I would like to prove the following:
"The nature of the equilibrium points (i.e. stability/instability) of a one-dimensional differential equation remains invariant under the effect of the linearization transformation."
Thank you all :)
2026-04-04 04:37:48.1775277468
Nature of Equilibrium Points
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This is a special case of the Hartman-Grobman theorem and depends on the equilibrium being hyperbolic. (Actually, in a single dimension the behavior is probably nicer but I haven't thought about 1-d in quite a while.)
Unfortunately, the proof of Hartman-Grobman in its usual generality could be considered quite technical by some, but in general it's hard to define what it means for "the nature of an equilibrium point to remain invariant" without referring to topological conjugacy, because in dimensions sufficiently greater than 1 the nature of an equilibrium point can be quite complicated.