MAT244-2014F > MAT244 Math--Lectures

Integrable/non-integrable systems

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Bruce Wu:
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:
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 cyclesâ€”which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).

Bruce Wu:
But from H(x,y) = c, how do we determine the directions of trajectories?

Bruce Wu:

--- Quote from: Victor Ivrii on December 05, 2014, 06:18:45 PM ---Then trajectories are level lines of $H$ (or their parts). This precludes nodes and spiral points (and limi cyclesâ€”which we have not studied) and allows only saddles and centers (provided at stationary points Hessian of $H$ is non-degenerate).

--- End quote ---

Also, all diagonal linear 2x2 systems are integrable, but those can be nodes, how is that explained?

Victor Ivrii:
Also, all diagonal linear 2x2 systems are integrable,

Wrong!! You confuse explicit solution with integrability. See definition