My Math Forum Linearizing ODEs

 Calculus Calculus Math Forum

March 25th, 2010, 02:34 AM   #1
Newbie

Joined: Mar 2010

Posts: 2
Thanks: 0

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:
$\frac{dx}{dt}= -y+(a-x^{2}-y^{2})x$, and $\frac{dy}{dt}= x+(a-x^{2}-y^{2})y$. 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:

Quote:
 Investigating the stability of this solution, we linearize equations (1) (the two equations above) for small x and y to find:
$\frac{dx}{dt}= -y + ax$, and $\frac{dy}{dt}= x + ay$

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.

Thanks

 March 25th, 2010, 01:38 PM #2 Global Moderator   Joined: May 2007 Posts: 6,788 Thanks: 708 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.
 March 25th, 2010, 04:19 PM #3 Newbie   Joined: Mar 2010 Posts: 2 Thanks: 0 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

 Tags linearizing, odes

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post zibb3r Calculus 1 September 21st, 2013 01:35 PM Execross02 Applied Math 9 July 1st, 2013 08:15 AM singapore Calculus 2 March 19th, 2012 06:48 PM FreaKariDunk Calculus 12 February 22nd, 2012 08:59 PM liakos Applied Math 0 January 1st, 2011 09:52 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top