jesus
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Reference needed: $C^r$ convergence of Euler's method
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I don't know if the following references (from lecture notes on the internet) clarify enough the subject, but your question seems to be related to what is called stability of a numerical method. http:...

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Stability of the Solar System
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in a conference in Paris, Jacques Féjoz said (and i quote from memory) that the big planets seem to be stable, while the small ones chaotic. if i remember well, it was based on numerical evidence, ...

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extension of the projectivized gradient of a harmonic function
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in fact, as can show the example $\sin x \sin y$ on the torus $\mathbb{T} ^2$ there can be a singularity of codimension $2$ where the projectivized gradient cannot be extended.

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