Timeline for Linearization of a vector field
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7 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Dec 8, 2011 at 17:09 | comment | added | Robert Bryant | @Kofi: That is more or less correct. The Hartman-Grobman theorem says that you can perform this linearization near $p$ as long as the linear part of the vector field has no purely imaginary (in particular, no zero) eigenvalues. You can see why these hypotheses are necessary, since, for example, you can't linearize the vector field $X = x^2\ \partial_x$ at $x=0$, and, similarly, you can't linearize the vector field $X = (y + x^3 + xy^2)\ \partial_x - (x - x^2y - y^3)\ \partial_y$ in the plane at $x=y=0$. | |
| Dec 8, 2011 at 17:05 | comment | added | Robert Bryant | @Kofi: By the way, I should also have said that it's worth taking a look at Sterberg's original paper, too: S. Sternberg, On contractions and a theorem of Poincare, Amer. J. Math. 79 (1957) 809–824. He does prove a linearization theorem with greater regularity than Lipschitz, in fact, in fact, he proves a smooth linearization theorem, but the hypotheses are more restrictive than just that the $a_i$ are all positive. (They have to be, as the example I quoted shows.) | |
| Dec 8, 2011 at 16:24 | comment | added | Matthias Ludewig | If I see this correctly, the sign of the coefficients doesn't make a difference then? | |
| Dec 8, 2011 at 16:23 | vote | accept | Matthias Ludewig | ||
| Dec 8, 2011 at 0:24 | history | edited | Robert Bryant | CC BY-SA 3.0 |
Added/corrected some information
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| Dec 8, 2011 at 0:00 | history | answered | Robert Bryant | CC BY-SA 3.0 |