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I need an example of a two variable linear system that's structurally stable. I know the definition (given below), I'm just not sure how to apply it.

A smooth dynamical system $x'=f(x)$, where $x \in \mathbb{R}^2$, is structurally stable in a region $D_0 \subset \mathbb{R}^2$ if and only if:

  1. It has a finite number of equilibria and limit cycles in $D_0$, and all of them are hyperbolic;
  2. There are no saddle separatrices returning to the same saddle or connecting two different saddles in $D_0$.
Sambo
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    Care to show us your definition ? –  Oct 04 '17 at 12:27
  • A smooth dynamical system $x'= f(x)$, $x ∈ R^2$, is structurally stable in a region $D_0 ⊂ R^2$ if and only if (i) it has a finite number of equilibria and limit cycles in $D_0$, and all of them are hyperbolic; (ii) there are no saddle separatrices returning to the same saddle or connecting two different saddles in $D_0$ – 3141 Oct 04 '17 at 12:33
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    Well, that's a theorem rather than a definition. But anyway, it's not hard to cook up a system which trivially satisfies the conditions, for example $(x_1,x_2)'=(1,0)$ (since it has no equilibria or limit cycles). – Hans Lundmark Oct 04 '17 at 13:24
  • @Evgeny: How? The trajectories of the original system are just horizontal lines. If you perturb the second component, you get curves with go to the right but may also wiggle up and down (but never intersect). Can't you just map each line to a corresponding curve to get the equivalence then? – Hans Lundmark Oct 04 '17 at 14:45

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