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We consider a slowly varying ODE \(\dot {x}= \varepsilon f(t, x)\) with terms periodic in time. A theorem is formulated to approximate the solutions by averaging over time with error \(O(\varepsilon )\) on a long timescale of order \(1/\varepsilon \) . The results can be extended to a sum of quasi-periodic vector fields as long as the sum is finite. The examples discuss the Van der Pol-equation and generalisations, the forced Duffing-equation, and problems with damping.

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First Order Periodic Averaging

  • 章节

  • 发表日期：

    2026年

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  • Ferdinand Verhulst

摘要

We consider a slowly varying ODE \(\dot {x}= \varepsilon f(t, x)\) with terms periodic in time. A theorem is formulated to approximate the solutions by averaging over time with error \(O(\varepsilon )\) on a long timescale of order \(1/\varepsilon \) . The results can be extended to a sum of quasi-periodic vector fields as long as the sum is finite. The examples discuss the Van der Pol-equation and generalisations, the forced Duffing-equation, and problems with damping.