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March 25th, 2010, 02:34 AM   #1
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Linearizing ODEs

Hi all, I hope this finds everyone well.

I'm having a bit of an issue with what seems to be a rather simple mathematical step. Yet, as hard as I try I'm still not coming out with the correct answer. Ok... as taken from the book Nonlinear Systems by P. G. Drazin, page 19, on Hopf Bifurcation:

I have two ODEs:
, and . First the author investigates their equilibirum points i.e., to find that x = y = 0. This is simple enough to understand. Next the author says:

Investigating the stability of this solution, we linearize equations (1) (the two equations above) for small x and y to find:
, and

So I've tried linearizing the two original ODEs yet I don't seem to get what the author does. If anyone can, could you please illustrate the steps. It's probably something so simple, yet it's driving me nuts.

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March 25th, 2010, 01:38 PM   #2
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Re: Linearizing ODEs

All the author is saying is, that for small x and y, x^2 and y^2 can be neglected, as long as a ? 0.
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March 25th, 2010, 04:19 PM   #3
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Re: Linearizing ODEs

Fantastic, thanks

After having made the post, I took some time out then came back and it suddenly clicked! Thanks for confirming
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