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Homoclinic connections and numerical integration. (English) Zbl 0885.65082

Numer. Algorithms 14, No. 1-3, 261-267 (1997); erratum ibid. 17, No. 2-4, 355 (1998).
It is known that the numerical integration of ordinary differential equations \(y'= f(y)\), \(f:\mathbb{R}^n\to \mathbb{R}^n\), \(f(0)=0\), on long time intervals becomes unstable and carries on “numerical chaos”. The author shows that Euler’s finite difference scheme does not preserve homoclinic connections.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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