Discrete Dynamics in Nature and Society

Volume 2015, Article ID 680970, 13 pages

http://dx.doi.org/10.1155/2015/680970

## Stability of Real Parametric Polynomial Discrete Dynamical Systems

^{1}Applied Mathematics, CIMAT, 36240 Guanajuato, GTO, Mexico^{2}Graduate School of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

Received 23 November 2014; Revised 22 January 2015; Accepted 23 January 2015

Academic Editor: Zhan Zhou

Copyright © 2015 Fermin Franco-Medrano and Francisco J. Solis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We extend and improve the existing characterization of the dynamics of general quadratic real polynomial maps with coefficients that depend on a single parameter *λ* and generalize this characterization to cubic real polynomial maps, in a consistent theory that is further generalized to real *m*th degree real polynomial maps. In essence, we give conditions for the stability of the fixed points of any real polynomial map with real fixed points. In order to do this, we have introduced the concept of *canonical polynomial maps* which are topologically conjugate to any polynomial map of the same degree with real fixed points. The stability of the fixed points of canonical polynomial maps has been found to depend solely on a special function termed *Product Position Function* for a given fixed point. The values of this product position determine the stability of the fixed point in question, when it bifurcates and even when chaos arises, as it passes through what we have termed *stability bands*. The exact boundary values of these stability bands are yet to be calculated for regions of type greater than one for polynomials of degree higher than three.

#### 1. Introduction

The theory of discrete dynamical systems with iteration functions given by polynomials is an intensive research subject where a wide variety of discrete models have been proposed to describe and to analyze different mechanisms in various areas of science. For example, in Biology and more specifically in Population Dynamics there are many simple models that are used to study the asymptotic behavior of some species that live in isolated generations; see, for instance, [1–7].

Although the dynamics of parametric polynomial discrete systems are very complex their bifurcation diagrams have proved to be a very useful visual tool. A new method for constructing a rich class of bifurcation diagrams for unimodal maps was presented in [8], where the behavior of quadratic maps was analyzed when the dependence of their coefficients was given by continuous functions of a parameter. Conditions on the coefficients of the quadratic maps were given in order to obtain regular reversal maps. Our first goal is to restate the results for more complex systems (cubic) than the quadratic systems analyzed in [8] and to state the results in the frame of a new formulation that would allow for generalization. Our second goal is to generalize the existing results on real quadratic maps for arbitrary real polynomial maps within a framework that allows us to understand the dynamics for a larger set of discrete systems. It is important to remark that our results are analytical and depend only on the parametric derivative of the system evaluated at equilibrium points. There are diverse results based on other approaches such as the linearized stability due to Lyapunov; see, for instance, [9, 10]. In our opinion, our approach is natural for polynomial iteration functions whereas the linearized stability can be used for more complex discrete systems with iteration functions such as piecewise functions. It is also important to notice that there is diverse numerical software specialized in the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems, such as Auto [11] and MatCont [12].

Before attempting to obtain general results for polynomial discrete systems, we want to motivate them with those for a nontrivial system. To do this, we propose in Section 2 analyzing the stability of a general cubic discrete dynamical system. Then in Section 3, we use a general framework in order to analyze the stability for real polynomial discrete dynamical systems by using some of the ideas introduced in the previous section. In order to illustrate the obtained results several examples are included in Section 4. Finally, conclusions are given in Section 5.

#### 2. Cubic Discrete Dynamical Systems

To motivate our results, we will start with a general cubic discrete system since it is in this case, besides linear and quadratic systems, when explicit calculations can be achieved. Then, for such system, we will define two particular forms, namely, the* Linear Factors Form* and the* Canonical Form*. We will show that these two forms are actually topologically conjugate, which in turn means that the property of chaos is preserved between the maps, which allows us to determine stability properties for any cubic map with real fixed points by analyzing only the Canonical Cubic Map.

Consider the cubic discrete dynamical system given in its General Form by , where the iteration function is given by the following definition.

*Definition 1 (general cubic map). *The* general cubic map (GCM)* is defined by
where
is called the* fixed points polynomial* of . All the coefficients , and are functions of the parameter .

It is evident that any cubic map can be put in this form by adjusting the corresponding values of the coefficients in the fixed points polynomial. By the fundamental theorem of algebra, we know that (2) has three roots, by which the GCM has three fixed points. The roots of are then , , and , where , , , , , , , and . The coefficients of the fixed points polynomial (2) and its roots are related by , , and . is called the discriminant and we have three cases.(i)If then one fixed point is real and the other two are complex conjugates.(ii)If then the three fixed points are real with at least two of them equal.(iii)If then all fixed points are real and distinct.

It is the last two cases (real fixed points) that will interest us most for the time being. Suppose in particular that . Then, we can write where . Using the previous notation we have the following definition.

*Definition 2 (linear factors form of the cubic map). *Let be a general cubic map with three fixed points, , , and . One can write as
where all , , , and are functions of the parameter , , ; one calls the* Linear Factors Form of the cubic map (LFFCM)*.

Now we will apply a linear transformation to (4) so that one fixed point is mapped to zero and the “amplitude” coefficient of the Linear Factors term is unity; this can be done since is cubic and at least one of the fixed points is real, so we can always map this fixed point to zero. The linear transformation can be chosen by each one of the following transformations: by taking , . Without loss of generality, we will use with the plus sign and call it simply , so that we get the following.

*Definition 3 (canonical cubic map). *The* Canonical Cubic Map (CCM)* is defined by
where it has been stressed out that both fixed points and depend upon the parameter .

So if then
and if

The relationship between the roots of the Linear Factors Form of the cubic map and the Canonical Cubic Map (CCM) is given by the following.

Corollary 4. *The fixed points of the Linear Factors Form of the cubic map and the Canonical Cubic Map are related by
*

*We have then reduced the parametric dependence to only two functions of the parameter : and . Notice is a homeomorphism between the domains of both maps; this will help us in Section 3 to prove that the Linear Factors Form and the Canonical Form of polynomial maps are actually topologically conjugate, which in turn means that the stability and chaos properties are preserved between the maps, which allows us to determine stability properties for any cubic map by analyzing only the CCM.*

*2.1. Stability for the Canonical Cubic Map*

*Let us determine the stability of the periodic points of the CCM. This analysis will suffice for any cubic map with real fixed points, by means of topological conjugacy. However, we can only explicitly give this for the fixed points. We already know, by construction, that the fixed points of the CCM are , , and . While the first is constant, the other two fixed points are set to be functions of the parameter . By evaluating in , we get the eigenvalue functions. For we have . So the stability condition for this fixed point is
*

*We can draw some conclusions from this. In order for zero to be a stable (attracting) fixed point one must have the following.*

*Lemma 5. The following are sufficient conditions for the asymptotic stability of the zero fixed point of the Canonical Cubic Map: (i)in magnitude, ;(ii)if , and must have different signs; or(iii)if , and must have the same sign.*

*Notice that the stability condition (10) states that the product of the relative positions from the other two fixed points to the zero fixed point must be within the range , for positive [or for negative ], for the zero fixed point to be asymptotically stable.*

*The case of , , is not included in the discussion here since this would represent repeated fixed points (multiplicity), which will be discussed in Section 2.3 below; likewise, in the remainder of this section we will avoid dealing with multiplicity of the fixed points. Now, for , its eigenvalue function is , so that the stability condition for this fixed point is
*

*This fact gives us the following.*

*Lemma 6. The following are sufficient conditions for the asymptotic stability of the fixed point.If then (i) and must have the same sign;(ii).On the other hand, if , (i);(ii)if , then ; or(iii)if , then ; or(iv)if , then .*

*Again, notice that the stability condition (11) for can be translated as that the product of the relative positions between the other two fixed points and must be within the range for positive [or for negative ]. Also notice that when the bound may be negative even if or positive even if , therefore the usefulness of the distinction. For we have analogous results since it is indistinguishable from in the present formulation.*

*We will later generalize these “stability conditions” to functions of the parameter which are different for each fixed point, but of whose value depends on the stability of not only the fixed points, but also higher period periodic points, through period doubling bifurcations. From the stability conditions for the three fixed points we have proved the following.*

*Corollary 7. A cubic polynomial map with three different real roots can only have a single attracting fixed point.*

*Proof. *Compare the stability conditions for the three fixed points.

*Also we have proved the following theorem.*

*Theorem 8. Then sufficient conditions for the stability of a fixed point of the Canonical Cubic Map are as follows.If , (i)the product of the relative positions between each unstable fixed point and the stable one must be negative, which means one position is positive and the other negative, which leads us to the following;(ii)the fixed point that lies between the other two will be stable, while the outer fixed points will be unstable, as long as the following holds;(iii)the product of the relative positions between each unstable fixed point and the stable one must be greater than −2.And if , (i)the product of the relative positions between each unstable fixed point and the stable one must be positive, which means that both relative positions are positive, which leads us to the following;(ii)either zero or the outer fixed point will be stable, while the other two fixed points will be unstable, as long as the following holds;(iii)the product of the relative positions between each unstable fixed point and the stable one must be less than 2.*

*2.2. Higher Period Periodic Points*

*2.2. Higher Period Periodic Points**Although, as previously stated, in general, we cannot calculate the values of the periodic points of period 2 or higher, we can calculate for which values of the stability conditions above the fixed points undergo period doubling bifurcations. We will see in Section 3 that these stability conditions can actually be generalized to something called the “Product Position Function,” which depends on the parameter and is different for each fixed point. An asymptotic parameterization of the fixed points allowed us to determine the bifurcation values, , of the fixed points of the CCM up to some precision. The values obtained are shown in Table 1. When the stability conditions of each fixed point cross these values, bifurcations take place. An estimation of the bifurcation value for the onset of chaos through a period doubling cascade has been calculated as .*