## Almost linear ODE: how node becomes a spiral

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Most introductory ODE books contain a discussion of almost linear systems, and there are two cases when the behavior of an almost linear system near an equilbrium point can differ from the behaviour of a linearlized system. One of these two cases is when the linearlized system has a repeated nonzero real eigenvalue; then the equilibrium solution of the linearlized system is a node, while the original almost linear system could be a node or a spiral, and I wish to see an explicit example of when the spiral occurs.

To give you an idea of what I need, let's discuss the other case when the linearized system is a center, but the almost linear system could be a center or a spiral, and the example when the spiral occurs is $x^\prime=y+x(x^2+y^2)$ and $y^\prime=-x+y(x^2+y^2)$.

UPDATE: I now feel much better about the question because it turned out not as silly as I initially feared. Apparently, given a $2\times 2$ almost linear system $X^\prime=F(X)$ if the solution $X=0$ of the linearized system $X^\prime=AX$ is a node, then it is also a node for $X^\prime=F(X)$, as long as $F$ is $C^2$. If $F$ is merely $C^1$ there is a counterexample indicated in the comments. In cartezian coordinates the counterexample is $x^\prime=-x-\frac{2y}{\ln(x^2+y^2)}$ and $y^\prime=-y+\frac{2x}{\ln(x^2+y^2)}$; the right hand side is not $C^2$.

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Maybe I don't understand, but does $x'= -x + y^3$, $y'= -y - x^3$ qualify?   However, the spiral and the node are always conjugated, so I don't see quite clearly what you mean by differ (when there are zero eigenvalues this is clear, since you can get, as in your example, really different behaviour). – rpotrie Nov 8 2010 at 10:03
In a small neighborhood of the equilibrium point $p$, the spiral goes around $p$ infinitely many times, the node does not, so the behaviour is quite different. However, I found an example that does the job. In polar coordinates it looks like $r^\prime=-r$ and $\theta^\prime=1/ln(r)$.   – Igor Belegradek Nov 8 2010 at 12:39
@rpotrie, your example looks like a node in a small neigborhood of the origin. In a larger neigborhood it starts looking like a spiral but my question was about local begavior. Still it is a nice example, thanks! – Igor Belegradek Nov 8 2010 at 18:26
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