The basic simplified dynamic system is defined by
Let be the current point in , and the matrix of the system. So we have and, more generally,
So the system is completely defined by M.
The eigenvalues of M are:
We see immediately that the value j=4 is special. We will see below what it means.
For j # 4 we can define a matrix A so that
(if j =4, A-1 doesnít exist).
For example, from the canonical form we find
In order to have simpler formulas, we can multiply by 2j, to produce a matrix A:
So if we define we can now write
that is to say we have, finally,
But L is a diagonal matrix, so we have simply
In particular, we have a cyclic behavior if and only if (or, more generally if ). This just means that we have the system of two equations:
For j<4, the eigenvalues are complex, and there is always at least one (real) solution for j.
More precisely we can write
and cycles are given by any q so that
So for each t, the solutions for j are given by
Table 1 gives some nontrivial values of j
for which the system is cyclic.
For any other value, the system is just quasi-cyclic (see Figure 4).
We can be a little bit more precise. Below, is the 2-norm (the Euclidean one for a vector).
We have here
For example, for v0=0 and y0=1, we have
Figure 1. 3-cycle.
Figure 2. Non convex 5-cycle.
Figure 3. Convex 5-cycle.
Figure 4. Quasi-cycle.
If j>4, then e1 and e2 are real numbers (and ), so we have either
So, and this is the point, there is no cyclic behavior
for j>4. And, in fact, the distance from
the point to the center
(0,0) is strictly increasing with t.
But we can also write
So, finally, is increasing "like".
This result can be used to prevent the "explosion" of the system by defining "constriction" coefficients.
We have here
In this particular case, the eigenvalues are both equal to -1, and there is just one family of eigenvectors, generated by . So we have .
So, if P0 is an eigenvector, proportional to V (that is to say if ), we just have two "symmetrical" points, for
In the case where P0 is not an eigenvector, we compute directly how is decreasing and/or increasing. Let us define .
It is easy to see (by recurrence) we have has the following form:
where at,bt,ct are integer numbers so that for .
Now, letís suppose for a particular t we have . What about ?
We easily compute .
This quantity is positive if and only if vt is not between (or equal to) the roots
Now, if we compute we have, and the roots are. As , it means that is also positive.
So as soon as begins to increase, it does so infinitely.
But it can be decreasing, at the beginning. How many times ?
Suppose we have D0 < 0.
It means v0 is between -2y0 and -12y0. For instance in the case y0>0, we can write
By recurrence, we have then
Finally, we can write
as long as
that is to say (for t is an integer) as long as
After that, increases.
We can do exactly the same analysis for y0<0. In this case e<0 too, so the formula is the same.
In fact, we can even be more precise. If we define
then we have
That is to say is decreasing/increasing almost linearly when t is big enough. In particular, even if it begins to decreases, after that it tends to increase almost like .