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Suppose $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse and has finitely many critical points. Is there an algorithm for determining whether there exists a saddle-saddle connection (an orbit of grad $f$ that connects two saddle points)?

Edit: As mentioned in the comments, the above problem is not decidable algorithmically unless certain conditions are imposed on $f$.

Consider the following example from

Chicone, Carmen C. "Quadratic gradients on the plane are generically Morse-Smale." Journal of Differential Equations 33.2 (1979): 159-166.

Let $f = x^3 + y^3 - 2xy^2 + x^2 - y^2$, then the vector field grad $f$ is shown below.

gradient field for f

The points (0, 0), (-2/3, 0) and (-2, -2) are hyperbolic saddles and (-4/5, -2/5) is a hyperbolic sink. The saddles (0, 0) and (-2/3, 0) are connected by a separatrix along the $x$-axis.

In this example, the algorithm would take $f$ as input and output true.


In the case where $f$ is a cubic polynomial in two variables, the reference given above states

If grad $f$ has a saddle connection then the connecting orbit lies on a straight line.

Certainly, this makes it easy to check whether a saddle connection exists.

My question is: are there any other references that show a specific function $f \colon \mathbb{R}^n \to \mathbb{R}$ is Morse-Smale; that is, the saddle connections can easily be broken (so the transversality condition holds)?

Specifically, I am looking at rational functions $f$ whose numerator is a polynomial and whose denominator is nowhere zero.

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Disclaimer: this is not my field of research. There is a vast literature on numerical algorithms trying to detect heteroclinic connections. Some algorithms not only compute a candidate connection, but also give a proof that a connection exists within a certain error bound. This might be interesting to pursue. It is however a hard problem, because the connections between saddles (in two dimensions) are not stable (small perturbations of the function, or the metric, destroy them). I doubt that you can find general analytic algorithms. –  Thomas Rot Feb 19 '13 at 14:41
    
Barring bringing in additional hypothesis, your question is basically equivalent to knowing some exact solutions to the DE. Approximations to such can be done algorithmically, but in general this is not an algorithmically-decidable problem. –  Ryan Budney Feb 19 '13 at 19:49
    
Fair enough. I will modify the question slightly. My main concern is finding papers with results similar to Chicone's. –  James Rohal Feb 21 '13 at 3:05
    
When you say "saddle connections can easily be broken" what do you mean? Now it sounds like you're not interested in detecting such connections, but you want to perturb the DE to remove them. Those are two very different questions. –  Ryan Budney Feb 21 '13 at 3:29
    
Ideally I would like to detect the connection, then show that I am able to perturb it in such a way that it disappears. –  James Rohal Feb 21 '13 at 5:56

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