Maps with vanishing denominators/first subpage

Figure 1: Example 1. Fan-shaped basin boundaries of lobes and crescents.

Exemple 1: Image of lobes and crescents

Figure 1 shows fan-shaped basin boundaries of lobes and crescents issuing from the focal points, generated by a two-dimensional map \((x',y')= (F(x,y),G(x,y))\) where at least one of the components \(F\) or \(G\) has a denominator which vanish in a one-dimensional subset of the phase plane. The map equation is (Bischi and Gardini [1997])\[T:\begin{cases}x'=(\rho xy+\mu x(1+x)^{2})/(1+\rho y)) \\ y'=1+\rho y \end{cases}\]

Figure 2: Example 2. (a) The two prefocal curves \(\delta_{Q_{i}}\) \(i\)=1,2, associated with the focal points \(Q_{i}\), are distinct. (b) At the bifurcation the two prefocal curves merge, the two focal points remaining distinct. The hyperbola \(LC_{-1}\) (cf. noninvertible maps) degenerates into two lines: the vertical branch gives the new critical set \(LC_{-1}\) and the other collapses into the set of nondefinition \(\delta_{s}\). The resulting "double" prefocal curve is an asymptote of the critical curve \(LC\) (cf. noninvertible maps).

The map has three focal points\[Q_{i}=(x_{i}^{*},-1/ \rho )\] for \(i=1,2,3\) and \(x_{1}^{*}=0, x_{2}^{*}=1-(\sqrt{\mu)}/\mu), x_{3}^{*}=1+(\sqrt{\mu)}/\mu)\), located on the line \(y=-(1/\rho)\) and for each of them the prefocal set \(\delta_{Q_{i}}\) is the line \(y=0\). In Fig.1 (\(\rho =0.3,\mu =10.3)\) the grey point are those having divergent trajectories, while the other points have trajectories converging to the line \(y=1/(1-\rho)\) which includes two coexisting attractors, whose basins are differently colored in Fig.1. Whenever an arc of the frontier crosses the prefocal set, it gives rise to three distinct preimages issuing from the focal points, leading to the characteristic pictures with lobes and crescents (when the lobes have contacts with the critical curves of the map), which increase more and more as the parameter \(\mu\) increases.

Example 2: Coincidence of prefocal curves without coincidence of focal points

The situations generated by focal points belonging to the closure of the set of merging preimages \(\overline {LC}_{-1}\) are easily observed from the following two-dimensional map \(T_{e}\) (taken from reference Bischi et al. [2003] of the master page)

\(T_{e}:\begin{cases}x'=y+\epsilon {x} \\ y'=(\alpha x^{2}+ \gamma{x}) /(y-\beta+\sigma{x})\end{cases}\)

Figure 3: Example 3. (a) \(\alpha\)=0.48. Situation before the homoclinic bifurcation for the stable manifold \(W^{s}\) and the unstable one \(W^{u}\) of the saddle \(P_{2}^{*}\). (b) \(\alpha \)=0.5 Situation after the homoclinic bifurcation\[W^{u}\] intersects \(\delta_{s}\). Infinitely many new (and unbounded) branches of \(W^{u}\) are created, asymptotic to the line \(\delta_{Q}\) and to its images.

Figure 4: Example 3. (a) \(\alpha\)=0.49. Situation before the homoclinic bifurcation. The basin of the stable fixed point \(P_{1}^{*}\) is red colored, and the basin of the period two cycle (two yellow points) is blue colored. (b) \(\alpha \)=0.497 Situation after the homoclinic bifurcation. Three attractors are in presence\[P_{1}^{*}\] now with a blue colored basin, a period two cycle (two yellow points) with a red colored basin, and also a period 8 chaotic attractor with a light brown basin

not defined in the points of the line \(\delta_{s}\) of equation \(y-\beta+\sigma{x}=0\), on which two focal points exist given by \(Q_{1}=(0,\beta)\), \(Q_{2}=-(\gamma/{\alpha},\beta+\gamma\sigma/{\alpha})\) and the corresponding prefocal curves \(\delta_{Q_{i}}\), of equation \(x=F(Q_{i})\), \(i=1,2\), are \(x=\beta\) and \(x=\beta-(\epsilon-\sigma)\gamma/{\alpha}\) respectively. The map \(T_{e}\) is a noninvertible map of \((Z_{0}-Z_2)\) (cf. noninvertible maps) type with inverses defined by

\(T_{e1,e2}^{-1}:\begin{cases}x=(1/(2{\alpha})((\sigma -\epsilon)y'-\gamma)\pm \sqrt{\Delta(x',y')} \\ y=x'-\epsilon x\end{cases}\)

\(\Delta(x',y')=(\gamma-\sigma{y'}+\epsilon{y'})^{2}-4{\alpha}(\beta y'-x'y')>0\) in the region \(Z_2\) and \(LC=\left \{(x,y)|\Delta(x',y')=0\right\}\) (cf. noninvertible maps). In Fig. 2a (the red region represents the basin of the stable fixed point \(O=(0,0)\) and the complementary region the basin of infinity), obtained with parameters \(\alpha =0.5, \gamma =0.5, \beta = \sqrt{2}\), \(\sigma =0.2, \epsilon =-0.2\), the two prefocal curves, associated with the focal points \(Q_{i}=(0,\beta)\), are distinct. In Fig. 2b, obtained with \(\sigma =\epsilon =0.1\) and the other parameters unchanged, the two focal points are still distinct, but the two prefocal lines merge and become an asymptote for \(LC\). Note that we have \(LC_{-1}=J_{C}=J_0\) before the bifurcation, while at the bifurcation the hyperbola \(LC_{-1}\) degenerates into two lines, the vertical branch gives the new critical set \(LC_{-1}\) and the other collapses into the singular set \(\delta_{s}\). At this bifurcation \(LC_{-1}=J_{0}\subset J_{C}\) (because \(J_{C}\) also includes the set of nondefinition \(\delta_{s}\)), the resulting "double" prefocal curve is an asymptote of \(LC\), the arcs \(\pi_{1}\) and \(\pi_{2}\) (see Fig. 2 of the master page) degenerate into the focal points, which now are not located on \(\overline {LC}_{-1}\).

Example 3: Homoclinic bifurcation resulting from the contact between the unstable manifold of a saddle point and a set of nondefinition

The two-dimensional map (taken from reference Bischi et al. [1999] of the master page)\[T:\begin{cases}x'=y \\ y'=(1-\alpha)y+\frac{\alpha(x-1.2)}{y-1.5}\end{cases}\]

is not defined on the line \(\delta_{s}\) of equation \(y=1.5\) on which the focal point \(Q=(1.2, 1.5)\) exists with the prefocal curve \(\delta_{Q}\) of equation \(x=1.5\). This map has two fixed points \(P_{1}^{*}=(x=y=0.648...)\) and \(P_{2}^{*}=(x=y=1.852...)\). For a parameter value \(\alpha=\alpha_{h}=0.494...\) a non classical homoclinic bifurcation occurs, illustrated by Fig. 3 which represents the unstable manifold \(W^{u}(P_{2}^{*})\) of the saddle \(P_{2}^{*}\), the stable manifold \(W^{s}(P_{2}^{*})\), the line of non definition \(\delta_{s}\), and also its rank-1 preimage \(\delta_{s}^{-1}\).

In Fig. 3a, obtained for \(\alpha =0.48<\alpha_{h}\), \(W^{u}(P_{2}^{*})\) does not intersect the singular line \(\delta_{s}\). It is entirely located above \(\delta_{s}\) and on the right of \(\delta_{Q}\). In Fig. 3b, obtained with \(\alpha=0.5>\alpha_{h}\), \(W^{u}(P_{2}^{*})\) intersects \(\delta_{s}\), so that new unbounded branches of \(W^{u}(P_{2}^{*})\) are created that intersect again \(\delta_{s}\) and so on. Hence just after the bifurcation infinitely many new (and unbounded) branches of \(W^{u}(P_{2}^{*})\) are created, asymptotic to the line \(\delta_{Q}\) and to its images as well.

Such a bifurcation has a strong effect on the basins of the attractors. Figure 4a (\(\alpha =0.49<\alpha_{h}\) represents the situation of the basins of the two attractors in presence: the stable fixed point \(P_{1}^{*}\) (red colored basin), and a period two cycle (two yellow points) with a blue colored basin. After the bifurcation, the period two cycle basin is strongly modified, as shown by Fig. 4b (\(\alpha =0.497>\alpha_{h}\). We have three attractors in presence: the stable fixed point \(P_{1}^{*}\) (now blue colored basin), the a period two cycle (two yellow points) with a red colored basin, and also a period 8 chaotic attractor (light brown basin), which disappears for \(\alpha =0.4975...\) via a bifurcation by contact of this attractor with its basin boundary.

Example 4: Polynomial invertible map whose inverse has a focal point

This map (taken from reference Bischi et al. [1999] of the master page) is given by\[T:\begin{cases}x'=y \\ y'=xy-by^{2}-ax+aby\end{cases}\]

with \(a=-1.5, b=0.5\). This map is defined in the whole plane. Its Jacobian vanishes on the line of equation \(y=a\). The image of this line is "focalized" by the map into a single point \((a,0)\). From this fact we argue that at least one inverse component must be a map with denominator, and with a focal point in \((a,0)\) related to a prefocal curve of equation \(y=a\). Indeed, the map can be easily inverted to obtain the fractional map\[T^{-1}:\begin{cases}x=(y'+bx'^{2}-abx')/(x'-a)\\ y=x'\end{cases}\]

not defined in the line \(x'=a\). It is easy to realize that the point \(Q=(a,0)\) is a focal point of \(T^{-1}\), with prefocal line \(\delta_{Q}\) of equation \(y=a\). For \(a=-1.5, b=0.5\) the map generates a chaotic attractor. Since the point \(Q=(a,0)\) is inside the basin of the attractor, also the whole line \(y=a\) (the prefocal line of the inverse, which is "focalized" by the map \(T\) into the point \(Q\)) must belong to the same basin, as well as its preimages of any rank (Fig. 5). This implies that the attractor basin cannot be a bounded set because it must necessarily include a whole line and its preimages, which are asymptotes of the basin boundary.

Moreover the chaotic attractor has a very peculiar shape, characterized by a sort of knot of infinitely many curves that shrink into a unique point, at the point \(Q=(a,0)\), which is the focal point of the inverse map. In fact, every trajectory contained inside the attractor is conveyed through \(Q\) whenever it crosses the line \(y=a\). In particular, \(Q\) is the centre of a fan of unstable sets of saddle cycles belonging to the chaotic area, such as the repelling fixed point node \(P\) \((a+1/(1-b),a+1/(1-b))\), or of unstable sets of cycles belonging to the basin boundary. However, the "structure" of such an attracting set must be more complex of what we see at a first glance because all the images of the knot-point, that belong to the attracting set, behave in the same way.

Figure 5: Example 4: Polynomial map whose inverse has a vanishing denominator. \(Q\) is the focal point of the inverse map. This focal point \(Q\) and its increasing rank images are the knots of the chaotic attractor. The attractor basin (white colored) is not a bounded set. It includes the whole prefocal line \(\delta_{Q}\) (\(y=a\)) of the inverse map, and its preimages, which are asymptotes of the basin boundary.

Another example of a two-dimensional invertible map with equivalent properties is given in Gu [2007]. An example of a two-dimensional noninvertible map, generating chaotic attractors with knots, and nonbounded basins, is dealt in Gardini et al. [2004].

Example 5: Polynomial noninvertible map whose inverse is fractional

Due to its fractional inverses, the two-dimensional map (taken from reference Bischi et al. [1999] of the master page)\[T:\begin{cases}x'=x^{2}-a \\ y'=bxy(1-y)\end{cases}\]

\(a,b\) being real parameters, generates a non critical line which in the plane separates regions with a different number of rank-one preimages. Its Jacobian determinant \(J_{0}=2bx^{2}(1-2y)\) vanishes on the lines of equation \(y=1/2\) and \(x=0\). The image of the line \(y=1/2\) is given by the parabola of equation \(x=16y^{2}/b^{2}-a\), whereas the whole y-axis is mapped by \(T\) into the point \(T(x=0)=(-a;0)\)

As in the previous example, it appears that at least one inverse of \(T\) must exist as a fractional map with a focal point at \((-a;0)\) and the y-axis as prefocal curve. However in this case the map \(T\) is not invertible, since the curve \(x=16y^{2}/b^{2}-a\) is a critical curve \(LC\), locus of points having two merging preimages, that separates regions characterized by a different number of inverses. In fact the map has up to four distinct inverses, given by

\( T_{i}^{-1}:\begin{cases}x=\sqrt{x'+a}\\ y=\frac{1}{2}\left [1\pm \sqrt{1-\frac {4y'}{b \sqrt{x'+a}}}\right ]\end{cases}\)

where \(i=1\) corresponds to the sign "+", \(i=2\) to the sign "-", defined for \(x'>-a\) and \(y \le (b/4) \sqrt {x'+a}\), and

\( T_{j}^{-1}:\begin{cases}x=-\sqrt{x'+a} \\ y=\frac{1}{2}\left [1\pm \sqrt{1+\frac {4y'}{b \sqrt{x'+a}}}\right ]\end{cases}\)

where \(j=3\) corresponds to the sign "+", \(j=4\) to the sign "-", defined for \(x'>-a\) and \(y \ge -(b/4) \sqrt {x'+a}\).

From the expressions of these inverses it is clear that not only the critical curve, \(x=16y^{2}/b^{2}-a\), separates regions whose points have different number of rank-one preimages, but also the vertical line of equation \(x=-a\). In fact, the plane can be divided into four regions:

• \(Z_{0}\)=\({(x,y)|(x<-a)}\) whose points have no preimages, i.e. no inverses are defined in this region;

• \(Z_{2}\)=\({(x,y)|[(x>-a)} \cup (y \ge -(b/4) \sqrt {x+a})]\), where the inverses \(T_{j}^{-1}\), \(j=3,4\) are defined;

• \(Z'_{2}\)=\({(x,y)|[(x>-a)} \cup (y \le (b/4) \sqrt {x+a}]\), where the inverses \(T_{i}^{-1}\), \(i=1,2\) are defined;

• \(Z_{4}\)=\({(x,y)|[(x>-a)} \cup [-(b/4) \sqrt {x+a} \le y \le (b/4) \sqrt {x+a}] ]\), where the four inverses are defined;

The line \(x=-a\) separates regions with different numbers of inverses, but it is not a critical curve. The only peculiarity of that line lies in the fact that it is the curve at which the denominator of the inverses vanishes. Roughly speaking, we can say that \(x=-a\) plays a role analogue to that of a horizontal asymptote in a one-dimensional map. In fact, as it is well known, a horizontal asymptote can separate, in the range of a one-dimensional map, regions characterized by a different number of inverses (cf. sec. 3.1 of the link noninvertible maps).

Examples arising in discrete dynamical systems

Maps with focal points and prefocal sets naturally arise in discrete dynamical systems of the plane found in several applications, such as economic modeling , numerical iterative methods, various equations (see the papers of the section "References", for which the titles give an idea of the map types). In such dynamic models, peculiar structures of the basins, characterized by the presence of lobes and crescents, have been observed, which can be explained in terms of contacts of two sets of different nature, such as prefocal sets with stable and unstable sets of saddles.

References

• Bastien G. and Rogalski M. [2004] "Global behavior of the solutions of the Lyness' difference equation \(u_{n+2}u_{n}=u_{n+1}+a\)" J. of Difference Equations and Applications, 10(11), pp. 977-1003.

• Bastien G. and Rogalski M. [2004]. "On some algebraic difference equations related to families of conics or cubics: generalization of the Lyness’ sequences". J. of Mathematical Analysis and Applications, 300, pp. 303-333.

• Billings, L. and Curry, J.H. [1996]. "On noninvertible maps of the plane: Eruptions", Chaos, 6, 108-119.

• L. Billings, J. H. Curry and E. Phipps, [1997] "Lyapunov Exponents, Singularities and a Riddling Bifurcation," Physical Review Letters, 79: 1018-1021.

• Bischi G.I. & Gardini L., [1996] "Focal points and basin fractalization in a class of triangular maps". Proceedings of European Conference on iteration theory, ECIT96, Urbino (8-14 Sept., 1996).

• Bischi, G.I. and L. Gardini [1997] "Basin fractalization due to focal points in a class of triangular maps", International Journal of Bifurcation and Chaos, 7(7), pp. 1555-1577.

• Bischi G.I. & Naimzada A., [1995] "A coweb model with learning effects". Atti XiX Convegno AMASES, Cacucci Editore, Bari, 162-177.

• Bischi, G.I. and A. Naimzada [1997]. "Global analysis of a nonlinear model with learning" Economic Notes, vol. 26, n.3, pp. 143-174.

• Bischi G.I., M. Kopel, and Naimzada A. [2001]. "On a rent-seeking game described by a non-invertible iterated map with denominator". Nonlinear Analysis, T.M.&A., 47, pp.5309-5324.

• Fischer P. and Gillis D.[2006] "Plane Maps, Singularities and Quasi-fixed points". International Journal of Bifurcation and Chaos, Vol. 16, No. 1, 179-­183.

• Gardini L., Bischi G.I., Fournier-Prunaret D. [1999] "Basin boundaries and focal points in a map coming from Bairstow's methods" CHAOS, vol.9 n.2.

• Gardini L., Foroni I. and Mira C. [2004]. "On a particular foliation associated with a polynomial family of noninvertible maps of the plane". International Journal of Bifurcation and Chaos, 14(5), pp. 1601-1624.

• Gardini L., Bischi G.I. and C. Mira [2003] "Invariant Curves and Focal Points in a Lyness Iterative Process" International Journal of Bifurcation and Chaos, vol. 13, No. 7, pp. 1841-1852.

• Gu En-Guo and Ruan Jiong [2005] "On some global bifurcations of the domains of feasible trajectories: an analysis of recurrence equations". International Journal of Bifurcation and Chaos, Vol. 15, No. 5, 1625-­1639.

• Gu En-Guo [2007]. "The feasable domains and their bifurcations in extended logistic model with an external interference". International Journal of Bifurcation and Chaos, 17(3), pp. 877-889.

• Janowski E.J. and Kulenovic R.S. [2007] "Stability of the kth order Lyness' equation with a periodic-k coefficient", International Journal of Bifurcation and Chaos, vol. 17, No. 1, pp. 143-152.

• Kulenovic M.R.S. and Merino O. [2006] "Competitive exclusion versus competitive-coexistence for systems in the plane" Discrete and Continuous Dynamical Systems, Series B, 6(5) 2006, pp. 1141-1156]

• Mira C. [1996] "Some properties of two-dimensional nondefined maps in the whole plane". Proceedings of European Conference on iteration theory, ECIT96, Urbino (8-14 Sept., 1996).

• Yee, H.C. and P.K. Sweby [1994]. "Global asymptotic behavior of iterative implicit schemes", International Journal of Bifurcation & Chaos, 4(6), pp. 1579-1611.