# On Some Properties of Focal Points

Authors: M. R. Ferchichi; I. Djellit
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## 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.


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## Body

## 1. Introduction

A specific class of maps, defined byx′=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 Section2 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:
(2.1)T:{x′=F(x,y),y′=N(x,y)D(x,y),
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
(2.2)δs={(x,y)∈ℝ2│D(x,y)=0}.
In the following, we will suppose that δs is a smooth curve in the plane. The initial conditions belong to E, given by(2.3)E=ℝ2-⋃k=0∞T-k(δs).
In order to define the concepts of focal point  and prefocal curve, we consider a smooth arc γ transverse to δs 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 limτ→0T(γ(τ)) is finite. The set of all these finite values, obtained in different arcs γ(τ), is called a prefocal curve δQ.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 J0′ for which det(DT-1) vanishes, and we calculate the images, under T-1, points of J0′. If J0′ 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.
LetT(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(δs) crosses transversely δs in a point Q=(x0,y0), where δs is the set of nondefinition of  T, then Q is a focal point of   T.Corollary 2.3.
LetQ be a focal point of T, if γ(τ)=T-1(δs) and γ(0)=Q, then limτ→0T(γ(τ)) belongs to δs∩δQ.Some of such behaviors can also be observed in maps without a vanishing denominator, but their inverses have a vanishing denominator.Corollary 2.4.
LetT be a polynomial map and T-1 its unique fractional inverse. If T(δs) crosses transversely δs in a point Q=(x0,y0), where δs 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 ofT and the fact that the focal points of  T-1 are inside or outside the basin.Proposition 3.1.
LetT be a smooth two-dimensional map, with a unique fractional and rational inverse T-1,δs being its set of nondefinition, and Qi, 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, δs,i∩D≠ϕ, such that δs,i is the branch of δs=∪i=1k(δs,i) containing the focal point Qi. Therefore, the basin D is connected, if and only if  all focal points Qi belong to D.Proof.
LetD0 be the immediate basin of A, F the boundary of the  basin D=∪n≥0T-n(D0), and δQi the prefocal curves of T-1 associated, respectively, with focal points Qi, i=1,2,…,k.
First, we recall the following implications:
(1)
δs∩D0≠ϕ⇒T(δs)∩D0≠ϕ,(2)
Qi∈D0⇔T-1{Qi}=δQi⊂T-1(D0),(3)
Qi∉D0⇔δQi∩T-1(D0)=ϕ,(4)
Qi∈F⇒T-1{Qi}=δQi⊂T-1(F)⊇F, and in this case T-1F=F (as T restricted to ℝ2-{∪iδQi} is invertible).Let us assume that all δs,i cross the immediate basin D0 of A. Otherwise, if a branch δs,i crosses an island Dn0 without crossing D0, we reason on this island as follows.
We denote byD0,i1 and D0,i2 the parts of D0 located on either side of δs,i with D0,i1∩D0,i2=D0∩δs,i, D0=D0,i1∪D0,i2=∪i=1k(D0,i1∪D0,i2). Since T has a unique inverse and that T-1(δs,i) is the set of points at infinity of T-1 plus δQi, then T-1(D0,i1) and T-1(D0,i2) are on either side of δQi.
( 1) We show that ifD is connected, then all Qi belong to D.
Suppose the contrary, in other words there existsQi0∉D, this implies that Qi0∉D0 and therefore, according to the equivalence (3) mentioned above, T-1{Qi0}=δQi0 has no intersection with T-1(D0)=T-1(D0,i01∪D0,i02)=T-1(D0,i01)∪T-1(D0,i02). Besides, as T-1(D0,i01) and T-1(D0,i02) are on either side of δQi0, this implies that T-1(D0,i01)∩T-1(D0,i02)=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 Qi0∈D.
( 2) Conversely, let us prove now that if allQi belong to D, then D is connected.
Suppose that allQi belong to D0. If a focal point Qi0 belongs to an island Dn0, we reason on this island as follows.
LetQi∈D0, from the equivalence (2), we have T-1{Qi}=δQi⊂T-1(D0)=T-1(D0,i1)∪T-1(D0,i2). Since T-1(D0,i1) and T-1(D0,i2) are on either side of δQi, therefore, in order that T-1(D0) contains δQi, it is necessary that T-1(D0,i1) and T-1(D0,i2) merge, this means that T-1(D0,i1)∩T-1(D0,i2)=δQi. This is true for all prefocal curves δQi, 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 periodk>1. Intuitively, some analogies with Proposition 3.1, as considering Tk, would be possible, but more work is needed to precise this fact.The proposition which follows, localizes a focal point ofT-1, when it is a fixed point of  T.Proposition 3.3.
LetT 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∈δQ.Proof.
Q fixed point of T⇔T(Q)=Q⇔T(Q)∈{Q}⇔Q∈T-1{Q}=δQ.The following proposition made the link between prefocal  curves ofT-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.
LetT be a smooth planar map and T-1 its unique and fractional inverse. Let δQ and ΔQ be the prefoal curves of T-1 and of  T-2 associated with the focal point Q. Then we can claim that:
(3.1)QfixedpointofT⇔δQ∩ΔQ≠ϕ.Proof.
( 1) Let us show thatδQ∩ΔQ≠ϕ is a necessary condition for Q to be a fixed point of  T. Let (x,y)∈δQ, then the  Jacobian JT(x,y)=0, thus JT(x,y)2=JTT(x,y)·JT(x,y)=JTQ·JT(x,y)=0. Besides, as Q is a fixed point of T, for all (x,y)∈δQ we have
(3.2)T2(x,y)=T(T(x,y))=T(Q)=Q.
Thus, one showed that δQ⊂ΔQ.
( 2)  Now let us show thatδQ∩ΔQ≠ϕ is a sufficient condition for Q to be a fixed point of  T. Let (x,y)∈δQ∩ΔQ then  we obtain
(3.3)T(x,y)=Q=T2(x,y)=T(T(x,y))=T(Q).
Consequently, Q is a fixed point of T.We can give another necessary condition so that a focal point ofT-1 is a fixed point of T, given by this proposition.Proposition 3.5.
LetT 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.
LetQ be a focal point of T-1, this implies that JT=0 on δQ. As Q is a fixed point of T, consequently, Q∈δQ and then JTQ=0. This last (JTQ=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
(4.1)T(x,y)={x'=(1-a)(b·x-x2)+x·y-b·ya+c,y'=x,
where a, b, and c are real parameters. Its unique inverse is given by
(4.2)T-1(x,y)={x'=y,y'=(1-a)y+ax-cy-b,
The nondefinition set δs of  T-1 is given by
(4.3)y=b,
However, T-1 has a prefocal curve δQ of equation
(4.4)x=b
associated with the focal point  Q=(c,b).
First let us fixa=0.5, c=1.5 and we vary the parameter b. We have the following situations.
( 1)  Forb=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 δQ 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 valueb=1.500, the focal point  Q=(1.500,1.500) is on the prefocal curve δQ. 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 valueb=1.501, the iterates of focal point Q converge toward the attractor, thus the focal  point Q and prefocal curve δQ 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:
(4.5)T(x,y)={x'=y,y'=x·y2-b·y2-a·x+a·b·y,
where a and b are real parameters. Its unique inverse is given by
(4.6)T-1(x,y)={x′=y+b·x2-a·b·x-a+x2,y′=x.
The set of nondefinition δs of T-1 is constituted of two branches δs,1={x=a} and δs,2={x=-a}:
(4.7)δs=δs,1∪δs,2,T-1 has two prefocal curves:
(4.8)δQ1:y=a,δQ2:y=-a,
associated, respectively, with the two focal points:
(4.9)Q1=(a,a·b(a-1)),Q2=(-a,-a·b(a+1)),
( I) Let us check thatδQi∩ΔQi=ϕ, i=1,2,  if and only if Qi is not a fixed point of  T (Proposition 3.4). We have JT(x,y)2=JTT(x,y)·JT(x,y)=((xy2-by2-ax+aby)2-a)·(y2-a)=0 this implies that
(4.10)y=±a,x=by2-aby±ay2-a.
The images of these curves, by T2, give us
(4.11)T2(x=by2-aby+ay2-a,y)=T(y,a)=Q1,T2(x,y=a)=T(Q1).
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, ΔQ1 is equal to
(4.12)ΔQ1={(x=by2-aby+ay2-a,y)}
with δQ1∩ΔQ1=ϕ. When Q1 is a fixed point of T, ΔQ1 is equal to
(4.13)ΔQ1={(x=by2-aby+ay2-a,y)}∪{(x,y=a)}.
Thus δQ1⊂ΔQ1.
The result is quite similar for the focal pointQ2.
(II) Study of the basin bifurcation “connected〈- -〉 nonconnected”
Fixb=-2 and we vary the parameter a.(1)
For the valuea=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 valuea=0.265, we see in Figure 5 that the basin of attraction is not more connected. δQ2 and Q2 are outside of the basin (Proposition 3.1).(3)
For the valuea=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 pointsQ1 and Q2 are fixed points of  T on the boundary of the basin.
T has three fixed points  A, B, C of coordinates
(4.14)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).
According to Proposition 3.3, Q1=(a,a·b(a-1)) is a fixed point of  T, if and only if Q1∈δQ1, then  a·b(a-1)=a, which gives the curve b1(a)=1/(a-a) for a≠0 and a≠1.
Fora=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 ofQ1 with the fixed points B, C  for b∈b1(a), we obtain
(4.15)Q1=(a,a)=Bfora∈]0,1[∪]1.6826,+∞[,Q1=(a,a)=Cfora∈]1,1.6826[.
Similarly, Q2=(-a,-a·b(a+1)) is a fixed point of T if and only if Q2∈δQ2, 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 ofQ2 with fixed points B, C  for b∈b2(a), we obtain
(4.16)Q2=(a,a)=Cfora∈]0,+∞[.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 mapT 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.



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