2. The Mappings and Relaxation to the Fixed Points Investigation
We consider in this section the behavior of the relaxation to the fixed points for two mappings, namely the logistic map given by the expression
and for the cubic map given by
where both
and
are control parameters. For our investigations in this paper we consider the ranges
and
. For either
and
yields the dynamics to go to
and is without interest for us.
Figure 1 shows the orbit diagram for the two mappings given by Equations (
1) and (
2).
As is well known in the literature, the logistic map has two fixed points for
namely
where, according to stability analysis,
is asymptotically stable for
while
is asymptotically stable for
for any initial conditions lying
. At
, the system experiences a transcritical bifurcation and fixed point
changes stability with
. For
the system exhibits a first period-doubling bifurcation following in a sequence of period-doubling until reaches chaotic behavior. At the bifurcations, the Lyapunov exponents are null given the eigenvalues at the bifurcation points are 1 or
. The sequence of period-doubling follows a Feingenbaum scaling [
17,
18].
Figure 1.
Bifurcation diagrams for: (a) logistic map and; (b) cubic map (for two different initial conditions). The names of some bifurcations are indicated in the figures.
Figure 1.
Bifurcation diagrams for: (a) logistic map and; (b) cubic map (for two different initial conditions). The names of some bifurcations are indicated in the figures.
For the cubic map, there is a fixed point
and two period one fixed points
and
both are born at
. At
the system experiences a pitchfork bifurcation when
loses stability and there is a birth of the
fixed points. Each one of them evolve independently suffering period-doubling bifurcations until reach the chaos. Again the sequences of period-doubling are described by the Feingenbaum scaling [
17,
18]. The two separate chaotic bands are merged only due to a merging chaotic attractors crisis [
13,
14].
The two regions we are interested in to discuss along this paper correspond to: (i) the region of the transcritical bifurcation in the logistic map and; (ii) the region of the pitchfork bifurcation in the cubic map. Indeed we are seeking to understand and describe how is the relaxation of orbits starting close to the fixed point near both bifurcations. We are then looking to describe the behavior of x approaching to denoting the fixed points at and . Exactly like any other variable, x is a function of two entities, i.e., n which is the number of iterations and for the logistic map and for cubic for both .
Following previous results in the literature [
19,
20], we start with two hypotheses:
For
it implies there is an algebraic decay in
x so that
where
β is a critical exponent and depends on the type of bifurcation.
For the parameter
, we assume the orbit relaxes to the equilibrium exponentially according to
where the relaxation time
τ has the following form
where
z is also a critical exponent.
Before showing some theoretical approaches to describe the critical exponents, let us first check what a numerical simulation provides.
Figure 2 shows a plot of the convergence to the fixed point considering the logistic map for: (a)
and (b)
.
Figure 2.
Convergence to the fixed point for the logistic map considering: (a) where a power law fit furnishes and; (b) with a slope of .
Figure 2.
Convergence to the fixed point for the logistic map considering: (a) where a power law fit furnishes and; (b) with a slope of .
After doing power law fittings for the two plots of
Figure 2 we obtain that
and
.
On the other hand the convergence to the fixed point for the cubic map is shown in
Figure 3 for: (a)
leading to a slope of decay given by
and; (b)
yielding in a slope of
.
Figure 3.
Convergence to the fixed point for the cubic map considering: (a) where a power law fit furnishes and; (b) with a slope of .
Figure 3.
Convergence to the fixed point for the cubic map considering: (a) where a power law fit furnishes and; (b) with a slope of .
Given the numerical results are now known, we can go ahead with the theoretical argumentation on the characterization of the relaxations. Let us start with the logistic map as example and considering the transcritical bifurcation,
i.e.,
. In this case the mapping is written as
Equation (
8) can be rewritten in a more convenient way as
that leads to the following approximation
The approach used in Equation (
10) is only valid in the limit of
very close to the fixed point. In such a limit, the discrete variables can be treated like a continuous variable, making the derivative possible.
Integrating Equation (
10) from both sides leads to
After doing the integration and rearranging the terms properly we end up with
As soon as
n grows attending to the condition
we obtain
After a comparison with Equation (
5) we see that for the logistic map at
the critical exponent
, in well agreement to the numerical results presented in
Figure 2.
The investigation for
is quite similar to the previous case with a minimal detail of subtracting from both sides of Equation (
1) a term
, that leads to
yielding at the end with an expression of the type
When
x is sufficiently close to the fixed point, the second term of Equation (
15) which is a quadratic term, becomes rather small as compared to the first one becoming then negligible. Because of this it indeed can be disregarded. Quoting the definition of
μ we can rewrite Equation (
15) as
which in terms of integral is given as
Doing the integral properly we obtain that
A comparison with Equation (
6) leads us to conclude that the critical exponent
, again is in well agreement with the simulation, as confirmed in
Figure 2.
Let us now continue the investigation but considering this time the cubic map for
. Doing similar procedure as made in Equation (
9) we obtain
After doing the integration and organize the terms properly we obtain that
for the limit of
. Comparing the result obtained from Equation (
20) with the one presented in Equation (
5) we find
which is confirmed by the numerical simulations shown in
Figure 3.
Considering the case of
but still close to 1, we end up with an expression of the type
Using similar arguments as used for the logistic map, the cubic term in Equation (
21) can be disregarded leading to an identical expression as given by Equation (
18). Therefore we conclude that the critical exponent is given by
as indeed confirmed by numerical simulations presented in
Figure 3.
The two mappings present a bifurcation in . At the bifurcation point, both maps exhibit algebraic relaxation to the fixed point but with different critical exponents. For the logistic map while for the cubic map it is given by . On the other hand, after the bifurcation takes place, the relaxation for the fixed point is given by the same law and with the same critical exponent .
Let us now discuss shortly on the behavior of the entropy for the bifurcations observed at
. To define a K-entropy, we follow same general discussion as made in [
21]. The procedure starts from the evolution in time of a single initial condition converging towards an attractor. The region defining the attractor is therefore covered by a set of discrete cells. An initial condition is started along such cells and its trajectory is followed and marked in the phase space saying what cell is visit at stage
n, as for example
. A second initial condition is started very close to the first one and that may lead to a different sequence of visits. The process continue to a very large number of initial conditions such that an ensemble average on the initial conditions can be made. From the average, a relative number of times a specific sequence of
N cells is visited can be defined. Then the entropy
is given as
where
gives the relative number for the ith sequence and with the summation taken over all possible sequences starting with
. From this, the K-entropy is then defined as
Because the convergence to the attractor at is indeed an evolution towards an attracting fixed point, the dynamics is regular. Therefore all sequences starting from the same sufficiently small cell are the same, all orbits follow each other as time passes. This leads to for all the ensemble of N. For a large enough N produces a because there is no change in S. This is only observed because the dynamics is regular and no chaos is present for .