Properties of differential and approximate finite-difference equations
Copyright policy )Properties of differential and approximate finite-difference equations
It is known [see, e.g., \textit{A. Iserles}, IMA J. Numer. Anal 10, No. 1, 1-30 (1990; Zbl 0686.65054); especially pp. 3-9, and \textit{L. Z. Fishman}, Differ. Uravn. 31, No. 4, 613-621 (1995; Zbl 0853.34036)] that certain (implicit) Runge-Kutta methods and explicit Adams-type methods preserve, for sufficiently small \(h>0\), the stability of structurally stable equilibria of the system \(\dot x=f(x,\alpha)\) (with smooth \(f\) satisfying \(f(0,0) =0)\) but that these equilibria may pass through false (spurious) bifurcations. The present paper analyzes the preservation of the stability and the multiplicity of such equilibria for arbitrary \(h\); in addition, sufficient conditions are given in the case of double and triple equilibria when \(h\) is sufficiently small. The results are then extended to scalar delay differential equations with constant delay.
- Yaroslav-the-Wise Novgorod State University Russian Federation
Bifurcation theory for ordinary differential equations, Runge-Kutta methods, delay differential equations, Stability theory of functional-differential equations, finite-difference equations, Numerical approximation of solutions of functional-differential equations, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, Structural stability and analogous concepts of solutions to ordinary differential equations, Adams-type methods, Numerical investigation of stability of solutions to ordinary differential equations, Numerical methods for functional equations, Stability and convergence of numerical methods for ordinary differential equations, preservation of stability and bifurcation, Bifurcation theory of functional-differential equations
Bifurcation theory for ordinary differential equations, Runge-Kutta methods, delay differential equations, Stability theory of functional-differential equations, finite-difference equations, Numerical approximation of solutions of functional-differential equations, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, Structural stability and analogous concepts of solutions to ordinary differential equations, Adams-type methods, Numerical investigation of stability of solutions to ordinary differential equations, Numerical methods for functional equations, Stability and convergence of numerical methods for ordinary differential equations, preservation of stability and bifurcation, Bifurcation theory of functional-differential equations
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