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P. Hartman (1965)
Ordinary Differential EquationsJournal of the American Statistical Association, 60
J. Cartwright, O. Piro (1992)
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Differential Equations, Vol. 36, No. 3, 2000, pp. 403-408. Translated from Differentsial'nye Uravneniya, Vol. 36, No. 3, 2000, pp. 359-364. Original Russian Text Copyright (~ 2000 by Fishman. ORDINARY DIFFERENTIAL EQUATIONS Properties of Differential and Approximate Finite-Difference Equations L. Z. Fishman Research Institute for Applied Mathematics and Cybernetics, Nizhnii Novgorod State University, Russia Received July 16, 1999 We consider the system ~= f(x, a), (1) where x is an n-vector, f(x, a) is a sufficiently smooth vector function of (x, a), f(0, 0) = 0, and a is a parameter. Let system (1) be approximated by the system of finite-difference equations 8 S xk+~=xk +h~-~bif(Y~,(x), Y~=xk +h~-~ciff(Y~,a) (2) i=l j=l constructed by the s-stage Runge-Kutta method of order p, bl + b2 +--. + bs = 1, and the system of finite-difference equations xk = Xk-1 + h ~ cJ (xk_i,a) (3) i=l constructed by the explicit m-step Adams method of order m, C 1 "q- C 2 -~- "'" -~- C m : 1, k > m. The stability of structurally stable equilibria of system (1) is preserved if the system is replaced by systems (2) and (3). This was proved in [1-3] for sufficiently small h. For
Differential Equations – Springer Journals
Published: Nov 15, 2007
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