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In some cases it is possible to establish a longer timescale of validity of approximation. If the average over time of the system \(\dot {x} = \varepsilon f(t, x)\) vanishes, the approximation can in the periodic case be extended to \(1/\varepsilon ^2\) . Another important case arises if the averaged equation contains a solution that is exponentially attracting. In such a case we can extend the timescale to the whole time axis.

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Approximations on Timescales Longer Than \(1/ \varepsilon \)

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  • 发表日期：

    2026年

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  • Ferdinand Verhulst

摘要

In some cases it is possible to establish a longer timescale of validity of approximation. If the average over time of the system \(\dot {x} = \varepsilon f(t, x)\) vanishes, the approximation can in the periodic case be extended to \(1/\varepsilon ^2\) . Another important case arises if the averaged equation contains a solution that is exponentially attracting. In such a case we can extend the timescale to the whole time axis.