Differ. Equations 45, 1218–1222 (2009).
V. V. Karpov, Strength and Stability of Stiffened Shells of Revolution, Part 1 (Fizmatlit, Moscow, 2010), Part 2 (Fizmatlit, Moscow, 2011).
S. S. Gavryushin, O. O. Baryshnikova, and O. F. Boriskin, Numerical Analysis of Construction Elements of Machines and Devices (Bauman Mosc. Gos. Technical Univ. Moscow, 2014) [in Russian].
D. F. Davidenko, “On the approximate solution of systems of nonlinear equations,” Ukr. Mat. Zh. 5, (2), 196–206 (1953).
V. I. Shalashilin and E. B. Kuznetsov, “The best parameter for the continuation of solution,” Dokl. Akad. Nauk 334, 566–568 (1994).
S. D. Krasnikov and E. B. Kuznetsov, “Numerical continuation of solution at singular points of codimension one,” Comput. Math. Math. Phys. 55, 1802–1822 (2015).
V. A. Trenogin, Functional Analysis (Nauka, Moscow, 2002) [in Russian].
A. M. Lyapunov, The General Problem of the Stability of Motion (Khar’kov, 1892) [in Russian].
E. Schmidt, “Zur Theorie der linearen und nichtlinearen Integralgleichungen. Teil 3. Über die Auflösungen der nichtlinear Integralgleichung und die Verzweigung ihrer Lösunger,” Math. Ann. 65, 370–399 (1908).
M. M. Vainberg and V. A. Trenogin, Bifurcation Theory for Solutions to Nonlinear Equations (Nauka, Moscow, 1969) [in Russian].
M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, et al., Approximate Solution of Operator Equations (Nauka, Moscow, 1969) [in Russian].
M. G. Crandall and P. H. Rabinowitz, “Bifurcation from simple eigenvalues,” J. Funct. Anal. 8, 321–340 (1971).
H. Kielhofer, Bifurcation Theory: An Introduction with Applications to PDEs, Ser. Appl. Math. Sciences (Springer, New York, 2004), Vol. 156.
E. B. Kuznetsov and V. I. Shalashilin, “Parametric approximation,” Zh. Vychisl. Mat. Mat. Fiz. 34, 1757–1769 (1994).
E. B. Kuznetsov and A. Yu. Yakimovich, “Modeling curves by parametric polynomials using the least squares method,” Mat. Model. 16, 48–51 (2004).
E. B. Kuznetsov and A. Yu. Yakimovich, “Optimal parametrization in approximation of curves and surfaces,” Comput. Math. Math. Phys. 45, 732–745 (2005).
E. B. Kuznetsov and A. Yu. Yakimovich, “The best parametrization for parametric interpolation,” J. Comput. Appl. Math. 191, 239–245 (2006).
S. D. Krasnikov and E. B. Kuznetsov, “On the parametrization of numerical solutions to boundary value problems for nonlinear differential equations,” Comput. Math. Math. Phys. 45, 2066–2076 (2005).
S. D. Krasnikov and E. B. Kuznetsov, “Parameterization of numerical solutions of nonlinear boundary value problems,” Mat. Model. 18, 3–16 (2006).
S. D. Krasnikov and E. B. Kuznetsov, “A remark on the parametrization of the numerical solution of boundary value problems,” Differ. Equations 43, 964–973 (2007).
E. B. Kuznetsov, “The best parameterization in curve construction,” Comput. Math. Math. Phys. 44, 1462–472 (2004).
E. B. Kuznetsov, “Optimal parametrization in numerical construction of curve,” J. Franklin Inst. 344, 658–671 (2007).
E. B. Kuznetsov, “On the best parametrization,” Comput. Math. Math. Phys. 48, 2162–2171 (2008).
E. B. Kuznetsov, “Multidimensional parametrization and numerical solution of systems of nonlinear equations,” Comput. Math. Math. Phys. 50, 244–255 (2010).
E. B. Kuznetsov, “Continuation of solutions in multiparameter approximation of curves and surfaces,” Comput. Math. Math. Phys. 52, 1149–1162 (2012).
J. Levin, “A parametric algorithm for drawing pictures of solid objects composed of quadric surfaces,” Commun. ACM 19, 555–563 (1976).
D. Kahaner, C. Moler, and S. Nash, Numerical Methods and Software (Prentice Hall, Englewood Cliffs, N.J., 1988; Mir, Moscow, 1998).
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Original Russian Text © S.D. Krasnikov, E.B. Kuznetsov, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 9, pp. 1571–1585
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Krasnikov, S.D., Kuznetsov, E.B. Numerical continuation of solution at a singular point of high codimension for systems of nonlinear algebraic or transcendental equations. Comput. Math. and Math. Phys. 56, 1551–1564 (2016). https://doi.org/10.1134/S0965542516090104
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DOI: https://doi.org/10.1134/S0965542516090104