Maps with vanishing denominators
Laura Gardini et al. (2007), Scholarpedia, 2(9):3277. | doi:10.4249/scholarpedia.3277 | revision #137336 [link to/cite this article] |
Let \(T\) be a two-dimensional map, denoted \((x',y') = T(x,y) = (F(x,y),G(x,y))\) (here \((x',y')\) denotes the image), where at least one of the components \(F\) or \(G\) has a denominator that vanishes in a one-dimensional subset of the phase plane. Consequently \(T\) is not defined in the whole plane and there can arise singular sets called focal points (Mira [1981], Mira et al. [1996], p. 26-33) and prefocal sets . These singularities give rise to topological structures of the attractors and their basins (see First Subpage|examples page) and to bifurcations that do not occur in continuous maps. At a focal point, one of the components has the form \(0/0\ .\) Roughly speaking, a prefocal curve is a set of points for which there exists at least one inverse, that maps (or "focalizes") the whole set into a single point, the focal point. Note that these singularities also play an important role for smooth maps which have an inverse with this property, i.e. having a denominator that vanishes on a subset of phase space (cf. examples 4 and 5 of polynomial maps in First Subpage) .
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Definitions and basic properties
In order to simplify the exposition, we assume that only one of the two functions defining the map \(T\) has a denominator which can vanish, say \[\tag{1} x'=F(x,y), \ \ \ \ \ y'=G(x,y)=N(x,y)/D(x,y) \]
where \(x\) and \(y\) real variables, \(F(x,y) \ ,\) \(N(x,y) \) and \(D(x,y) \) are continuously differentiable functions (\(N\) and \(D\) without common factors) and defined in the whole plane \(\mathbb{R}^2\ .\) The "set of nondefinition" of the map \(T\) (given by the set of points where at least one denominator vanishes) reduces to \[\tag{2} \delta_{s}={(x,y)\in \mathbb{R}^{2}\big|D(x,y)=0}. \]
Let us assume that \(\delta_{s}\) is given by the union of smooth curves of the plane. The successive iterations of the two-dimensional map \(T\) are well defined provided that the initial condition belongs to the set \(E\) given by \[ E=\mathbb{R}^{2}\setminus\textstyle\bigcup_{k=0}^{\infty}T^{-k}(\delta_{s}) \] where \(T^{-k}(\delta_{s})\) denotes the set of the rank-\(k\) preimages of \(\delta_{s}\ ,\) i.e. the set of points which are mapped into \(\delta_{s}\) after \(k\) applications of \(T\) (\(T^{0}(\delta_{s})=\delta_{s}\)). Indeed, in order to generate full forward orbits by the iteration of the map \(T\ ,\) the points of \(\delta_{s}\ ,\) as well as all their preimages of any rank, constitute a set of zero Lebesgue measure which must be excluded from the set of initial conditions so that \(T:E\rightarrow E\ .\)
Let us consider a bounded and smooth simple arc \(\gamma\ ,\) parametrized as \(\gamma(\tau)\ ,\) transverse to \(\delta_{s}\ ,\) such that \(\gamma\cap\delta_{s}=(x_{0}, y_{0})\) and \(\gamma(0)=(x_{0},y_{0})\ .\) Note that \(\gamma\) is nontangent to the set of nondefinition. The tangent case is discussed in more detail below. We are interested in the \(\gamma\) image \(T(\gamma)\ .\) As \((x_{0}, y_{0})\in\delta_{s}\) we have, according to the definition of \(\delta_{s}\ ,\) \(D(x_{0},y_{0})=0\ .\) If \(N(x_{0},y_{0})\neq0\ ,\) then \(\textstyle\lim_{\tau\to 0\pm}T(\gamma(\tau))=F((x_{0}, y_{0}),\infty)\) where \(\infty\) means either \(+\infty\) or \(-\infty\ .\) This means that the image \(T(\gamma)\) is made up of two disjoint unbounded arcs asymptotic to the line of equation \(x=F(x_{0}, y_{0})\ .\) A different situation may occur if the point \((x_{0}, y_{0})\in \delta_{s}\) is such that not only the denominator but also the numerator vanishes in it, i.e. \(D(x_{0}, y_{0})=N(x_{0}, y_{0})=0\ .\) In this case, the second component of \(T\) assumes the form \(0/0\ .\) This implies that the limit above may give rise to a finite value, so that the image \(T(\gamma)\) is a bounded arc (see fig. 1a) crossing the line \(x=F(x_{0}, y_{0})\) in the point \(F((x_{0}, y_{0}),y)\ ,\) where\[ y=\textstyle\lim_{\tau\to 0}G(x(\tau),y(\tau)) \] It is clear that the limiting value \(y\) must depend on the arc \(\gamma\ .\) Furthermore it may have a finite value along some arcs and be infinite along other ones. This leads to the following definition of focal point and prefocal curve Bischi et al. [1999]:
Definition. Consider the map \(T(x,y)\rightarrow(F(x,y),N(x,y)/D(x,y))\ .\) A point \(Q=(x_{0}, y_{0})\) is a focal point of \(T\) if \(D(x_{0}, y_{0})=N(x_{0}, y_{0})=0\) and there exist smooth simple arcs \(\gamma(\tau)\ ,\) with \(\gamma(0)=Q\ ,\) such that \(\textstyle\lim_{\tau\to 0}T(\gamma(\tau))\) is finite. The set of all such finite values, obtained by taking different arcs \(\gamma(\tau)\) through \(Q\ ,\) is the prefocal set \(\delta_{Q}\ ,\) which belongs to \(x=F(Q)\).
Here we shall only consider simple focal points, i.e. points which are simple roots of the algebraic system \(N(x,y)=0,D(x,y)=0\ .\) Thus a focal point \(Q=(x_{0}, y_{0})\) is simple if \(\overline{N}_{x}\overline{D}_{y}-\overline{N}_{y}\overline{D}_{x}\neq 0\ ,\) where \(\overline{N}_{x}=((\partial N)/(\partial x))(x_{0}, y_{0})\) and analogously for the other partial derivatives. In this case (of a simple focal point), there exists a one-to-one correspondence between the point \((F(Q),y)\ ,\) in which \(T(\gamma)\) crosses \(\delta_{Q}\ ,\) and the slope \(m\) of \(\gamma\) in \(Q\) (as shown in Bischi et al. [1999]): \[ m\rightarrow(F(Q),y(m)), \ \ \ \ \ with \ \ \ \ \ y(m)=(\overline{N}_{x}+m \overline{N}_{y})/(\overline{D}_{x}+m\overline{D}_{y}) \] and \[ (F(Q),y)\rightarrow m(y), \ \ \ \ \ with \ \ \ \ \ m(y)=(\overline{D}_{x}y- \overline{N}_{x})/(\overline{N}_{y}-\overline{D}_{y}y) \]
From the definition of the prefocal curve, it follows that the Jacobian \(det(DT^{-1})\) must necessarily vanish in the points of \(\delta_{Q}\ .\) Indeed, if the map \(T^{-1}\) is defined in \(\delta_{Q}\ ,\) then all the points of the line \(\delta_{Q}\) are mapped by \(T^{-1}\) into the focal point \(Q\ .\) This means that \(T^{-1}\) is not locally invertible in the points of \(\delta_{Q}\ ,\) being it a many-to-one map, and this implies that its Jacobian cannot be nonzero in the points of \(\delta_{Q}\ .\) From the relations given above it results that different arcs \(\gamma_{j}\ ,\) passing through a focal point \(Q\) with different slopes \(m_{j}\ ,\) are mapped by \(T\) into bounded arcs \(T(\gamma_{j})\) crossing \(\delta_{Q}\) in different points \((F(Q),y(m_{j}))\) (fig. 1b). Interesting properties are obtained if the inverse of \(T\) (or the inverses, if \(T\) is a noninvertible map) is (are) applied to a curve that crosses a prefocal curve.
Nonsimple focal points are considered in Bischi et al. [2005], where it is shown that they are generally associated with particular bifurcations (called of class two).
Case of an invertible map
Let \(T\) be invertible, and \(\delta_{Q}\) a prefocal curve whose corresponding focal point is \(Q\) (and several prefocal curves may exist, each having a corresponding focal point). Then each point sufficiently close to \(\delta_{Q}\) has its rank-1 preimage in a neighborhood of the focal point \(Q\ .\) If the inverse \(T^{-1}\) is continuous along \(\delta_{Q}\) then all the points of \(\delta_{Q}\) are mapped by \(T^{-1}\) in the focal point \(Q\ .\) Roughly speaking we can say that the prefocal curve \(\delta_{Q}\) is "focalized" by \(T^{-1}\) in the focal point \(Q\ ,\) i.e. \(T^{-1}(\delta_{Q})=Q\ .\) We note that the map \(T\) is not defined in \(Q\ ,\) thus \(T^{-1}\) cannot be strictly considered as an inverse of \(T\) in the points of \(\delta_{Q}\ ,\) even if \(T^{-1}\) is defined in \(\delta_{Q}\ .\)
The relation given above implies that the preimages of different arcs crossing the prefocal curve \(\delta_{Q}\) in the same point \((F(Q),y)\) are given by arcs all crossing the singular set through \(Q\ ,\) and all with the same slope \(m(y)\) in \(Q\ .\) Indeed, consider different arcs \(\omega_{n}\ ,\) crossing \(\delta_{Q}\) in the same point \((F(Q),y)\) with different slopes, then these arcs are mapped by the inverse \(T^{-1}\) into different arcs \(T^{-1}(\omega_{n})\) through \(Q\ ,\) all with the same tangent, of slope \(m(y)\ ,\) according to the formula given above (cf. fig. 1c). They must differ by the curvature at the point \(Q\ .\)
Case of a non invertible map
General considerations
In the case of continuous noninvertible maps \(T\ ,\) several focal points may be associated with a given prefocal curve \(\delta_{Q}\ ,\) each with its own one-to-one correspondence between slopes and points. The phase space of a noninvertible map is subdivided into open regions (or zones) \(Z_{k}\ ,\) whose points have \(k\) distinct rank-1 preimages, obtained by the application of \(k\) distinct inverse maps \(T_{j}^{-1}\) (i.e. such that \(T_{j}^{-1}(x,y)= (x_{j},y_{j})\ ,\) \(j=1,...,k\)). A specific feature of noninvertible maps is the existence of the critical set \(LC\) defined as the locus of points having at least two coincident rank-1 preimages, located on the set of merging preimages denoted by \(LC_{-1}\ .\) In any neighborhood of a point of \(LC_{-1}\) there are at least two distinct points mapped by \(T\) into the same point, so that the map \(T\) is not locally invertible in the points of \(LC_{-1}\ ,\) which implies that for differentiable maps the set \(LC_{-1}\) is included in the "set \(J_0\)" of points in which the Jacobian of \(T\) vanishes: \[ J_{0}={(x,y)\in \mathbb{R}^{2}\big|det(DT)=0} \] and \(LC_{-1}\subseteq J_0\ .\) Segments of the critical curve \(LC=T(LC_{-1})\) are boundaries that separate different regions \(Z_{k}\ ,\) but the converse is not generally true, that is boundaries of regions \(Z_{k}\ ,\) which are not portions of \(LC\ ,\) may exist (this happens, for example, in polynomial maps having an inverse function with a vanishing denominator, as shown in Bischi et al. [1999]). This fact is related to the existence of a set which is mapped by \(T\) in one point, such a set belongs to \(J_0\) but is not critical, so that we have a strict inclusion\[LC_{-1}\subset J_0\ .\] Another distinguishing feature in many noninvertible maps is the existence of a set of points, which we shall denote by \(J_{C}\ ,\) crossing which we have a change in the sign of the Jacobian of \(T\ ,\) \(det(DT)\ .\) From the geometric action of the foliation of the Riemann plane we can also say that the critical set \(LC_{-1}\) must belong to \(J_{C}\ .\) In fact, a plane region \(U\) which intersects \(LC_{-1}\) is "folded" along \(LC\) into the side with more preimages, and the two folded images have opposite orientation; this implies that the map has different sign of the Jacobian in the two portions of \(U\) separated by \(LC_{-1}\ .\) So, \(LC_{-1}\subseteq J_C\ .\) From the properties of maps with a vanishing denominator it results that generally a focal point \(Q\) belongs to the set \(\overline {LC}_{-1}\cap \delta_S\ ,\) where \(\overline {LC}_{-1}\) denotes the closure of \(LC_{-1}\ ,\) but in particular bifurcation cases, in which \(\delta_S\) belongs to \(J_{C}\ ,\) it happens that a focal point \(Q\) may not belong to \(LC_{-1}\ .\) The geometric behavior and the plane's foliation are different in the two cases. This leads to two different situations, according to the fact that the focal points belong or not to the set \(LC_{-1}\ .\)
The focal points do not belong to the closure of the set of merging preimages \(\overline {LC}_{-1}\ .\)
The following properties have been shown in Bischi et al. [1999].
- (a) For each prefocal curve \(\delta_{Q}\) we have \(LC \cap\delta_{Q}=\varnothing\).
- (b) If all the inverses are continuous along a prefocal curve \(\delta_{Q}\ ,\) then the whole prefocal set \(\delta_{Q}\) belongs to a unique region \(Z_{k}\) in which \(k\) inverse maps \(T_{j}^{-1}\ ,\) \(j=1,...,k\) are defined (cf. the link noninvertible maps).
It is plain that for a prefocal \(\delta_{Q}\) at least one inverse is defined that "focalizes" it into a focal point \(Q\ .\) However, other inverses may exist that "focalize" it into distinct focal points, all associated with the same prefocal curve \(\delta_{Q}\ .\) These focal points are denoted as \(Q_{j}=T_{j}^{-1}(\delta_{Q})\ ,\) \(j=1,...,n\ ,\) with \(n\le k\ .\) For each focal point \(Q_{j}\) the same results given above can be obtained with \(T^{-1}\) replaced by \(T_{j}^{-1}\ ,\) so that for each \(Q_{j}\) a one-to-one correspondence \(m_{j}(y)\) in the form given above is defined. With similar arguments it is easy to see that an arc \(\omega\) crossing \(\delta_{Q}\) in a point \((F(Q),y)\ ,\) where \(F(Q)=F(Q_{j})\) for any \(j\ ,\) is mapped by each \(T_{j}^{-1}\) into an arc \(T_{j}^{-1}(\omega)\ ,\) through the corresponding \(Q_{j}\) with the slope \(m_{j}(y)\ .\) If different arcs are considered, crossing \(\delta_{Q}\) in the same point, then these are mapped by each inverse \(T_{j}^{-1}\) into different arcs through \(Q_{j}\ ,\) all with the same tangent. We note that property (a) given above implies that the critical curve \(LC\) is generally asymptotic to the prefocal curves (see Figure 2b of First Subpage, also several examples are shown in Bischi et al. [1999]).
The focal points belong to the closure of the set of merging preimages \(\overline {LC}_{-1}\)
When the focal points belong to \(\overline {LC}_{-1}\) (closure of \(LC_{-1}\)) the "geometrical" situations of the phase plane, and the bifurcation types, are more complex (see Bischi et al. (2003)) with respect to the previous case. This is due to the fact that now \(LC\) has contact points at finite distance with the prefocal curves. The property \(Q_{j}=T_{j}^{-1}(\delta_{Q})\ ,\) \(j=1,...,n\ ,\) with \(n\le k\ ,\) does not occur. Now in the generic case a given prefocal curve \(\delta_{Q}\) is not associated with several focal points \(Q_{j}\ .\) Only one of the inverses \(T_{j}^{-1}\) maps a non critical point of a given prefocal curve into its related focal point, so that we can write \(Q=T_{j}^{-1}(F(Q),y)\) (or \(Q=T_{j}^{-1}(\delta_{Q})\) for short), but the index \(j\) depends on the non critical point \((F(Q),y)\) considered on \(\delta_{Q}\ .\) For this reason the previous situation of \(\delta_{Q}\) (focal points do not belong to \(\overline {LC}_{-1}\)) appears as non generic (indeed it may result from the merging of two prefocal curves \(\delta_{Q}^{r}\) and \(\delta_{Q}^{s}\) without merging of the corresponding focal points, as shown in Bischi et al. [2003, 2005]).
A qualitative illustration is given in Fig. 2, where a situation with two prefocal curves is represented for a noninvertible map \(T\ ,\) \((x,y)\rightarrow (x',y')\ ,\) of type \(Z_{0}-Z_{2}\) (see noninvertible maps). The inverse relation \(T^{-1}(x',y')\) has two components in the region \(Z_{2}\ ,\) denoted by \(T_{1}^{-1}\) and \(T_{2}^{-1}\ ,\) and no real components in the region \(Z_{0}\ .\) The set of nondefinition \(\delta_{s}\) is a simple straight line, and there are two prefocal lines, \(\delta_{Q_{i}}\ ,\) of equation \(x=F(Q_{i})\ ,\) associated with the focal points \(Q_{i}\ ,\) \(i=1,2\ ,\) respectively, and \(V_{i}=LC\cap \delta_{Q_{i}}\) are the points of tangency between \(LC\) and the two prefocal curves. Let \(\delta^{'}_{Q_{i}}\) be the segment of \(\delta_{Q_{i}}\) such that \(y<y(V_{i})\) (continuous line in Fig.2), and \(\delta^{''}_{Q_{i}}\) the segment of \(\delta_{Q_{i}}\) such that \(y>y(V_{i})\) (segmented line in Fig.2). The "focalization" occurs in the following way:\[ T_{1}^{-1}(\delta^{'}_{Q_{i}})=Q_{i}, \ \ \ \ \ \ \ \ T_{2}^{-1}(\delta^{''}_{Q_{i}})=Q_{i} \]
with \(T_{2}^{-1}(\delta^{'}_{Q_{i}})\cup T_{1}^{-1}(\delta^{''}_{Q_{i}})=\pi_{i}\ ,\) \(i=1,2\ ,\) being the two lines passing through the focal points \(Q_{i}\) and tangent to \(LC_{-1}\) at these points. When \(\delta_{Q_{1}}\rightarrow \delta_{Q_{2}}\ ,\) due to a parameter variation, without merging of the focal points, the points \(V_{i}\) on the prefocal curves tend to infinity, i.e. \(\delta_{Q_{1}}=\delta_{Q_{2}}\) becomes an asymptote for \(LC\ .\)
These situations are illustrated by the Example 2 of First Subpage.
Some dynamic properties of focal points
Important effects on the geometric and dynamical properties of the map \(T\) can be observed, due to the existence of a vanishing denominator. Indeed, a contact between a curve segment \(\gamma\) and the singular set \(\delta_{s}\) causes noticeable qualitative changes in the shape of the image \(T(\gamma)\ .\) Moreover, a contact of an arc \(\omega\) with a prefocal curve \(\delta_{Q}\ ,\) gives rise to important qualitative changes in the shape of the preimages \(T_{j}^{-1}(\omega)\ .\) When the arcs \(\omega\) are portions of phase curves of the map \(T\ ,\) such as invariant closed curves, stable or unstable sets of saddles, basin boundaries, we have that contacts between singularities of different nature generally induce important qualitative changes, which constitute new types of global bifurcations that change the structure of the attracting sets, or of their basins. In order to simplify the description of geometric and dynamic properties of maps with a vanishing denominator, and their particular global bifurcations, we assume that \(\delta_{s}\) and \(\delta_{Q}\) are made up of branches of simple curves of the plane. Let us describe what happens to the images of a small curve segment \(\gamma\) when it has a tangential contact with \(\delta_{s}\) and then crosses it in two points, and what happens to the preimages of a small curve segment \(\omega\) when it has a contact with a prefocal curve \(\delta_{Q}\) and then crosses it in two points.
Action of the map
Consider first a bounded curve segment \(\gamma\) that lies entirely in a region in which no denominator of the map \(T\) vanishes, so that the map is continuous in all the points of \(\gamma\ .\) As the arc \(\gamma\) is a compact subset of \(\mathbb{R}^{2}\ ,\) also its image \(T(\gamma)\) is compact (see the upper qualitative sketch in Fig.3). Suppose now to move \(\gamma\) towards \(\delta_{s}\ ,\) until it becomes tangent to it in a point \(A_{0}=(x_{0},y_{0})\) which is not a focal point. This implies that the image \(T(\gamma)\) is given by the union of two disjoint and unbounded branches, both asymptotic to the line \(\sigma\) of equation \(x=F(x_{0},y_{0})\ .\) Indeed, \(T(\gamma)=T(\gamma_{a})\cup T(\gamma_{b})\ ,\) where \(T(\gamma_{a})\) and \(T(\gamma_{b})\) are the two arcs of \(\gamma\) separated by the point \(A_{0}=\gamma\cap\delta_{s} \ .\) The map \(T\) is not defined in \(A_{0}\) and the limit of \(T(x,y)\) assumes the form \((F(x_{0},y_{0}),\infty)\) as \((x,y)\rightarrow A_{0}\) (along \(T(\gamma_{a})\ ,\) as well as along \(T(\gamma_{b})\)). In such a situation any image of \(\gamma\) of rank \(k>1\ ,\) given by \(T^{k}(\gamma)\ ,\) includes two disjoint unbounded branches, asymptotic to the rank-\(k\) image of the line \(\sigma\ ,\) \(T^{k}(\sigma)\ .\) When \(\gamma\) crosses through \(\delta_{s}\) in two points, say \(A_{1}=(x_{1},y_{1})\) and \(A_{2}=(x_{2},y_{2})\ ,\) both different from focal points, then the asymptote \(\sigma\) splits into two disjoint asymptotes \(\sigma_{1}\) and \(\sigma_{2}\) of equations \(x=F(x_{1},y_{1})\) and \(x=F(x_{2},y_{2})\) respectively, and the image \(T(\gamma)\) is given by the union of three disjoint unbounded branches (see the lower sketch in Fig.3).
When \(\gamma\) is, for example, the local unstable manifold \(W^{u}\) of a saddle point or saddle cycle, the qualitative change of \(T(\gamma)\ ,\) due to a contact between \(\gamma\) and \(\delta_{s}\ ,\) as described above, may represent an important contact bifurcation of the map \(T\ .\) Indeed the creation of a new unbounded branch of \(W^{u}\ ,\) due to a contact with \(\delta_{s}\ ,\) may cause the creation of homoclinic points, from new transverse intersections between the stable and unstable sets, \(W^{s}\) and \(W^{u}\ ,\) of the same saddle point (or cycle). In such a case it is worth noting that the corresponding homoclinic bifurcation does not come from a tangential contact between \(W^{u}\) and \(W^{s}\ .\) For maps with a vanishing denominator, this implies that homoclinic points can be created without a homoclinic tangency between \(W^{u}\) and \(W^{s}\ ,\) from the sudden creation of unbounded branches of \(W^{u}\) when it crosses through \(\delta_{s}\) (see Bischi et al. [1999]). If before the bifurcation \(W^{u}\) is associated with a chaotic attractor, the homoclinic bifurcation resulting from the contact between \(W^{u}\) and \(\delta_{s}\) may gives rise to an unbounded chaotic attractive set made up of unbounded, but not diverging, chaotic trajectories (see Bischi et al. [2000]). If before the bifurcation \(W^{u}\) is not associated with a chaotic attractor, the homoclinic bifurcation resulting from the contact between \(W^{u}\) and \(\delta_{s}\) may gives rise to global bifurcations of the basin (cf. Example 3 of First Subpage).
If the map is noninvertible, a direct consequence of the above arguments concerns the action of the curve of nondefinition \(\delta_{s}\) on \(LC_{-1}\ .\) If \(\overline {LC}_{-1}\) has \(n\) transverse intersections with the set \(\delta_{s}\) in non focal points \(P_{i}=(x_{i},y_{i})\ ,\) \(i=1,..,n\ ,\) then the critical set \(LC=T(LC_{-1})\) includes \((n+1)\) disjoint unbounded branches, separated by the \(n\) asymptotes \(\sigma_{i}\) of equation \(x=F(x_{i},y_{i})\ ,\) \(i=1,..,n\).
Action of the inverses
- (a) Let \(T\) be an invertible map, \(T(x,y)=(F(x,y),N(x,y)/D(x,y))\ .\) Consider a smooth curve segment \(\omega\) that moves towards a prefocal curve \(\delta_{Q}\) until it crosses through \(\delta_{Q}\) (see Fig.4) so that only a focal point \(Q=T^{-1}(\delta_{Q})\) is associated with \(\delta_{Q}\ .\) The prefocal set \(\delta_{Q}\) belongs to the line of equation \(x=F(Q)\ ,\) and the one-to-one correspondences between slopes and points hold, as given in Section 1. When \(\omega\) moves toward \(\delta_{Q}\ ,\) its preimage \(\omega_{-1}=T^{-1}(\omega)\) moves towards \(Q\ .\) If \(\omega\) becomes tangent to \(\delta_{Q}\) in a point \(C=(F(Q),y_{c})\ ,\) then \(\omega_{-1}\) has a cusp point at \(Q\ .\) The slope of the common tangent to the two arcs, that join at \(Q\ ,\) is given by \(m(y_{c})\ .\) If the curve segment \(\omega\) moves further, so that it crosses \(\delta_{Q}\) at two points \(F(Q,y_{1})\) and \(F(Q,y_{2})\ ,\) then \(\omega_{-1}\) forms a loop with a double point at the focal point \(Q\ .\) Indeed, the two portions of \(\omega\) that intersect \(\delta_{Q}\) are both mapped by \(T^{-1}\) into arcs through \(Q\ ,\) and the tangents to these two arcs of \(\omega_{-1}\ ,\) issuing from the focal point, have different slopes, \(m(y_{1})\) and \(m(y_{2})\) respectively, according to the formulas given in Section 1.
- (b) Now let \(T\) be a noninvertible map with focal points not located on \(LC_{-1}\ .\) In this case, \(k\ge 1\) distinct focal points \(Q{j}\ ,\) \(j=1,...,k\ ,\) may be associated with a prefocal curve \(\delta_{Q}\ .\) Then each inverse \(T_{j}^{-1}\ ,\) \(j=1,...,k\ ,\) gives a distinct preimage \(\omega_{-1}^{j}=T_{j}^{-1}(\omega)\) which has a cusp point in \(Q_{j}\ ,\) \(j=1,...,k\ ,\) when the arc \(\omega\) is tangent to \(\delta_{Q}\ .\) Each preimage \(\omega_{-1}^{j}\) gives rise to a loop in \(Q_{j}\) when the arc \(\omega\) intersects \(\delta_{Q}\) in two points (see fig.5, concerning the case \(k=2\)).
When \(\omega\) is an arc belonging to a basin boundary \(\mathcal{F}\ ,\) the qualitative modifications of the preimages \(T_{j}^{-1}(\omega)\) of \(\omega\ ,\) due to a tangential contact of \(\omega\) with the prefocal curve, can be particularly important for the global dynamical properties of the map \(T\ .\) As a frontier \(\mathcal{F}\ ,\) generally is backward invariant, i.e. \(T^{-1}(\mathcal{F})=\mathcal{F}\ ,\) if \(\omega\) is an arc belonging to \(\mathcal{F}\ ,\) then all its preimages of any rank must belong to \(\mathcal{F}\ .\) This implies that if a portion \(\omega\) of \(\mathcal{F}\) has a tangential contact with a prefocal curve \(\delta_{Q}\ ,\) then necessarily at least \(k\) cusp points, located in the focal points \(Q_{j}\ ,\) are included in the boundary \(\mathcal{F}\ .\) Moreover, if the focal points \(Q_{j}\) have preimages, then also they belong to \(\mathcal{F}\ ,\) so that further cusps exist on \(\mathcal{F}\ ,\) with tips at each of such preimages. It results that if the basin boundary \(\mathcal{F}\) was smooth before the contact with the prefocal curve \(\delta_{Q}\ ,\) such a contact gives rise to points of non smoothness, which may be infinitely many if some focal point \(Q_{j}\) has preimages of any rank, with possibility of fractalization of \(\mathcal{F}\) when it is nowhere smooth. When \(\mathcal{F}\) crosses through \(\delta_{Q}\) in two points, after the contact \(\mathcal{F}\) must contain at least \(k\) loops with double points in \(Q_{j}\ .\) Also in this case, if some focal point \(Q_{j}\) has preimages, other loops appear (even infinitely many, with possibility of fractalization) with double points in the preimages of any rank of \(Q_{j}\ ,\) \(j=1,...,n\ .\)
- (c) Whatever be the map \(T\) (invertible, or not, with focal points on \(\overline {LC}_{-1}\) or not) a contact of a basin boundary with a prefocal curve gives rise to a new type of basin bifurcation that causes the creation of cusp points and, after the crossing, of loops called lobes (this usage of "lobe" is distinct from that used in transport theory), along the basin boundary. This may give rise to a very particular fractalization of the basin boundary (see Example 1 of Maps_with_vanishing_denominators/First Subpage|First Subpage).
- (d) Let \(T\) be a noninvertible map with focal points not located on \(\overline {LC}_{-1}\). In this case, the contact of two lobes on \(LC_{-1}\) (related to a contact of \(LC\) with the basin boundary) gives rise to a crescent (Bischi et al. [1999]) bounded by the two focal points, from which lobes appeared. The creation of "crescents", resulting from the contact of lobes, is specific to noninvertible maps with denominator, when the focal points are not located on \(\overline {LC}_{-1}\). It requires the intersection of the boundary with a prefocal curve (located in a region with more than one inverse), at which the lobes are created, followed by a contact with a critical curve, causing the contact and merging of the lobes. At the contact the lobes are not tangent to \(LC_{-1}\ .\) After the contact, they merge creating the crescent (See Example 1 of Maps_with_vanishing_denominators/First Subpage|First Subpage).
- (e) If \(T\) is a noninvertible map with focal points located on \(\overline {LC}_{-1}\), then in the generic case we have a behavior similar to that of the invertible case, in which only one focal point is associated with \(\delta_{Q}\ ,\) but in a more complex situation with respect to the role of the components of the inverse map on \(\delta_{Q}\ ,\) and the presence of the arcs denoted \(\delta '_{Q_{i}}\)and \(\delta ''_{Q_{i}}\) in Fig.2. Details on this situation are given in Bischi et al. [2003]. Now a crescent does not results from the contact of two lobes, but from the contact of a lobe (issuing from a focal point) with another focal point. This situation is specific to noninvertible maps with denominator, when the focal points are located on \(\overline {LC}_{-1}\ .\) It requires the intersection of a basin boundary with a prefocal curve, followed by the contact of the resulting lobe with a focal point.
Further remarks
The theory of focal points and prefocal curves is also useful in understanding some properties of maps defined in the whole plane \(\mathbb{R}^{2}\ ,\) having at least one inverse map with vanishing denominator. Such maps may have the property that, among the points at which the Jacobian vanishes, there exists a curve which is mapped into a single point (see Bischi et al. [1999]). Another noticeable property of these maps is that a curve, at which the denominator of some inverse vanishes may separate regions of the phase plane characterized by a different number of preimages, even if it is not a critical curve of rank-1 (a critical curve of rank-1 is defined as a set of points having at least two merging rank-1 preimages). Such a case is shown in First Subpage with Example 5. At least one inverse is not defined on these non-critical boundary curves, due to the vanishing of some denominator. In a two-dimensional map, the role of such a curve is the analogue of an horizontal asymptote in a one-dimensional map, separating the range into intervals with different numbers of rank-1 preimages Bischi et al. [1999]. The existence of focal points of an inverse map can also cause the creation of particular attracting sets. Indeed a focal point, generated by the inverse map, may behave like a knot , where infinitely many invariant curves of an attracting set shrink into a set of isolated points. Example 4 of First Subpage shows this situation.
Concerning the relations between the concepts of focal points and prefocal curve proposed in the framework of the theory of iteration of two-dimensional real maps, using the style and terminology of the theory of dynamical systems, and the concepts of exceptional locus and blow-up in the framework of the study of (single application of) rational maps in the literature on algebraic geometry, it would be very interesting to create a link between these two literature streams. Indeed it seems that the first sign of such a possible link is given in Harris [1992], where the loop situation of figure 4 is described. Nevertheless it is underlined that the article topic is not limited to rational maps (cf. the sec. 1 hypothesis). So the numerator and the denominator of the map can be transcendental functions. With complementary precautions it would be even possible to widen the sec. 1 hypothesis, for example with piecewise smooth (piecewise linear for example) functions, for which the concepts of focal point and prefocal set remain the same.
Examples
They are given in First Subpage.
References
The papers, quoted in this main page, only concern the theory. Here only the tools, for a study of the dynamic effects due to a vanishing denominator, are outlined. The concept of focal point was given first in Mira [1981]. In the algebraic geometry framework, an equivalent formulation, related to the creation of a loop, is given in Harris [1992]. If previous references from other authors about this theory are found, they will be added. The references given in Maps_with_vanishing_denominators/First Subpage|First Subpage are related to applications to particular maps, using or not the concepts of focal point and prefocal set.
- Bischi, G.I., L. Gardini and C. Mira [1999]. "Maps with denominator. Part 1: some generic properties", International Journal of Bifurcation & Chaos, 9(1), 119-153.
- Bischi, G.I., L. Gardini and C. Mira [2000]. "Unbounded sets of attraction", International Journal of Bifurcation & Chaos, 10(9), pp. 1437-1470.
- Bischi G.I., L. Gardini and C. Mira [2003]. "Plane maps with denominator. Part II: noninvertible maps with simple focal points", International Journal of Bifurcation and Chaos 13(8), pp 2253-2277, .
- Bischi, G.I., L. Gardini and C. Mira [2005]. "Plane Maps with Denominator. Part III: Non simple focal points and related bifurcations ", International Journal of Bifurcation and Chaos, 15(2), 451-496.
- Harris J. [1992]. "Algebraic Geometry: a first course". Springer-Verlag New-York.
- Mira C. [1981]. "Singularités non classiques d'une récurrence et d'une équation différentielle". Comptes Rendus Acad. Sci. Paris, Série I, 292,146-151.
- Mira C., L. Gardini, A. Barugola and J.C. Cathala [1996]. Chaotic dynamics in two-dimensional noninvertible maps. World Scientific, Singapore, Series on Nonlinear Science, Series A, vol. 20.
Internal references
- John W. Milnor (2006) Attractor. Scholarpedia, 1(11):1815.
- Edward Ott (2006) Basin of attraction. Scholarpedia, 1(8):1701.
- John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
- James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
- Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
- Christian Mira (2007) Noninvertible maps. Scholarpedia, 2(9):2328.
- Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
External links
See Also
Attractor, Bifurcation, Dynamical systems, Iterated map, Noninvertible maps