On Some Properties of Focal Points

Abstract

We consider some dynamical properties of two-dimensional maps having an inverse with vanishing denominator. We put in evidence a link between a fixed point of a map with fractional inverse and a focal point of this inverse.

1. Introduction

A specific class of maps, defined by 𝑥^′=𝐹(𝑥,𝑦), 𝑦^′=𝐺(𝑥,𝑦), with at least one of the components 𝐹 or 𝐺 defined by a fractional rational function, has been studied in the work in [1–10], and has very interesting properties. Some peculiar dynamical properties have been evidenced and observed in iterated maps (or in one of the inverses) having a vanishing denominator or having the form 0/0 in a point of ℝ². This characteristic has revealed new types of singularities in the phase plane, such as focal points and prefocal curves. The presence of these sets may cause new kinds of bifurcations generated by contacts between them and other singularities, which give rise to new dynamic phenomena and new structures of basins and invariant sets.

In this paper, which is a continuation of a former study [10], we present results on such singularities. We can distinguish typical scenarios between the map and its inverse. We recall definitions and main properties as given in some papers by Bischi et al. (see, e.g., [2–8]). A prefocal curve of a map is a set of points, whose image by one inverse is reduced into a single point called focal point.

We put in evidence necessary conditions and/or sufficient for a focal point to be a fixed point of the inverse map.

This paper is organized as follows. In Section 2 we present some basic definitions and facts about the key role of focal points and prefocal curves. In Section 3, we give our main results about the link between the property of the basin of attraction of a map 𝑇 being connected (resp., nonconnected) and focal points of 𝑇⁻¹ being inside (resp., outside) of this attraction basin, and the link existing between fixed point of this map and the focal point of its inverse. Section 4 is devoted to applications to illustrate our results, two examples are considered with one focal point and two focal points.

2. Definitions and Geometric Localization of Focal Point, Prefocal Curve

In this paper, we use the notation introduced in [6–8]. Definitions and generic properties given here concern fractional maps (𝑥,𝑦)→(𝑥^′,𝑦^′)=𝑇(𝑥,𝑦) of the form: 𝑥𝑇∶^′𝑦=𝐹(𝑥,𝑦),_′=𝑁(𝑥,𝑦),𝐷(𝑥,𝑦) (2.1) where it is assumed that the functions 𝐹, 𝑁, and 𝐷 are polynomial functions defined from ℝ² to ℝ². The set of nondefinition of 𝑇 is given by 𝛿_s={(𝑥,𝑦)∈ℝ²0𝑥02502𝐷(𝑥,𝑦)=0}. (2.2) In the following, we will suppose that 𝛿_𝑠 is a smooth curve in the plane. The initial conditions belong to 𝐸, given by

𝐸=ℝ²−^∞_𝑘=0𝑇^−𝑘(𝛿_𝑠). (2.3) In order to define the concepts of focal point and prefocal curve, we consider a smooth arc 𝛾 transverse to 𝛿_𝑠 and we study the shape of its image under 𝑇, that is, 𝑇(𝛾).

Definition 2.1 (see [6]). Consider the map (2.1). A point 𝑄=(𝑥₀,𝑦₀) is a focal point if at least one component of 𝑇 takes the form 0/0 in 𝑄 and there exist smooth simple arcs 𝛾(𝜏), with 𝛾(0)=𝑄, such that lim_𝜏→0𝑇(𝛾(𝜏)) is finite. The set of all these finite values, obtained in different arcs 𝛾(𝜏), is called a prefocal curve 𝛿_𝑄.

We can compute prefocal curve and the focal point analytically by the following method (see [6]), provided that the inverse map is known explicitly. We search the set 𝐽^′₀ for which det(𝐷𝑇⁻¹) vanishes, and we calculate the images, under 𝑇⁻¹, points of 𝐽^′₀. If 𝐽^′₀ contains a curve 𝛿 such that 𝑇⁻¹(𝛿) is reduced to a point 𝑄, thus 𝛿 is a prefocal curve for the map 𝑇 and 𝑄 is then the associated focal point.

The presence of a vanishing denominator induces important effects on the geometrical and dynamical properties of the map. In this case, we need to locate geometrically the focal point in the phase plane. This concept is stated and proved in the following proposition [10].

Proposition 2.2. Let 𝑇(𝑥,𝑦)=(𝐹(𝑥,𝑦),𝑁(𝑥,𝑦)/𝐷(𝑥,𝑦)) be a fractional rational map with a unique polynomial inverse 𝑇⁻¹. If 𝑇⁻¹(𝛿_𝑠) crosses transversely 𝛿_𝑠 in a point 𝑄=(𝑥₀,𝑦₀), where 𝛿_𝑠 is the set of nondefinition of 𝑇, then 𝑄 is a focal point of 𝑇.

Corollary 2.3. Let 𝑄 be a focal point of 𝑇, if 𝛾(𝜏)=𝑇⁻¹(𝛿_𝑠) and 𝛾(0)=𝑄, then lim_𝜏→0𝑇(𝛾(𝜏)) belongs to 𝛿_𝑠∩𝛿_𝑄.

Some of such behaviors can also be observed in maps without a vanishing denominator, but their inverses have a vanishing denominator.

Corollary 2.4. Let 𝑇 be a polynomial map and 𝑇⁻¹ its unique fractional inverse. If 𝑇(𝛿_𝑠) crosses transversely 𝛿_𝑠 in a point 𝑄=(𝑥₀,𝑦₀), where 𝛿_𝑠 is the set of nondefinition of 𝑇⁻¹, then 𝑄 is a focal point of 𝑇⁻¹.

3. Properties of Focal Points and Prefocal Curves

We now describe our main results concerning several focal points in more detail, we confirm and extend some others for the simplest case with one focal point as stated in [10], and similar properties continue to hold.

The following proposition shows the link between the structure of the basin of attraction of an attractor of 𝑇 and the fact that the focal points of 𝑇⁻¹ are inside or outside the basin.

Proposition 3.1. Let 𝑇 be a smooth two-dimensional map, with a unique fractional and rational inverse 𝑇⁻¹,𝛿_𝑠 being its set of nondefinition, and 𝑄_𝑖, 𝑖=1,2,…,𝑘, its focal points. Let 𝐷 be the basin of attraction of an attracting set 𝐴 of 𝑇 of period-1. Suppose that for any 𝑖=1,2,…,𝑘, 𝛿_𝑠,𝑖∩𝐷≠𝜙, such that 𝛿_𝑠,𝑖 is the branch of 𝛿_𝑠=∪^𝑘_𝑖=1(𝛿_𝑠,𝑖) containing the focal point 𝑄_𝑖. Therefore, the basin 𝐷 is connected, if and only if all focal points 𝑄_𝑖 belong to 𝐷.

Proof. Let 𝐷₀ be the immediate basin of 𝐴, 𝐹 the boundary of the basin 𝐷=∪_𝑛≥0𝑇^−𝑛(𝐷₀), and 𝛿_𝑄𝑖 the prefocal curves of 𝑇⁻¹ associated, respectively, with focal points 𝑄_𝑖, 𝑖=1,2,…,𝑘.
First, we recall the following implications:
(1)𝛿_𝑠∩𝐷₀≠𝜙⇒𝑇(𝛿_𝑠)∩𝐷₀≠𝜙,(2)𝑄_𝑖∈𝐷₀⇔𝑇⁻¹{𝑄_𝑖}=𝛿_𝑄𝑖⊂𝑇⁻¹(𝐷₀),(3)𝑄_𝑖∉𝐷₀⇔𝛿_𝑄𝑖∩𝑇⁻¹(𝐷₀)=𝜙,(4)𝑄_𝑖∈𝐹⇒𝑇⁻¹{𝑄_𝑖}=𝛿_𝑄𝑖⊂𝑇⁻¹(𝐹)⊇𝐹, and in this case 𝑇⁻¹𝐹=𝐹 (as 𝑇 restricted to ℝ²−{∪_𝑖𝛿_𝑄𝑖} is invertible).Let us assume that all 𝛿_𝑠,𝑖 cross the immediate basin 𝐷₀ of 𝐴. Otherwise, if a branch 𝛿_𝑠,𝑖 crosses an island 𝐷_𝑛0 without crossing 𝐷₀, we reason on this island as follows.
We denote by 𝐷¹_0,𝑖 and 𝐷²_0,𝑖 the parts of 𝐷₀ located on either side of 𝛿_𝑠,𝑖 with 𝐷¹_0,𝑖∩𝐷²_0,𝑖=𝐷₀∩𝛿_𝑠,𝑖, 𝐷₀=𝐷¹_0,𝑖∪𝐷²_0,𝑖=∪^𝑘_𝑖=1(𝐷¹_0,𝑖∪𝐷²_0,𝑖). Since 𝑇 has a unique inverse and that 𝑇⁻¹(𝛿_𝑠,𝑖) is the set of points at infinity of 𝑇⁻¹ plus 𝛿_𝑄𝑖, then 𝑇⁻¹(𝐷¹_0,𝑖) and 𝑇⁻¹(𝐷²_0,𝑖) are on either side of 𝛿_𝑄𝑖.
( 1) We show that if 𝐷 is connected, then all 𝑄_𝑖 belong to 𝐷.
Suppose the contrary, in other words there exists 𝑄_𝑖0∉𝐷, this implies that 𝑄_𝑖0∉𝐷₀ and therefore, according to the equivalence (3) mentioned above, 𝑇⁻¹{𝑄_𝑖0}=𝛿_𝑄𝑖0 has no intersection with 𝑇⁻¹(𝐷₀)=𝑇⁻¹(𝐷¹_0,𝑖0∪𝐷²_0,𝑖0)=𝑇⁻¹(𝐷¹_0,𝑖0)∪𝑇⁻¹(𝐷²_0,𝑖0). Besides, as 𝑇⁻¹(𝐷¹_0,𝑖0) and 𝑇⁻¹(𝐷²_0,𝑖0) are on either side of 𝛿_𝑄𝑖0, this implies that 𝑇⁻¹(𝐷¹_0,𝑖0)∩𝑇⁻¹(𝐷²_0,𝑖0)=𝜙. Therefore, 𝑇⁻¹(𝐷₀) is not connected and then 𝐷=∪_𝑛≥0𝑇^−𝑛(𝐷₀) is not connected. This contradicts the hypothesis, D is connected, then 𝑄_𝑖0∈𝐷.
( 2) Conversely, let us prove now that if all 𝑄_𝑖 belong to D, then D is connected.
Suppose that all 𝑄_𝑖 belong to 𝐷₀. If a focal point 𝑄_𝑖0 belongs to an island 𝐷_𝑛0, we reason on this island as follows.
Let 𝑄_𝑖∈𝐷₀, from the equivalence (2), we have 𝑇⁻¹{𝑄_𝑖}=𝛿_𝑄𝑖⊂𝑇⁻¹(𝐷₀)=𝑇⁻¹(𝐷¹_0,𝑖)∪𝑇⁻¹(𝐷²_0,𝑖). Since 𝑇⁻¹(𝐷¹_0,𝑖) and 𝑇⁻¹(𝐷²_0,𝑖) are on either side of 𝛿_𝑄𝑖, therefore, in order that 𝑇⁻¹(𝐷₀) contains 𝛿_𝑄𝑖, it is necessary that 𝑇⁻¹(𝐷¹_0,𝑖) and 𝑇⁻¹(𝐷²_0,𝑖) merge, this means that 𝑇⁻¹(𝐷¹_0,𝑖)∩𝑇⁻¹(𝐷²_0,𝑖)=𝛿_𝑄𝑖. This is true for all prefocal curves 𝛿_𝑄𝑖, then 𝑇⁻¹(𝐷₀) is connected. Since, by the definition of the attraction basin, 𝑇⁻¹(𝐷₀) contains 𝐷₀, and taking into account that 𝐷₀ is the biggest connected component containing the attractor, then 𝐷₀ contains 𝑇⁻¹(𝐷₀). This means that 𝑇⁻¹(𝐷₀)=𝐷₀. Therefore, 𝑇^−𝑛(𝐷₀)=𝐷₀ and then 𝐷=∪_𝑛≥0𝑇^−𝑛(𝐷₀)=𝐷₀ is connected.

Remark 3.2. The proposition does not apply for the case of the periodic attractors of period 𝑘>1. Intuitively, some analogies with Proposition 3.1, as considering 𝑇^𝑘, would be possible, but more work is needed to precise this fact.

The proposition which follows, localizes a focal point of 𝑇⁻¹, when it is a fixed point of 𝑇.

Proposition 3.3. Let 𝑇 be a smooth planar map, let 𝑇⁻¹ be, its unique and fractional inverse and let 𝑄 be a focal point of 𝑇⁻¹. Then, 𝑄 is a fixed point of 𝑇 if and only if 𝑄∈𝛿_𝑄.

Proof. 𝑄 fixed point of 𝑇⇔𝑇(𝑄)=𝑄⇔𝑇(𝑄)∈{𝑄}⇔𝑄∈𝑇⁻¹{𝑄}=𝛿_𝑄.

The following proposition made the link between prefocal curves of 𝑇⁻¹ and of 𝑇⁻² associated with the point focal 𝑄 and the fact that this last is a fixed point of 𝑇.

Proposition 3.4. Let 𝑇 be a smooth planar map and 𝑇⁻¹ its unique and fractional inverse. Let 𝛿_𝑄 and Δ_𝑄 be the prefoal curves of 𝑇⁻¹ and of 𝑇⁻² associated with the focal point 𝑄. Then we can claim that: 𝑄fixedpointof𝑇⇔𝛿_𝑄∩Δ_𝑄≠𝜙. (3.1)

Proof. ( 1) Let us show that 𝛿_𝑄∩Δ_𝑄≠𝜙 is a necessary condition for 𝑄 to be a fixed point of 𝑇. Let (𝑥,𝑦)∈𝛿_𝑄, then the Jacobian 𝐽𝑇_(𝑥,𝑦)=0, thus 𝐽𝑇²_(𝑥,𝑦)=𝐽𝑇_{𝑇(𝑥,𝑦)}⋅𝐽𝑇_(𝑥,𝑦)=𝐽𝑇_𝑄⋅𝐽𝑇_(𝑥,𝑦)=0. Besides, as 𝑄 is a fixed point of 𝑇, for all (𝑥,𝑦)∈𝛿_𝑄 we have
𝑇²(𝑥,𝑦)=𝑇(𝑇(𝑥,𝑦))=𝑇(𝑄)=𝑄. (3.2) Thus, one showed that 𝛿_𝑄⊂Δ_𝑄.
( 2) Now let us show that 𝛿_𝑄∩Δ_𝑄≠𝜙 is a sufficient condition for 𝑄 to be a fixed point of 𝑇. Let (𝑥,𝑦)∈𝛿_𝑄∩Δ_𝑄 then we obtain
𝑇(𝑥,𝑦)=𝑄=𝑇²(𝑥,𝑦)=𝑇(𝑇(𝑥,𝑦))=𝑇(𝑄). (3.3) Consequently, 𝑄 is a fixed point of 𝑇.

We can give another necessary condition so that a focal point of 𝑇⁻¹ is a fixed point of 𝑇, given by this proposition.

Proposition 3.5. Let 𝑇 be a smooth planar map and 𝑇⁻¹ its unique and fractional inverse, and let 𝑄 be a focal point of 𝑇⁻¹. If 𝑄 is a fixed point of 𝑇, then an eigenvalue of the Jacobian matrix of 𝑇 in 𝑄 is null.

Proof. Let 𝑄 be a focal point of 𝑇⁻¹, this implies that 𝐽𝑇=0 on 𝛿_𝑄. As 𝑄 is a fixed point of 𝑇, consequently, 𝑄∈𝛿_𝑄 and then 𝐽𝑇_𝑄=0. This last (𝐽𝑇_𝑄=0) is necessary and sufficient to have a null eigenvalue, being the determinant of a matrix equal to the product of their eigenvalues.

4. Applications

Example 4.1. Case of a polynomial mapping, having a unique fractional inverse with a unique focal point.
We consider this map
𝐱𝐓(𝐱,𝐲)=^′=(𝟏−𝐚)(𝐛⋅𝐱−𝐱^𝟐)+𝐱⋅𝐲−𝐛⋅𝐲𝐚𝐲+𝑐,_′=𝑥, (4.1) where 𝑎, 𝑏, and 𝑐 are real parameters. Its unique inverse is given by 𝐓^−𝟏𝐱(𝐱,𝐲)=^′𝐲=𝑦,_′=(𝟏−𝐚)𝑦+𝑎𝐱−𝐜,𝐲−𝐛 (4.2) The nondefinition set 𝛿_𝑠 of 𝑇⁻¹ is given by 𝑦=𝑏, (4.3) However, 𝑇⁻¹ has a prefocal curve 𝛿_𝑄 of equation 𝑥=𝑏 (4.4) associated with the focal point 𝑄=(𝑐,𝑏).
First let us fix 𝑎=0.5, 𝑐=1.5 and we vary the parameter 𝑏. We have the following situations.
( 1) For 𝑏=1.490, the map 𝑇 has a saddle fixed point 𝐶 of coordinates (1.469,1.469), whose invariant stable manifold delimits the basin of attraction of a stable focus 𝐹 of coordinates (1.021,1.021). We can see in Figure 1 that the prefocal curve 𝛿_𝑄 and the focal point 𝑄=(1.500,1.490) are outside the basin that this last is not connected and is constituted of an immediate basin containing 𝐹 and of islands with a shape of crescents.
( 2) For the value 𝑏=1.500, the focal point 𝑄=(1.500,1.500) is on the prefocal curve 𝛿_𝑄. From Proposition 3.3, 𝑄 is then a fixed point of 𝑇. It is merged with the saddle fixed point 𝐶=(1.500,1.500), whose invariant stable manifold delimits the basin of attraction of the stable focus 𝐹. in Figure 2, we can see that there exists a contact between the boundaries of the immediate basin and islands, so the closure of the total basin is connected. We can see also that the prefocal curve delimits a part of the basin of attraction, which implies that 𝑄 is on the boundary of the basin.
( 3) For the value 𝑏=1.501, the iterates of focal point 𝑄 converge toward the attractor, thus the focal point 𝑄 and prefocal curve 𝛿_𝑄 are inside the basin. We can see, in Figure 3, that the attractor is now an invariant closed curve, resulting from a Neimark-Hopf bifurcation for the value 𝑏=1.500. We see that islands aggregate to the immediate basin. Therefore, the basin is connected. This value of the parameter 𝑏 is at the same time, a basin-bifurcation value “connected ⟨--⟩ nonconnected” (as established in Proposition 3.1) and Neimark-Hopf bifurcation value.

Figure 1 

𝑏=1.490 basin of attraction non connected.

Figure 2 

𝑏=1.500 basin-bifurcation value.

Figure 3 

𝑏=1.501 basin of attraction connected.

Example 4.2. Case of polynomial mapping, having a unique fractional inverse with two focal points is considered.
We consider the following example:
𝐱𝐓(𝐱,𝐲)=^′𝐲=𝑦,_′=𝑥⋅𝐲^𝟐−𝑏⋅𝐲^𝟐−𝑎⋅𝑥+𝑎⋅𝑏⋅𝑦, (4.5) where a and b are real parameters. Its unique inverse is given by 𝑇⁻¹𝑥(𝑥,𝑦)=^′=𝑦+𝑏⋅𝑥²−𝑎⋅𝑏⋅𝑥−𝑎+𝑥²,𝑦^′=𝑥. (4.6) The set of nondefinition 𝛿_𝑠 of 𝑇⁻¹ is constituted of two branches 𝛿_𝑠,1={𝑥=√𝑎} and 𝛿_𝑠,2={𝑥=−√𝑎}: 𝛿_𝑠=𝛿_𝑠,1∪𝛿_𝑠,2, (4.7)𝑇⁻¹ has two prefocal curves: 𝛿_𝑄1∶𝑦=√𝛿𝑎,_𝑄2∶𝑦=−√𝑎, (4.8) associated, respectively, with the two focal points: 𝑄₁=(√𝑎,𝑎⋅𝑏(√𝑄𝑎−1)),₂=(−√𝑎,−𝑎⋅𝑏(√𝑎+1)), (4.9)
( I) Let us check that 𝛿_𝑄𝑖∩Δ_𝑄𝑖=𝜙, 𝑖=1,2, if and only if 𝑄_𝑖 is not a fixed point of 𝑇 (Proposition 3.4). We have 𝐽𝑇²_(𝑥,𝑦)=𝐽𝑇_{𝑇(𝑥,𝑦)}⋅𝐽𝑇_(𝑥,𝑦)=((𝑥𝑦²−𝑏𝑦²−𝑎𝑥+𝑎𝑏𝑦)²−𝑎)⋅(𝑦²−𝑎)=0 this implies that
𝑦=±√𝑎,𝑥=𝑏𝑦²−𝑎𝑏𝑦±√𝑎𝑦²−𝑎. (4.10) The images of these curves, by 𝑇², give us 𝑇²(𝑥=𝑏𝑦²−𝑎𝑏𝑦+√𝑎𝑦²√−𝑎,𝑦)=𝑇(𝑦,𝑎)=𝑄₁,𝑇²(𝑥,𝑦=√𝑎)=𝑇(𝑄₁). (4.11) We deduce that 𝑦=√𝑎 is a prefocal curve of 𝑇⁻² associated with the focal point 𝑇(𝑄₁), and hence if 𝑄₁ is not fixed point of 𝑇, Δ_𝑄1 is equal to Δ_𝑄1={(𝑥=𝑏𝑦²−𝑎𝑏𝑦+√𝑎𝑦²−𝑎,𝑦)} (4.12) with 𝛿_𝑄1∩Δ_𝑄1=𝜙. When 𝑄₁ is a fixed point of 𝑇, Δ_𝑄1 is equal to Δ_𝑄1={(𝑥=𝑏𝑦²−𝑎𝑏𝑦+√𝑎𝑦²√−𝑎,𝑦)}∪{(𝑥,𝑦=𝑎)}. (4.13) Thus 𝛿_𝑄1⊂Δ_𝑄1.
The result is quite similar for the focal point 𝑄₂.
(II) Study of the basin bifurcation “connected ⟨--⟩ nonconnected”
Fix 𝑏=−2 and we vary the parameter 𝑎.(1)For the value 𝑎=0.264, we see in Figure 4 that the basin of attraction is connected, and the two prefocal curves and the two focal points are inside the basin.(2)For the value 𝑎=0.265, we see in Figure 5 that the basin of attraction is not more connected. 𝛿_𝑄2 and 𝑄₂ are outside of the basin (Proposition 3.1).(3)For the value 𝑎=0.583, we can see in Figure 6 that the basin is not connected and the two prefocal curves do not belong to the basin.
(III) We can obtain basin bifurcations curves, when the focal points 𝑄₁ and 𝑄₂ are fixed points of 𝑇 on the boundary of the basin.
𝑇 has three fixed points 𝐴, 𝐵, 𝐶 of coordinates 𝐴=(0,0),𝐵=(𝑏+𝑏²−4𝑎𝑏+4𝑎+42,𝑏+𝑏²−4𝑎𝑏+4𝑎+42),𝐶=(𝑏−𝑏²−4𝑎𝑏+4𝑎+42,𝑏−𝑏²−4𝑎𝑏+4𝑎+42). (4.14) According to Proposition 3.3, 𝑄₁=(√𝑎,𝑎⋅𝑏(√𝑎−1)) is a fixed point of 𝑇, if and only if 𝑄₁∈𝛿_𝑄1, then 𝑎⋅𝑏(√𝑎−1)=√𝑎, which gives the curve 𝑏₁(𝑎)=1/(𝑎−√𝑎) for 𝑎≠0 and 𝑎≠1.
For 𝑎=0, 𝑄₁=(0,0) is an attractive fixed point (with eigenvalues 𝜆₁=𝜆₂=0), hence 𝑄₁ is not on the boundary of a basin. For a = 1, 𝑄₁=(1,0) is not a fixed point of 𝑇.
By identification of 𝑄₁ with the fixed points 𝐵, 𝐶 for 𝑏∈𝑏₁(𝑎), we obtain
𝑄₁=(√𝑎,√𝑄𝑎)=𝐵for𝑎∈]0,1[∪]1.6826,+∞[,₁=(√𝑎,√𝑎)=𝐶for𝑎∈]1,1.6826[. (4.15) Similarly, 𝑄₂=(−√𝑎,−𝑎⋅𝑏(√𝑎+1)) is a fixed point of 𝑇 if and only if 𝑄₂∈𝛿_𝑄2, then −𝑎⋅𝑏(√𝑎+1)=−√𝑎, which gives 𝑏₂(𝑎)=1/(𝑎+√𝑎) for 𝑎≠0. For a =0, 𝑄₂=(0,0) is an attractive fixed point.
By identification of 𝑄₂ with fixed points 𝐵, 𝐶 for 𝑏∈𝑏₂(𝑎), we obtain
𝑄₂=(√𝑎,√𝑎)=𝐶for𝑎∈]0,+∞[. (4.16)

Figure 4 

𝑎=0.264 basin of attraction connected.

Figure 5 

𝑎=0.265 basin of attraction non connected.

Figure 6 

𝑎=0.583 basin of attraction non connected.

In the parameter plane (𝑎,𝑏), the curves 𝑏₁(𝑎) and 𝑏₂(𝑎) can be bifurcation curves “connected ⟨--⟩ nonconnected” (when an attractor exists and that the fixed points 𝐵, 𝐶 are on the boundary of the basin of this attractor), if between these two curves, a focal point cannot reach the boundary of the attraction basin without being a fixed point. Otherwise, it will be basin bifurcations curves, in the sense that the number of connected components of the basin changes while crossing them.

5. Conclusion

Fractional maps have suggested interesting analysis of the dynamic behavior and showed that they have their own characteristics, and have revealed new types of singularities considered as theoretical tools for analyzing the bifurcation phenomena. In this paper, we have presented complementary results that appear not to have been given in the works cited therein. We have proved some results, the first permits to localize geometrically the focal point in the phase plane. The second result makes the link between the property of the basin of attraction of an attractor of a map 𝑇 with a unique inverse being connected (resp., nonconnected) and focal points of 𝑇⁻¹ being inside (resp., outside) of this attraction basin. The third and last result gives the link between fixed point and focal point of 𝑇 and 𝑇⁻¹. It would be interesting to study these same properties for maps with several inverses.

Acknowledgment

This work has been performed under the activity of the National Research Project B01120060034.