Integrable/non-integrable systems

[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 cycles—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]

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?

[Victor Ivrii]

Also, all diagonal linear 2x2 systems are integrable,

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

[Bruce Wu]

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]

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]

Thanks