Toronto Math Forum

MAT244-2014F => MAT244 Math--Lectures => Topic started by: Bruce Wu on December 05, 2014, 03:23:08 PM

Title: Integrable/non-integrable systems
Post by: Bruce Wu on December 05, 2014, 03:23:08 PM
What is the difference between integrable and non-integrable systems of first order ODEs?

I remember the professor talking about it in class but I cannot find it in the textbook. I know what it means, but how does this property affect its solutions?
Title: Re: Integrable/non-integrable systems
Post by: Victor Ivrii on December 05, 2014, 06:18:45 PM
2x2-ystem is integrable if there exists non-trivial (not identically constant) function $H(x,y)$ which is preserved along trajectories.

Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cycles—which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).
Title: Re: Integrable/non-integrable systems
Post by: Bruce Wu on December 06, 2014, 01:41:43 PM
But from H(x,y) = c, how do we determine the directions of trajectories?
Title: Re: Integrable/non-integrable systems
Post by: Bruce Wu on December 06, 2014, 02:09:55 PM
Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cycles—which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).

Also, all diagonal linear 2x2 systems are integrable, but those can be nodes, how is that explained?
Title: Re: Integrable/non-integrable systems
Post by: Victor Ivrii on December 06, 2014, 05:49:18 PM
Also, all diagonal linear 2x2 systems are integrable,

Wrong!! You confuse explicit solution with integrability. See definition
Title: Re: Integrable/non-integrable systems
Post by: Bruce Wu on December 06, 2014, 08:52:31 PM
So please tell me if I am understanding it correctly now,

y = c*x^2 for all real numbers c represents a particular node. However, H(x,y) = y/x^2 = c is not preserved at x = 0, so it is not integrable, even though it has an explicit solution.
Title: Re: Integrable/non-integrable systems
Post by: Victor Ivrii on December 06, 2014, 11:50:38 PM
However, $H(x,y) = y/x^2 = c$ is not preserved at $x = 0$, so it is not integrable, even though it has an explicit solution.
It is undefined at $(0,0)$ and cannot be defined as continuous function. But your feeling is correct
Title: Re: Integrable/non-integrable systems
Post by: Bruce Wu on December 07, 2014, 12:13:45 AM
Thanks