# Noninvertible maps

Post-publication activity

Curator: Christian Mira Figure 1: One-dimensional map $$x'=f(x)\ .$$ $$Z_{0}\ :$$ $$x<C$$ where $$C$$ is the point $$C$$ abscissa. $$Z_{2}:$$ $$x>C\ .$$ $$M \in Z_2$$ has two inverses (rank-one preimages $$M^1_{-1}$$ and $$M^2_{-1}\ .$$ A point of $$Z_{0}$$ has no preimage. A point out of x'=0 has the same name as its projection on this axis. The first iterate $$C$$ (critical point) of the $$f$$ extremum $$C_{-1}$$ has two coincident inverses at this point. The point $$M^2_{-1}$$ has no preimage. Figure 2: One-dimensional map $$x'=f(x)\ .$$ The extrema of $$f$$ are mapped into the critical points $$C, \ C'\ .$$ $$Z_{1}: x<C$$,$$x>C'\ ,$$ a point has only one inverse (rank-one preimage). $$Z_{3}:$$ $$C<x<C'\ ,$$ a point $$M \in Z_3$$has three rank-one preimages $$M^1_{-1}$$, $$M^2_{-1}\ ,$$ and $$M^3_{-1}\ .$$ The points$$M^2_{-1},M^3_{-1} \in Z_1$$have only one rank-one preimage $$M^{2,2}_{-2}$$, $$M^{3,1}_{-2}\ ,$$ being also rank-two preimages of $$M\ .$$

Let $$T$$ be a map defined by $$X'=F(X)\ ,$$ where $$X$$ represents a point in a real-valued phase space. If dimX=1, then $$X$$ will be written $$x\ .$$ We emphasize that here $$X'$$ is not a derivative, but the iterate of $$X\ .$$ A map is invertible when its inverse, $$X=F^{-1}(X')\ ,$$ exists and is unique for each point in the range. If this is not the case the map is said noninvertible . In this latter case the phase space can be divided into open regions $$Z_i\ ,$$ such that for each point in $$Z_i$$ the map has $$i$$ distinct inverses (cf. Figure 1,Figure 2, when $$dimX '=1$$). For example, if the map $$T$$ is one-dimensional, defined by $$x'=f(x)\ ,$$ $$f(x)$$ being a polynomial of degree $$d\ ,$$ its inverse $$x=f^{-1}(x')$$ is determined from the real roots of an algebraic equation of degree $$d\ ,$$ whose solutions depend on the parameter $$x' \in Z_i\ .$$ So the quadratic map $$x'=x^2-\lambda\ ,$$ which has two inverses $$x=\pm\sqrt{x'+\lambda}\ ,$$ generates two regions $$Z_0$$ defined by $$x'<-\lambda\ ,$$ and $$Z_2$$ by $$x'>-\lambda\ .$$ We say that it belongs to the $$Z_0$$-$$Z_2$$ class. Various classes of noninvertible maps can be defined in a similar manner. So the map $$x'=\lambda x-x^3\ ,$$ $$\lambda >0\ ,$$ belongs to the $$Z_1$$-$$Z_3$$-$$Z_1$$ class. But this map is invertible if $$\lambda\le{0}\ ,$$ because only one inverse is real. The map $$x'=x^4-\lambda x^2+\mu\ ,$$ $$\lambda >0\ ,$$ belongs to the $$Z_0$$-$$Z_2$$-$$Z_4$$ class, because this quartic equation has either four, two, or zero real roots. When $$dimX>1$$ the degree of the polynomial is not sufficient to characterize its properties. For example, a two-dimensional quadratic map can either be invertible (the Henon family), or belong to the $$Z_0$$-$$Z_2$$ class, or to the $$Z_0$$-$$Z_2$$-$$Z_4$$ class. In contrast to the invertible maps class, noninvertible maps generate a very large set of map classes. The article is limited to some of the main aspects of this topic.

## Discrete models and noninvertible maps. Some definitions

The solutions of dynamical models with discrete time, are characterized by sequences of points, each of which are determined from an initial point (initial condition). Such models are called recurrence relationships, or simply recurrences. Their explicit form is written: $\tag{1} X_{n+1}=F(X_n,\Lambda), \ \ \ \ \ X(n=0)=X_0$

$$X$$ is a p-dimensional real vector in the phase space (or state space), $$n$$ the discrete time, $$\Lambda$$ a parameter vector, $$X_0$$ the initial condition. Since the discrete time $$n$$ does not appear explicitly (i.e. $$F$$ does not depend on $$n$$), the equation (1) is said to be autonomous. In this article the real function $$F(X_n,\Lambda)$$ is supposed to be continuous with respect to its arguments. Depending upon the area of science such an equation is also called an iteration , map , or point-mapping . Sometimes it is wrongly called a difference equation , for which the solution $$X(t)$$ is no longer a point sequence, but is continuous, and defined from an initial function $$X(t)=X_{0}(t)$$ for $$-1\le{t}<0$$ (Sharkovskij et al. ). In this article (1) will be called map , and denoted $$T\ .$$ Its equation is symbolically represented by: $\tag{2} X_{n+1}=TX_{n}, \qquad \mathbf{or} \qquad X'=TX$

The point $$X'$$ (resp. $$X_{n+1}$$) is called the rank-one image (or rank-one consequent) of $$X$$(resp. $$X_n$$). It is worth noting that the single-valuedness ($$X'$$ exists and is unique) of the function $$F(X_n,\Lambda) \ ,$$ defining the map $$T\ ,$$ does not imply anything about the existence and uniqueness of its inverse $$X=T^{-1}X' \ .$$ This condition is only satisfied for invertible maps , for which $$X=T^{-1}X'$$ exists and is unique for each point in the range. If the functions defining the map are continuous and have a continuous inverse, then it is said to be a homeomorphism. When the functions are smooth (of class at least $$C^1$$), then $$T$$ is called a diffeomorphism. For the noninvertible class of maps, depending upon which phase space region the point $$X'$$ lies in, the inverse map may be either multivalued, or single-valued, or may correspond to a void set. The set of points $\{X : X'=TX \}$ is the set of rank-one preimages (or rank-one antecedent) of $$X'\ .$$ This set may consist of one or several points, or be void, according to the region of the phase space to which $$X'$$ belongs. For the inverse map the form $$X=T^{-1}X'$$ is also used indicating that the inverse may have several determinations. In this form the sign "=" is written for convenience of writing, here "=" not having a strict sense. This in the same way as the inverse of $$x'=x^2-\lambda$$ is written $$x=\pm\sqrt{x'+\lambda}\ .$$

The recurrence $X_{n+r}=F_r{(X_n,\Lambda)} =F(X_{n+r-1},\Lambda)$ deduced from (1) after $$r$$ iterations is associated with a map denoted $$T^r\ ,$$ $$X_{n+r}=T^r{X_n}\ ,$$ $$X_{n+r}$$ being the rank-r image (or rank-r consequent) of $$X_n\ .$$ Each point $$X_n$$ that satisfies $$X_{n+r}=T^{r}X_n$$ belongs to the set of rank-r preimages (or rank-r antecedents) of $$X_{n+r}\ .$$

The one-dimensional example $$x'=x^2-\lambda$$ (Myrberg's map , equivalent to the logistic map studied long after in the English literature), with parameter $$\lambda\ ,$$ illustrates the simplest case of a noninvertible map. Here the inverse map is given by $$x=\pm\sqrt{x'+\lambda}\ .$$ The rank-one preimage of a point $$x'$$ is double-valued for $$x'>-\lambda\ ,$$ and is not real for $$x'<-\lambda\ .$$ The boundary point $$x'=-\lambda$$ is called a critical point (in the Julia-Fatou sense). Though in many publications the extremum point, $$x=0\ ,$$ of two coincident rank-one preimages is also called critical, this will not be the case in this article.

More generally a discrete model of a process in engineering does not have a simple explicit form like (1). The model appears either in an implicit form (for example $$F(X_{n+1},X_n{,\Lambda})=0\ ,$$ or in a parametric form, frequently as a noninvertible map (cf. p. 441-460 of Gumowski & Mira [1980a]).

For the initial condition $$X(n=0)=X_0\ ,$$ the solution of (1) is a sequence of points: $\tag{3} X_n=X(n,X_{0},\Lambda), \qquad n=1,2,3,.....$

called iterated sequence, or discrete phase trajectory, or orbit. The map $$T$$ can be considered as an implicit definition of the function $$X(n,X_{0},\Lambda) \ .$$ Though theoretically quite satisfactory, such a definition practically is almost useless, because in general the function $$X$$ is unknown, except for the linear case, and for very few examples in the nonlinear case. In all non-contrived cases it cannot be expressed explicitly in terms of known elementary and transcendental functions. For such situations, qualitative methods of nonlinear dynamics are used to characterize the non classical transcendental function $$X(n,X_{0},\Lambda)\ .$$ Then a meaningful characterization consists in the identification of its singularities (in a way similar to that of functions of the complex variable), and the behavior of the latter as the parameter $$\Lambda$$ varies. Any change in the nature of the singularities so-obtained, or any change of their qualitative properties, is called a bifurcation.

## Singularities common to invertible and non invertible maps

• The simplest singularities are zero-dimensional: period (or order) $$k$$ cycles (or periodic orbit ), made up of $$k$$ different consequent points solution of $$X_{n+k}=X_{n}\ ,$$ $$X_{n+r}\ne X_{n}\ ,$$ $$0<r<k\ .$$ When $$k=1$$ the point $$X^{\star}$$ is called a fixed point (period one cycle). When the map $$T$$ is smooth at a fixed point $$X^{\star}\ ,$$ the $$p$$ eigenvalues $$S_{j}$$ of the Jacobian matrix, $$j=1,...,p\ ,$$ are called the fixed point multipliers . Similarly the $$p$$ eigenvalues $$S_{j}$$ of the Jacobian matrix of $$T^{k}$$ at the point of a period $$k$$ cycle $$X_{i}^{\star}$$ (the eigen-values are the same whatever be the index i) are called the cycle multipliers. A hyperbolic cycle is attracting (or asymptotically stable) if all the multipliers are such that $$|S_{j}|<1\ .$$ When at least one of the multipliers exceeds 1 in modulus, then the cycle is repelling (or unstable, i.e; as not a synonym of expanding). When at least one of the multiplier is $$|S_{l}|=1\ ,$$ for a parameter value $$\Lambda =\Lambda _{b}\ ,$$ the cycle is non hyperbolic. Crossing through this value by a $$\Lambda$$ variation gives rise to a local bifurcation . An unstable cycle with $$|S_{j}|>1\ ,$$ $$|S_{m}|<1\ ,$$ $$j=1,2,...,r\ ,$$ $$m=1,2,...,s\ ,$$ $$r+s=p\ ,$$ is called a saddle. Then $$r \ ,$$ $$s \ ,$$ $$|S_{j}|\ ,$$ and$$|S_{m}|$$ define different types of saddle. A fixed point, or a cycle, with all the multipliers $$|S_{j}|>1\ ,$$ $$j=1,...,p\ ,$$ is said expanding '.
• A manifold (or set) of dimension $$d=1,2,...,p-1$$ $$(dimX=p)\ ,$$ invariant (or mapped into itself) by $$T$$ or $$T^{-1}$$ (resp. $$T^{k}\ ,$$ or $$T^{-k}$$), and passing through a cycle point, constitutes a a more complex singularity with respect to fixed points and cycles. Locally it is defined from the eigenvectors associated with the cycle multipliers. For an hyperbolic saddle cycle $$X^{\star}\ ,$$ the manifold (or set) associated with multiplier(s) $$|S_{m}|<1$$ is called the stable manifold $$W^{s}(X^{\star})$$ of this cycle. The manifold (or set) associated with multiplier(s) $$|S_{j}|>1$$ is called the unstable manifold $$W^{u}(X^{\star})$$ of the saddle cycle.
• An important $$d$$-dimensional, $$(p-1)\le{d}<p$$ singularity is the basin boundary , separating open regions of the phase space, $$d=p-1$$ when the boundary is not fractal. These open regions correspond to a set of initial conditions giving rise to a convergence toward a stable steady state, i.e; a well defined attractor $$A\ .$$ Generally each region constitutes the influence domain (called basin , or basin of attraction) $$D(A)$$ of $$A\ .$$ When the map $$T$$ is invertible, a basin is always simply connected . This is not always the case when $$T$$ is noninvertible (cf. sec. 3.2, and Fig. 4 of First Subpage).

A map may also generate singularities the dimension of which is not an integer $$d\le p\ .$$ Such a singularity constitutes what is called a "fractal set", either attracting ( strange attractor ), or repelling ( strange repellor , or repellor) for the points located in a sufficiently small neighborhood of this set. Whatever be the map, invertible (with $$p\ge{2}$$), noninvertible (with $$p\ge{1}$$), a basin boundary $$\partial D(A)$$ may be also fractal (cf. fractal basin boundaries), i.e. having a non-integer dimension, $$\partial D(A)$$ containing a strange repellor. Then the set $$\partial D(A)$$ is sometimes called chaotic basin boundary.

• The set $$W^{s}{(X^{\star})}\cap W^{u}{(X^{\star})}$$ is said homoclinic, if it is made up of infinitely many points. Let $$X^{\star}$$ and $$Y^{\star}$$ be two fixed points (or cycles) the set $$W^{s}{(X^{\star})}\cap W^{u}{(Y^{\star})}$$ is said heteroclinic.

In the homoclinic case, when transverse intersections occur between the stable and unstable set of a same cycle, (stable, or unstable) chaotic behaviors are possible. While in the heteroclinic case this happens when both $$W^{s}{(X^{\star})}\cap W^{u}{(Y^{\star})}$$ and $$W^{s}{(Y^{\star})}\cap W^{u}{(X^{\star})}$$ holds. Bifurcations by homoclinic or heteroclinic tangency are global bifurcations which may correspond to bifurcations of an ordered dynamics toward a chaotic one.

## Singularities specific to noninvertible maps

### Definitions and some properties

With respect to invertible maps, noninvertible maps $$T$$ introduce a singularity of a different nature: the critical set (Gumowski & Mira [1980a,b], Mira et al. , Abraham et al. , Mira & Shilnikov ). The rank-one critical set $$CM$$ is the geometric locus of points $$X$$ having at least two coincident rank-one preimages. Such preimages are located on a set $$CM_{-1}\ ,$$ the set of merging (or coincident) rank-one preimages . The set$$CM$$satisfies the relations: $T^{-1}(CM) \supseteq{CM_{-1}}, \qquad T(CM_{-1})=CM$ A rank-$$q$$ critical set $$CM_{q-1}$$ is given by the rank-$$q$$ image $$CM_{q-1}=T^{q}(CM_{-1})\ ,$$ $$CM_{0}\equiv CM\ .$$ If $$dimX=p=1\ ,$$ $$CM$$ is made up of one, or several points (all are rank-one critical points).

Such new singularities play a fundamental role in the attractors and basins structure, and also for their bifurcations. It is the case of contact bifurcations , resulting from the meeting of two singularities of different nature: an invariant manifold (or set) by $$T\ ,$$ or by $$T^{-1}$$ with a critical set. This situation generally gives rise to global bifurcations, which may be related to homoclinic and heteroclinic bifurcations (Mira et al. ). Most of the results obtained till now concern the general class of maps $$T$$ of the line ($$dimX=p=1\ ,$$ cf. Mira  and chap. 2 of Mira et al. ), and of the plane ($$dimX=p=2$$). For plane maps the critical set$$CM$$generally becomes a critical curve $$LC$$ (for Ligne Critique in French). In exceptional cases the critical set may include isolated points. This may occur when the inverse map $$T^{-1}$$ has a vanishing denominator (cf. Bischi et al.  and the link "maps with vanishing denominators"). When the two functions defining $$T$$ satisfy the Cauchy-Riemann conditions (case of one-dimensional maps of the complex variable, related to a Julia set),the set $$LC$$ in the plane is generally made up of isolated points.

In general the critical curve $$LC$$ is made up of several branches separating the plane into open regions. So the plane $$\mathbb{R}^2$$ can be subdivided into open regions $$Z_{i}$$ ($$\mathbb{R}^2$$=$$\cup \overline{Z_i}\ ,$$ $$\overline{Z_i}$$ being the closure of $$Z_i$$), each point of $$Z_i$$ having $$i$$ distinct rank-one preimages. Generally the boundaries of the regions $$Z_i$$ are branches of the rank-one critical curve $$LC\ ,$$ locus of points such that at least two determinations of $$T^{-1}$$ are merging. The locus of these coincident first rank preimages is a curve $$LC_{-1}\ ,$$ called rank-one curve of merging preimages . If $$T$$ is smooth, $$LC_{-1}$$ is a set of points for which the$$T$$ Jacobian determinant vanishes. If the map is nonsmooth, $$LC_{-1}$$ may include points for which $$T$$ is not smooth. The curve $$LC$$ satisfies the relations $$T^{-1}(LC) \supseteq{LC_{-1}}$$ and $$T(LC_{-1})=LC\ .$$ The simplest case is that of maps in which $$LC$$ (made up of only one branch) separates the plane into two open regions $$Z_0$$ and $$Z_2\ .$$ In more complex cases a classification of noninvertible maps can be made from the structure of the set of $$Z_i$$ regions (cf. Mira et al. ).

It is worth noting that there is a class of maps giving rise to sets of points separating regions with distinct rank-one preimages, without being critical in the above sense. It is the case when the map, or its inverse, has a vanishing denominator (cf. Bischi et al. , and the link "maps with vanishing denominators"). A one-dimensional illustrative example is given by a map defined by a function having an extremum and an arc with a horizontal asymptote. It is the case of the map $$x'=1/(1+x^{2})$$ having the horizontal asymptote $$x'=0\ ,$$ and the critical point $$x=1$$ rank-one image of the maximum $$x=0\ .$$ The inverse map $$x=\pm\sqrt{(1-x')/x'}$$ has a vanishing denominator at $$x'=0\ .$$ The interval $$0<x<1$$ corresponds to the region $$Z_2\ ,$$ separated from $$Z_0$$ ($$x>1$$) by the critical point $$x=1\ .$$ The other boundary of $$Z_2$$ is the non critical point $$x=0$$ separating $$Z_2$$ from the region $$Z'_0$$ ($$x<0$$) without preimage. Two-dimensional examples of this situation are given in sec. 3.2 of Bischi et al.  and with the example 5 in the subpage of the link maps with vanishing denominators.

The non invertibility of the map induces the following properties:

• The stable set $$W^{s}(p ^{\star})$$ of a saddle $$p ^{\star}$$ (Mira et al. )
• is backward invariant $$T^{-1}[W^{s}(p^{\star})]=W^{s}(p^{\star})\ ,$$
• it is mapped into itself by $$T\ ,$$ i.e. $$T[W^{s}(p^{\star})]\subseteq {W^{s}(p^{\star})}\ .$$
• It is invariant ($$T[W^{s}(p^{\star})]=W^{s}(p^{\star})$$) if $$T$$ is invertible , while for a noninvertible map it may be strictly mapped into itself i.e. $$T[W^{s}(p^{\star})]\subset {W^{s}(p^{\star})}\ .$$
• When $$T$$ is noninvertible with $$p=2\ ,$$ $$W^{s}(p ^{\star})$$ may be non-connected and made up of infinitely many closed curves. An equivalent property holds for higher dimensions $$p>2\ .$$
• The invariant unstable set $$W^{u}(p ^{\star})$$ of a saddle point $$p ^{\star}$$ generated by a continuous and noninvertible map $$T$$ with $$p>1$$
• is invariant and connected,
• but self intersections are possible (so that it may not be a manifold), which cannot happen for invertible maps (Gumowski & Mira [1980a,b], Mira et al. ).

The role of the critical set $$LC$$ and of merging preimages sets $$LC_{-1}\ ,$$ is again essential in the understanding of the formation of self intersections of the unstable set of a saddle fixed point, and properties of invariant closed curves (Mira et al. , Frouzakis et al. ).

### Basins and their bifurcations

Let $$D$$ be the basin of an attractor $$A\ ,$$ generated by a noninvertible map $$T\ .$$ $$D$$ is invariant under backward iteration $$T^{-1}$$ of $$T\ ,$$ but not necessarily invariant by $$T\ .$$ The basin $$D$$ and its boundary $$\partial D$$ satisfy the relations: $T(\partial D) \subseteq \partial D, \qquad T^{-1}(\partial D) = \partial D, \qquad T(D)\subseteq D,\qquad T^{-1}D=D$ Figure 3: Two-dimensional map. Global bifurcation simply-multiply connected basin. Out of (D) (brown region) and its boundary, the orbits are diverging.$$Z_0$$ (no preimage) and $$Z_2$$ (two rank-one preimages) are separated by$$L$$(critical curve), locus of points having two coincident rank-one preimages located on$$L_{-1}\ .$$ The increasing rank preimages of$$H_{0}\subset Z_2$$ create the basin (D) holes, here limited to the rank three, each rank being differently colored.

Here, the strict inclusion holds iff $$D$$ contains points of a $$Z_0$$ region (i.e. with no real preimage). As we noted above, $$D$$ may be simply connected (cf. Example 1 in First Subpage) as in the invertible case, but also non connected , or multiply connected (Fig. 4 in First Subpage), or both non connected and multiply connected . Its boundary $$\partial D$$ may contain repelling sets related to the presence of strange repellors $$SR\ .$$ Such an unstable set $$SR$$ is made up of infinitely many unstable cycles with increasing period, their limit sets in increasing classes of accumulation in the Pulkin' sense  (Gumowski & Mira [1980a] p. 129-131, Mira  p. 99-100), the preimages of increasing rank of all these points. This situation may give rise to fractal basin boundaries (or fuzzy boundaries) separating the domain of influence of different attractors, and chaotic transient toward a defined attractor (Mira et al. , Mira et al. , Mira & Shilnikov , Bischi et al. ). When the two functions of a plane map satisfy the Cauchy-Riemann conditions $$\partial D$$ may become a Julia set.

Critical curves play a basic role in the bifurcations of type simply connected basin $$\leftrightarrow$$ non-connected basin. It is the same for bifurcations of type simply connected basin $$\leftrightarrow$$ multiply connected basin. These bifurcations always result from the contact of a basin boundary with a critical curve arc. They are generated by the same mechanism, but in many different ways. One of them is illustrated by Fig. 3, for the global bifurcation : simply connected basin $$\leftrightarrow$$ multiply connected basin. Here $$D$$ is the basin of an attractor $$A\ ,$$ $$\partial D$$ its boundary, which limits a domain of diverging orbits, and the domain of convergence toward $$A\ .$$ The regions $$Z_2$$,$$Z_0$$ are separated by the critical curve $$L$$ locus of points having two coincident rank-one preimages, which are located on $$L_{-1}\ .$$ In Fig. 3a $$D$$ is simply connected. In Fig. 3c it is multiply connected, due to the presence of the region $$H_{0}\subset Z_2\ ,$$ which belongs to the domain of diverging orbits. The increasing rank preimages of $$H_{0}$$ also belong to the domain of diverging orbits. They are represented until the rank-N, $$N=1,2,3\ ,$$ for $$N=1,2$$ defined by$H_{1}=T^{-1}(H_0)\ ,$ $$H_{2}^{1}\cup H_{2}^{2}=T^{-1}(H_1)\ .$$ The bifurcation, i.e. tangential contact of $$L$$ with $$\partial D\ ,$$ is given by Fig. 3b.$$H_{0}$$ and its increasing rank preimages turn into a points set belonging to $$\partial D\ .$$ Figure 3c also shows a case of a nonconnected basin, here domain of diverging orbits (basin of the attractor at infinity), with the bifurcation simply connected basin $$\leftrightarrow$$ non-connected basin occurring with Fig. 3b.

## Absorbing areas, chaotic areas, their bifurcations.

Roughly speaking an absorbing area $$(d')$$ is a region bounded by critical curves arcs of finite rank $$LC_{n}\ ,$$ $$n=0,1,2,...l, \ ,$$ $$LC_{0} \equiv LC\ ,$$ such that the successive images of all points of a neighborhood $$U(d')\ ,$$ from a finite number of iterations, enter into $$(d')$$ and cannot get away after entering. Except for some bifurcation cases, a chaotic area $$(d)$$ (Gumowski & Mira [1980a,b], Mira et al. , Mira & Shilnikov , Bischi et al. ) is an invariant absorbing area, containing a fractal set made up of repelling cycles with increasing period, and their limit sets (as the Pulkin's ones ), giving rise to orbits with the property of sensitivity to initial conditions. Its boundary $$\partial d$$ is made up of $$LC_{n}$$ arcs (see Fig. 5 of Example 2 in First Subpage). A chaotic area may be periodic of period $$k\ ,$$ i.e. k-cyclic chaotic area constituted by $$k$$ nonconnected chaotic areas invariant by $$T^{k}\ .$$ An extended notion of absorbing area and chaotic area, that of mixed absorbing area, mixed chaotic area, is described in (Mira et al. ). For non invertible maps, as for invertible ones, when the inverse map has a vanishing denominator, a chaotic attractor may present "points of focalization", called knots (Bischi et al. , Gardini et al. ).

The role of critical curves is fundamental in the definition of the bifurcation leading to the destruction of a chaotic area $$(d)\ .$$ Such a bifurcation corresponds to the contact of the boundary $$\partial D(d)$$ of the basin $$D(d)$$ with the boundary $$\partial d$$ of $$(d)\ .$$ After this contact $$(d)$$ is destroyed. Then the fractal set of repelling cycles (with increasing period) turns into a strange repellor contained into a region of chaotic transient before a regular convergence toward an other attractor (see Fig. 6 in First Subpage).

Other type of chaotic area bifurcation, for which the critical curve plays the fundamental role, concerns: transitions simply connected chaotic area $$\leftrightarrow$$ doubly connected chaotic area (or annular chaotic area), nonconnected chaotic area $$\leftrightarrow$$ doubly connected chaotic area (Mira et al. ).

Example 2 (see First Subpage), related to a $$Z_{0}-Z_2$$ map, illustrates several bifurcations of basins and chaotic areas.

## Important remark

A chaotic area $$(d)$$ is not always a strange attractor in a mathematical sense. It may occur that $$(d)$$ contains numerous, and even infinitely many attractors. In this case let $$A$$ be the union of these attractors. A numerical simulation of a map solution is always made from a limited number of iterations. Let $$(d)$$ be the chaotic area obtained after a transient, i.e. the simulation is made after $$N$$ iterations, $$N$$ being sufficiently large to attain what at first view appears as a steady state. Then either the numerical simulation reproduces points of a chaotic area related to a "strict" strange attractor, or it represents a transient toward $$A$$ including attracting cycles of large period, a large part of them with a period larger than the simulation duration. The first case for example is that of some piecewise smooth maps, not permitting attracting cycles (i.e. the Jacobian determinant absolute value is not sufficiently small). Supposing an ideal case (numerical iterations without error), in the second case the transient would be that toward one of the attractors of $$A$$ with a period larger than the number of iterations. Thus this transient, which is chaotic, occurs inside a very narrow fractal basin, tangled with similar basins of the other attractors with large period. In presence of unavoidable numerical errors, the iterated points cannot remain inside the same narrow basin. They sweep across the narrow tangled basins of other attractors of $$A\ .$$ Then a map orbit occupies the largest place inside the chaotic area $$(d)\ ,$$ bounded by subsets of critical sets $$CM_{q}\ .$$ This means that $$(d)$$ coincides with the numerical observation, in the smooth case as a very long chaotic transient toward $$A \subset (d)\ ,$$ in the nonsmooth case as a true strange attractor. This shows the interest of the notion of chaotic area, even if in the smooth case it is impossible to discriminate numerically a situation of a strange attractor in the mathematical sense, from that of a very long chaotic transient toward an attractor of $$A\ .$$

## Examples

Two examples (simply connected basins, multiply connected ones, chaotic areas, their bifurcations) are presented in First Subpage.

Acknowledgments. Profs. G.I. Bischi and L. Gardini corrected inaccuracies in this article, and the three reviewers contributed to the article improvement.