Algebraic view
The basic simplified dynamic system is defined by
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Equ. 1
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where .
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:
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Equ. 2
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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 doesn’t 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:
Case j<4
For j<4, the eigenvalues are complex, and there is always at least one (real) solution for j.
More precisely we can write
with and
and then
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.
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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
Case j>4
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.
We have
So
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.
Case j=4
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, let’s 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
, with
By recurrence, we have then
, with
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
.