Author Topic: Integrable/non-integrable systems  (Read 714 times)

Bruce Wu

  • APM346
  • Sr. Member
  • *
  • Posts: 57
  • Karma: 0
    • View Profile
Integrable/non-integrable systems
« 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?

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 1332
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Integrable/non-integrable systems
« Reply #1 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).

Bruce Wu

  • APM346
  • Sr. Member
  • *
  • Posts: 57
  • Karma: 0
    • View Profile
Re: Integrable/non-integrable systems
« Reply #2 on: December 06, 2014, 01:41:43 PM »
But from H(x,y) = c, how do we determine the directions of trajectories?

Bruce Wu

  • APM346
  • Sr. Member
  • *
  • Posts: 57
  • Karma: 0
    • View Profile
Re: Integrable/non-integrable systems
« Reply #3 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?
« Last Edit: December 06, 2014, 02:11:41 PM by Fei Fan Wu »

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 1332
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Integrable/non-integrable systems
« Reply #4 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

Bruce Wu

  • APM346
  • Sr. Member
  • *
  • Posts: 57
  • Karma: 0
    • View Profile
Re: Integrable/non-integrable systems
« Reply #5 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.

Victor Ivrii

  • Administrator
  • Elder Member
  • *****
  • Posts: 1332
  • Karma: 0
    • View Profile
    • Personal website of Victor Ivrii
Re: Integrable/non-integrable systems
« Reply #6 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

Bruce Wu

  • APM346
  • Sr. Member
  • *
  • Posts: 57
  • Karma: 0
    • View Profile
Re: Integrable/non-integrable systems
« Reply #7 on: December 07, 2014, 12:13:45 AM »
Thanks