4
$\begingroup$

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 $C^2$.

$\endgroup$
3
  • $\begingroup$ 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). $\endgroup$
    – rpotrie
    Nov 8, 2010 at 10:03
  • $\begingroup$ 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 $\theta^\prime=1/ln(r)$. $\endgroup$ Nov 8, 2010 at 12:39
  • $\begingroup$ @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! $\endgroup$ Nov 8, 2010 at 18:26

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.