On the length of validity of averaging and the uniform approximation of elbow orbits, with an application to delayed passage through resonance
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Consider a system of differential equations of the form\(\dot x = \varepsilon f(x,t,\varepsilon )\), wheref is 2π-periodic int and the averaged system\(y' = \bar f(y)\) is Morse-Smale. We show by geometrical resoning that in general no asymptotic approximation to the initial value problem is valid for larger thant=O(1/ε). This comes about because the stable manifolds vary withε. But in the neighborhood of the stable and unstable manifolds of a hyperbolic periodic solution, every orbit (a so-called elbow orbit) has an approximation which is valid for as long as it remains in the neighborhood, although it does not satisfy the same initial conditions. This result is applied to passage through resonance.
KeywordsDifferential Equation Manifold Periodic Solution Mathematical Method Asymptotic Approximation
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