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The H-G Theorem say that if the singularity is hyperbolic (all the eigenvalues of the "linearized" system have a not null real part), then in an environment of the singularity the phase portrait is topologically conjugated to that of the linearized system.

In all the books I have read, one of the hypotheses assumed in the proof of this theorem is that the singular point is at the origin. However, intuition tells me that the Theorem also works with singularities that are not at the origin. Is this so?

Daniele Tampieri
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Vandermonde
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    Yes, it holds also in other points. If your system is expressed in the variables $(x_1,x_2,\ldots,x_n)$ and the stationary point is $(\bar x_1,\bar x_2,\ldots,\bar x_n)$, then you just need to rewrite the system in the "local" variables $(y_1,y_2,\ldots,y_n)=(x_1-\bar x_1,x_2-\bar x_2,\ldots,x_n-\bar x_n)$. – Gerhard S. Jan 28 '19 at 19:04

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Yes, certainly, since you can just apply the simple change of variables $y=x-x^*$ to move the singular point from $x=x^*$ to $y=0$.

Hans Lundmark
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