In this paper, we study the dynamics of some two-dimensional mappings which arise when standard numerical integration schemes are applied to an unforced oscillator with a cubic stiffness nonlinearity, i.e., the Duffing equation. While the continuous time problem is integrable and is solved analytically in terms of Jacobi elliptic functions, the discrete versions of this simple system arising from standard integration schemes exhibit very complicated dynamics due to the presence of homoclinic tangles. We present an alternative scheme for discretizing the nonlinear term which preserves the integrable dynamics of the continuous system and derive analytic expressions for the orbits and invariant curves of the resulting mapping.

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