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This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata.

It seems to be related in certain case to the Painlevé Property first discovered for Painlevé equations and their solutions.

I have seen several definitions, notations, criteria, which do not always seem to match each other or to be easily applicable to all context. I would appreciate some clarification about it.

I am mostly interested in the discrete applications but all explanations about the continuous cases are welcome especially if there are links to the discrete setting.

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Perhaps it would help you to try to explain some of the definitions you have seen in your own words here in your question. –  j.c. May 4 '10 at 21:24
Perhaps the following paper could be of some help: math.arizona.edu/~goriely/Papers/2004-JMP(map-singularity).pdf –  mathphysicist May 10 '10 at 20:32

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The singularity confinement property refers to a property of discrete integrable systems. I am unaware of this property in the context for continuous systems. I can understand why you might have difficulty in getting a definition, since it is rather oddly defined all too often. Since the paper of Goriely and La Fortune have been referenced I will assume that you are interested in this property for maps rather than partial difference equations.

Let me start with an example that hopefully illustrates what is going on, before generalizing: Let us take a simple second order difference equation, which we right as a sequence, satisfying the recurrence relation:

$x_{n+1}x_{n-1} = 1+ x_n$

If we let $x_1 = -1$, then $x_2 = 0$, regardless of what $x_0$ is. Then $x_3 = -1$ and $x_4 = \infty$. Worst of all, you have also lost your initial conditions, namely $x_0$. Suppose, instead of $x_1 = -1$, we take $x_1 = -1+e$, where e is assumed very small. Then

$x(2) = \frac{e}{x_0}$

$x(3) = \frac{e+x_0}{(e-1)x_0}$

$x(4) = \frac{1+x_0}{e-1}$

Notice that in the limit as $e \to 0$, $x_2 \to 0$, $x_3 \to -1$, $x_4 \to -1 -x_0$. Somehow, in the limit around a "bad" point, or "singularity", we recover initial conditions, namely $x_0$. We say that the singularity is confined because, despite a loss of information at $x_2$ and $x_3$ in the limit, we can regain the information again at $x_4$ in the same limit. We say the equation has the singularity confinement property if we can always regain the initial conditions in the limit somewhere along the line. This is the usual, however you may want something more general.

The generalization of the above is a map

$f: (x_n, x_{n-1}) \to (x_{n+1},x_n) = \left(\frac{x_n+1}{x_{n-1}},x_n\right)$,

however, we generally take a map of $R^n$ and then take the corresponding projective space, $P^n$, then we consider a map

$f: P^n \to P^n$.

This function now describes a discrete dynamical system, via $y_{n+1} = f(y_n)$. A singularity in this context is where the function, $f$, fails to be invertible. For example, in the above, we had all initial conditions of the for $(x_0,-1)$ being sent to $(-1,0)$. However, we say the singularity is confined if some power of $f$ is invertible. Note that in the above example, the third power of the map can be continuously extended to the map $(x_0,-1) \to (-1,-1-x_0)$, hence the singularity, $(-1,0)$ is confined. A system has the singularity confinement property if all singularities are confined.

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