mappings-2003

International Journal of Bifurcation and Chaos, Vol. 13, No. 2 (2003) 287–310c© World Scientific Publishing Company

COMPLEX PATTERNS ON THE PLANE:

DIFFERENT TYPES OF BASIN FRACTALIZATION

IN A TWO-DIMENSIONAL MAPPING

RICARDO LOPEZ-RUIZFacultad de Ciencias – Edificio B, Area de Ciencias de la Computacion,

Universidad de Zaragoza, 50009 – Zaragoza, Spain

DANIELE FOURNIER-PRUNARETInstitut National des Sciences Appliquees, Systemes Dynamiques (SYD), L.E.S.I.A.,

Avenue de Rangueil, 31077 Toulouse Cedex, France

Received November 13, 2000; Revised January 28, 2002

Basins generated by a noninvertible mapping formed by two symmetrically coupled logisticmaps are studied when the only parameter λ of the system is modified. Complex patterns onthe plane are visualized as a consequence of the basins’ bifurcations. According to the alreadyestablished nomenclature in the literature, we present the relevant phenomenology organized indifferent scenarios: fractal islands disaggregation, finite disaggregation, infinitely disconnected

basin, infinitely many converging sequences of lakes, countable self-similar disaggregation andsharp fractal boundary. By use of critical curves, we determine the influence of zones withdifferent number of first rank preimages in the mechanisms of basin fractalization.

Keywords : Complex patterns; two-dimensional noninvertible maps; invariant sets; basins; crit-ical curves; contact bifurcations; absorbing and chaotic areas; fractalization; self-similarity.

First, we give a glossary, which introduces the spe-cific vocabulary used in this paper.

Glossary

INVARIANT: A subset of the plane is invariant un-der the iteration of a map if this subset is mappedexactly onto itself.

ATTRACTING: An invariant subset of the plane isattracting if it has a neighborhood, every point ofwhich tends asymptotically to that subset or arrivesthere in a finite number of iterations.

CHAOTIC AREA: An invariant subset that ex-hibits chaotic dynamics. A typical trajectory fillsthis area densely.

CHAOTIC ATTRACTOR: A chaotic area, whichis attracting.

BASIN: The basin of attraction of an attracting setis the set of all points, which converge towards theattracting set.

IMMEDIATE BASIN: The largest connected partof a basin containing the attracting set.

ISLAND: Nonconnected region of a basin, whichdoes not contain the attracting set.

LAKE: Hole of a multiply connected basin. Sucha hole can be an island of the basin of another at-tracting set.

HEADLAND: Connected component of a basinbounded by a segment of a critical curve and a seg-ment of the immediate basin boundary of anotherattracting set, the preimages of which are islands.

BAY: Region bounded by a segment of a criticalcurve and a segment of the basin boundary, the

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288 R. Lopez-Ruiz & D. Fournier-Prunaret

successive images of which generate holes in thisbasin, which becomes multiply connected.

COAST: Basin boundary.

SEA: An open domain of divergent iteratedsequences.

ROADSTEAD: The coast situation obtained aftera bifurcation opening a lake in a sea.

DEGREE OF A NONINVERTIBLE MAP: Maxi-mum of rank-one preimages generated by the map.

CONTACT BIFURCATION: Bifurcation involvingthe contact between the boundaries of different re-gions. For instance, the contact between the bound-ary of a chaotic attractor and the boundary of itsbasin of attraction or the contact between a basinboundary and a critical curve LC.

AGGREGATION: The situation obtained after twoor more disconnected components of a basin form asingle connected component.

EXTERNAL BOUNDARY: Boundary of the im-mediate basin (and other basin components if theyexist) obtained by taking out the lakes.

CUSP POINT: (1) Point on the critical curve LC

where three first rank preimages coincide. (2) Re-pelling node belonging to the external basin bound-ary, the eigenvalues, η1, η2, of which satisfy η1 > 1,η2 < −1 and |η2| > |η1|.SHARP FRACTAL BOUNDARY: Basin boundary,the external boundary of which is made up of arcshaving a fractal structure containing an arbores-cent sequence of preimages of a cusp point. Thusinfinitely many cusp points belong to the boundary.

1. Introduction

Mappings are simple models that have been ex-tensively studied as independent objects of interest[Mira, 1987] or as ingredients of other more com-plex systems [Kapral, 1985; Crutchfield & Kaneko,1987]. The unimodal one-dimensional case is wellunderstood and the main results are summarizedin [Collet & Eckmann, 1980] and in [Mira, 1980,1987]. Two-dimensional endomorphisms are oftenobtained in different fields and, up to now, theyhave been mainly investigated by numerical simu-lations and analytical approach [Mira et al., 1996].

Attractors and time evolution in two-dimensional mappings can be analyzed by a setof measures such as Lyapunov exponents, powerspectrum, invariant measures and fractal dimen-

sions [Parker & Chua, 1989]. In the past years,an analytical instrument, furnished by the criticalcurves, has been introduced [Gumowski & Mira,1980, and references therein]. This tool allows tostudy the bifurcations and some geometrical prop-erties of basins, particularly the phenomenon offractalization [Mira et al., 1996; Abraham et al.,1997].

Basins constitute an interesting object of studyby themselves. If a color is given to the basin ofeach attractor, we obtain a colored figure, which isa phase-plane visual representation of the asymp-totic behavior of the points of interest. The strongdependence on the parameters of this colored figuregenerates a rich variety of complex patterns on theplane and gives rise to different types of basin frac-talization as a consequence, for instance, of contactbifurcations between a critical curve segment andthe basin boundary. Taking into account the com-plexity of the matter and its nature, the study ofthese phenomena can be carried out only via the as-sociation of numerical investigations guided by fun-damental considerations that can be found in [Miraet al., 1996].

Different models of two-dimensional coupledmaps are scattered in the literature of several fields(physics, engineering, biology, ecology and eco-nomics). See, for example [Kaneko, 1983; Yuanet al., 1983; Hogg & Huberman, 1984; Van Biskirk& Jeffries, 1985; Schult et al., 1987; De Sousa Vieiraet al., 1991] and [Aicardi & Invernizzi, 1992]. Inthese works the symmetries, dynamics, bifurcationsand transition to chaos are investigated and inter-preted within the standards of the theory of dynam-ical systems. The role played by critical curves indetermining the properties and global bifurcationsof basins of a double logistic map has been studiedin [Gardini et al., 1994].

In this paper, we explain the basin behaviorof a two-dimensional mapping proposed initially asa coupled pair of logistic oscillators [Lopez-Ruiz &Perez-Garcia, 1991]. There, its dynamics, stabilityand attractors were presented for some range of theparameter in parallel with the study of two othermodels built under similar insights. The metricand statistical properties of that system were alsocomputed and explained in [Lopez-Ruiz & Perez-Garcia, 1992]. It is our objective in the presentwork to continue the investigation of that systemand to analyze the fractalization and parameter de-pendence of basins by using the technique of criticalcurves.

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Complex Patterns on the Plane 289

The plan is as follows. Section 2 is devoted tothe recalls of symmetry properties and local stabil-ity for the parameter range of interest. In Sec. 3some notions concerning the critical curves andtheir application to the present case are given. InSec. 4 we explain the different kinds of fractaliza-tion that are found in our system. The last sectioncontains the discussion and conclusion.

2. Symmetry and Bifurcations

A logistic map whose parameter µn is not fixed,xn+1 = µnxn(1 − xn), but itself follows a dynam-ics forced to remain in the interval [1, 4] has a spe-cific dynamics [Lopez-Ruiz, 1991]. The existenceof a nontrivial fixed point at each step n ensuresthe nontrivial evolution of the system. Thus, thedifferent choices of µn give a wide variety of dy-namical behaviors. For instance, the applicationof this idea produces the on–off intermittence phe-nomenon when µn is chosen random [Platt et al.,1993] or the adaptation to the edge of chaos whenµn is a constant with a small time perturbation[Melby et al., 2000]. Other systems built under thismechanism are models (a)–(c) presented in [Lopez-Ruiz, 1991]. In these cases µn is forced to followa logistic dynamics expanded to the whole inter-val [1, 4]. The result corresponds to three differenttwo-dimensional endomorphisms of coupled logisticmaps.

We will concentrate our attention in model (b).This application can be represented by Tλ : R2 →R2, Tλ(xn, yn) = (xn+1, yn+1). It takes the form:

xn+1 = λ(3xn + 1)yn(1 − yn) ,

yn+1 = λ(3yn + 1)xn(1 − xn) ,(1)

where λ is a real and adjustable parameter. In thefollowing, we shall write T instead of Tλ as the de-pendence on the parameter λ is understood.

In this section, some results on the behavior ofT gathered in [Lopez-Ruiz & Perez-Garcia, 1991]are recalled and some new bifurcation and dynami-cal aspects not collected in that publication are ex-plained. The nomenclature is maintained.

2.1. Symmetry

This model has reflection symmetry P through thediagonal ∆ = (x, x), x ∈ R. If P (x, y) = (y, x)then T commutes with P :

T [P (x, y)] = P [T (x, y)] .

Note that the diagonal is T -invariant, T (∆) = ∆.In general, if Ω is an invariant set of T , T (Ω) = Ω,so is P (Ω) due to the commutativity property:T [P (Ω)] = P [T (Ω)] = P (Ω). It means that if pi,i ∈ N is an orbit of T , so is P (pi), i ∈ N. Infact, if some bifurcation happens in the half planebelow the diagonal, it remains in the above halfplane, and vice versa. The dynamical properties ofthe two halves of phase space separated by the di-agonal are interconnected by the symmetry. Alsoif the set Γ verifies P (Γ) = Γ, so does T (Γ). Thenthe T -iteration of a reflection symmetrical set con-tinues to keep the reflection symmetry through thediagonal.

2.2. Fixed points, 2-cycles andclosed invariant curves

We focus our attention on bifurcations playing animportant role in the dynamics, those happening inthe interval −1.545 < λ < 1.0843. In this range,there exist stable attractors for each value of λ andit makes sense to study their basins of attraction.

The restriction of T to the diagonal is a one-dimensional cubic map, which is given by the equa-tion xn+1 = λ(3xn + 1)xn(1 − xn). Thus the so-lutions of xn+1 = xn are the fixed points on thediagonal,

p0 = (0, 0) ,

p3 =1

3

1 −(

4 − 3

λ

)12

, 1 −(

4 − 3

λ

)12

,

p4 =1

3

1 +

(

4 − 3

λ

)12

, 1 +

(

4 − 3

λ

)12

.

For −1 < λ < 1, p0 is an attractive node. For allthe rest of parameter values, p0 is a repelling node.If λ < 0, the fixed points p3,4 are repelling nodes.For 0 < λ < 0.75, p3,4 are not possible solutions.When λ = 0.75 a saddle-node bifurcation on the di-agonal generates p3,4. For 0.75 < λ < 0.866, p3 is asaddle point and p4 is a node. In this parameter in-terval, the whole diagonal segment between p3 andp4 is the locus of points belonging to heteroclinictrajectories connecting the two fixed points.

When λ =√

3/2 ∼= 0.866 the point p4 changesits stability via a pitchfork bifurcation generating

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two new stable fixed points p5,6 outside the diag-onal. These points are obtained by solving thequadratic equation λ(4λ + 3)x2 − 4λ(λ + 1)x + 1 +λ = 0. The solutions are:

p5 =

(

2λ(λ + 1) +√

λ(λ + 1)(4λ2 − 3)

λ(4λ + 3),

2λ(λ + 1) −√

λ(λ + 1)(4λ2 − 3)

λ(4λ + 3)

)

,

p6 =

(

2λ(λ + 1) −√

λ(λ + 1)(4λ2 − 3)

λ(4λ + 3),

2λ(λ + 1) +√

λ(λ + 1)(4λ2 − 3)

λ(4λ + 3)

)

.

For λ = 0.957, these two symmetric points lose sta-bility via a Neimark–Hopf bifurcation. Each pointp5,6 gives rise to a stable closed invariant curve. Thesize of these symmetric invariant curves grow whenλ increases into the interval 0.957 < λ < 1, and,for some values of λ, frequency locking windows areobtained.

The period two cycles of the form (x, 0) ↔(0, y) play an important role when λ < 0, and arefound by solving the cubic equation: λ3x3−2λ3x2+(λ3 + λ2)x + 1 − λ2 = 0. The solutions are

p1 =

(

λ − 1

λ, 0

)

↔ p2 =

(

0,λ − 1

λ

)

,

p7 =

(

(λ + 1) −√

(λ + 1)(λ − 3)

2λ, 0

)

↔ p8 =

(

0,(λ + 1) +

√

(λ + 1)(λ − 3)

2λ

)

,

p9 =

(

(λ + 1) +√

(λ + 1)(λ − 3)

2λ, 0

)

↔ p10 =

(

0,(λ + 1) −

√

(λ + 1)(λ − 3)

2λ

)

.

Observe that the restriction of the map T 2 to theaxes is invariant and reduces to the double logis-tic map xn+1 = f2(xn) with f(x) = λx(1 − x), sothat its dynamics is completely known and it givesrise to a known cyclic behavior on the axes for themap T .

The 2-cycle (p1, p2) exists for every parametervalue and is unstable for every value of λ.

The pair of 2-cycles (p7, p8) and (p9, p10) has alimited existence to the parameter intervals: λ > 3or λ < −1. For λ = 3, they appear simultaneouslyby a pitchfork bifurcation from the period 2 orbit(p1, p2). They are always unstable for λ > 3. In therange −1.45 < λ < −1, they are stable. This pair of2-cycles grows from the origin p0 when λ = −1. Itis the result of a codimension-two bifurcation givingrise also to a repelling 2-cycle in the direction of theinvariant diagonal.

2.3. Transition to chaos

We follow the presentation of the former section.Remark that the considered system presents twodifferent routes to chaos depending on if λ > 0 or ifλ < 0.

λ > 0: For λ slightly larger than 1, the twoclosed invariant curves approach the stable invari-ant set of the hyperbolic point p4 on the diagonal[Fig. 1(a)]. At first sight, for λ 1.029, the sys-tem still seems quasi-periodic but a finer analysisreveals the fingerprints of chaotic behavior. Ef-fectively, the crossing of the invariant curves with

LC(2)−1 produces a folding process in the two invari-

ant sets [Fig. 1(b)] cf. [Frouzakis et al., 1997], whichgives rise to the phenomenon of weakly chaotic ringswhen the invariant set intersects itself [Fig. 1(c)](cf. [Mira et al., 1996, p. 529]). The existence ofweakly chaotic ring is also confirmed by the posi-tivity of the largest Liapunov exponent for the con-sidered parameter values as computed and shownin Fig. 3(a) of [Lopez-Ruiz & Perez-Garcia, 1992].For λ ≈ 1.032, the tangential contact of the twosymmetric invariant sets with the stable set of thesaddle p4 on the diagonal leads to the disappear-ance of those two weakly chaotic rings. Just af-ter the contact, infinitely many repulsive cycles ap-pear due to the creation of homoclinic points anda single and symmetric chaotic attractor appears[Fig. 2(a)]. For 1.032 < λ < 1.0843, this chaoticinvariant set folds strongly around p4, and the dy-namics becomes very complex [Fig. 2(b)]. Whenthe limit value λ = 1.084322 is reached, the chaoticarea becomes a tangent to its basin boundary, theiterates can escape to infinity and the attractor dis-appears by a contact bifurcation [Mira et al., 1987,Chap. 5] [Fig. 9(d)].

λ < 0: The pair of 2-cycles (p7, p8) and (p9, p10)are the germ of two period doubling cascades ofbifurcations, respectively, one below the diagonal

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xn

yn

LC(2)

λλλλ=1.03

LC(3)

LC-1(3)

LC-1(2)

(∆∆∆∆)

(a)

LC(2)

xn

yn

λλλλ=1.03

LC-1(2)

(∆∆∆∆)

(b)

LC(2)

xn

yn

λλλλ=1.0318

LC-1(2)

(∆∆∆∆)

(c)

Fig. 1. (a) Attractive closed invariant curves for λ = 1.03, and (b) Enlargment of (a); (c) Weakly chaotic rings limited bysegments of critical curves LCn.

and the other above it. Thus, for λ = −1.45, apair of stable 4-cycles is created on the axes whenthe 2-cycle pairs (p7, p8) and (p9, p10) lose theirstability, after which an eigenvalue of each 2-cyclecrosses through the value −1. For λ = −1.51, each

4-cycle bifurcates and a stable 8-cycle pair grow out-side the axes. For λ = −1.51, a new flip bifurca-tion generates the pair of 16-cycles, and so on untilthe dynamics becomes chaotic. For λ = −1.5131,the cascade of flip bifurcations is completed and a

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xn

yn

λλλλ=1.083

LC(2)

LC(3)

LC-1(3)

LC-1(2)

(∆∆∆∆)

(a)

xn

yn

λλλλ=1.083

LC(2)

(∆∆∆∆)

(b)

Fig. 2. (a) Symmetric chaotic attractor for λ = 1.083. (b) Complex folding process around p4 for λ = 1.08.

xn

yn

λλλλ = -1.52

LC(1)

LC(3)

LC-1(3)

LC-1(2)

LC-1(2)

(a)

xn

yn

λλλλ = -1.52

LC(3)

LC-1(3)

xn=0

(b)

Fig. 3. (a) Pair of chaotic areas for λ = −1.52 as a result of two period doubling cascades. (b) Enlargment of a chaotic area.

symmetric pair of chaotic areas has merged inthe region around p0 (see Figs. 3(a) and 3(b) forλ = −1.52). When λ = −1.545 the chaotic areashave contact with their basin boundaries and theydisappear.

3. Stable Attractors

For the sake of clarity, we summarize the

dynamical behavior of the system explained

above when the parameter is inside the interval

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−1.545 < λ < 1.0843. The different parameter regions where the mapping has stable attractors are givenin two tables, one for λ > 0 and the other for λ < 0.

λ > 0:

Number ofInterval Attractors Attractors

0 < λ < 0.75 1 p0

0.75 < λ < 0.866 2 p0, p4

0.866 < λ < 0.957 3 p0, p5,6

0.957 < λ < 1 3 p0, pair of invariant closed curves

1 < λ < 1.03 2 pair of invariant closed curves

1.03 < λ < 1.032 2 pair of weakly chaotic rings

1.032 < λ < 1.0843 1 symmetric chaotic attractor(or frequency lockings)

λ < 0:

Number ofInterval Attractors Attractors

−1 < λ < 0 1 p0

−1.45 < λ < −1 2 (p7, p8), (p9, p10)

−1.5 < λ < −1.45 2 pair of 4-cycles

−1.51 < λ < −1.5 2 pair of 8-cycles

−1.512 < λ < −1.51 2 pair of 16-cycles

−1.5131 < λ < −1.512 2 pair of 2n-cycles, n > 5

−1.545 < λ < −1.5131 2 symmetric pair of chaotic areas

4. Critical Curves

4.1. Definitions and properties

An important tool used to study noninvertible mapsis that of critical curves, introduced by Mira in 1964(see [Mira et al., 1996] for further details). The mapT is said to be noninvertible if there exist pointsin state space that do not have a unique rank-onepreimage under the map. Thus the state space is di-vided into regions, which we call Zi, in which pointshave i rank-one preimages under T. These regionsare separated by the so-called critical curves LC,and so the number of first rank preimages changesonly when the LC are crossed, except for singu-

lar points. LC is the image of a set LC−1. If themap T is continuous and differentiable, LC−1 is asubset of the locus of points where the determinantof the Jacobean matrix of T is zero, and is the two-

dimensional analogue of the set of local extrema ofa one-dimensional map. When LC is crossed rank-one preimages appear or disappear in pairs. (Seealso the glossary for different technical terms usedalong this work).

4.2. Critical curves of T

The map T defined in (1) is noninvertible. It has anonunique inverse. Each point in the state planecan possess up to five rank-one preimages. Thepreimages can only be calculated numerically, bysolving a polynomial of degree five. The criticalcurve LC of T (the locus of points having at leasttwo coincident rank-one preimages) is the imageof LC−1, the locus of points where the Jacobean|DT (p)| vanishes. That is, LC = T (LC−1). TheJacobean matrix of T depends on (x, y) and the

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parameter λ and has the following form:

DT (x, y; λ)

=

[

3λy(1 − y) λ(3x + 1)(1 − 2y)

λ(3y + 1)(1 − 2x) 3λx(1 − x)

]

.

LC−1 is the curve verifying |DT (x, y)| = 0. Itis formed by the points (x, y) that satisfy theequation:

27x2y2 + 3x2y + 3xy2 − 6x2 − 6y2

− 8xy + x + y + 1 = 0 . (2)

LC−1 is independent of parameter λ and isquadratic in x and y. Thus we can solve one ofthe two variables as a function of the other one.For instance, the variable y as function of x givesus:

(−3x2 + 8x − 1)

y =±√

657x4 − 84x3 − 194x2 − 4x + 25

54x2 + 6x − 12.

Numerical calculations allow us to discover that forevery value of x, or equivalently for every value ofy, there are two points belonging to two differentbranches of LC−1 [Fig. 4(a)]. Observe that LC−1 isa quartic curve of four branches, with two horizontal

and two vertical asymptotes. The branches LC(1)−1

and LC(2)−1 have as horizontal asymptote the line

y = 0.419 and the vertical asymptote in x = 0.419.

The other two branches, LC(3)−1 and LC

(4)−1, have the

horizontal asymptote in y = −0.530 and the verti-cal one is the line x = −0.530. The values 0.419and −0.530 are the roots of the polynomial fac-tor, 27x2 + 3x − 6, that multiply the term y2 inEq. (2). It also follows that the critical curve of

rank-1, LC(i) = T (LC(i)−1), i = 1, 2, 3, 4, consists

of four branches. The shape of LC is shown inFigs. 4(b) and 4(c). LC depends on λ and sepa-rates the plane into three regions that are locus ofpoints having 1, 3 or 5 distinct preimages of rank-one. They are respectively named as Zi, i = 1, 3, 5.Observe that the set of points with three preimagesof rank-one, Z3, is not connected and is formed byfive disconnected zones in the plane [see Figs. 4(b)and 4(c)]. The next section is devoted to the studyof this state space organisation.

Remark that LC−1 has the reflection sym-metry through the diagonal: P (LC−1) = LC−1.Then every critical curve of rank-(k + 1),

LCk = T k+1(LC−1), will conserve this symmetry:P (LCk) = LCk.

4.3. Zi-regions

LC separates the plane into seven disjoint and openzones, which are locus of points having distinctnumber of preimages of rank-one [Figs. 4(b) and4(c)]. We name each region depending on the num-ber of preimages of rank-one. Thus a Zi-zone meansthe set of points with i preimages of rank-one. Ob-serve that four arcs of LC-curve divide the diagonal∆ in five intervals, each one associated only withthe Zi-zone including it. Then to know the numberi of preimages of rank-one of each segment on thediagonal, it is necessary to determine the number ofpreimages of rank-one of each Zi-zone of the plane.Then we proceed to this calculation.

The set of rank-one preimages of the diagonalincludes the diagonal itself and another set formedby an hyperbola of two branches. Thus, a point onthe diagonal (x′, x′) can have preimages of rank-oneon the diagonal or on that hyperbola:

(a) Preimages belonging to the diagonal: It followsfrom Eq. (1) that the preimages (x, x) of the point(x′, x′) verify the cubic equation:

3λx3 − 2λx2 − λx + x′ = 0 . (3)

This equation can have one or three solutions de-pending on x′ and λ. We define the limit values:x′

1d ≈ 0.65λ and x′

2d ≈ −0.1λ. If λ > 0, Eq. (3)presents one root if x′ > x′

1d or if x′ < x′

2d, andthree roots if x′

2d < x′ < x′

1d. If λ < 0, Eq. (3) hasone solution if x′ > x′

2d or if x′ < x′

1d, and threesolutions if x′

1d < x′ < x′

2d.

(b) Preimages belonging to the hyperbola: Equa-tion (1) tells us that the image of a point (x, y) ison the diagonal when (x, y) verifies:

y =1 − x

1 + 3x. (4)

Equation (4) represents an hyperbola of twobranches with asymptotes in x = −1/3 and y =−1/3. The point (x′, x′) has preimages (x, y) onthe hyperbola if x′ = λ(3y + 1)x(1 − x). If we in-troduce relation (4) between x and y in the formerexpression we obtain the equation:

4λx2 + (3x′ − 4λ)x + x′ = 0 . (5)

If the radicand of this equation is positive, the point

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xn

yn

LC-1(2)

LC-1(1)

LC-1(3)

LC-1(4)

λλλλ = 0.3

(a)

xn

yn

LC(1)

LC(2)

LC(3)

LC(4)

λλλλ = 0.3

Z1

Z3

Z3

Z3

Z5

Z3

Z3

(b)

xn

yn

LC(1)

LC(2)

LC(3)

LC(4)

λλλλ = -0.3

Z1

Z3

Z3

Z3Z5

Z3

Z3

Z5

(c)

Fig. 4. (a) Critical curves LC(i)−1, i = 1, 2, 3, 4. (b and c) Critical curves LC (i), i = 1, 2, 3, 4, for λ = 0.3 and for λ = −0.3,

respectively. Observe the different Zj-zones, j = 1, 3, 5.

(x′, x′) has two preimages on the hyperbola (4), andif the radicand is negative, it has no preimages onthat hyperbola. The roots of the radicand of Eq. (5)give us the behavior of its sign:

9x′2 − 40λx′ + 16λ2 = 0 . (6)

The roots of this equation are x′

1h = 4λ and x′

2h =0.44λ. Then, if λ > 0, Eq. (5) presents two solu-tions if x′ > x′

1h or x′ > x′

2h, and it has no solutionswhen x′

2h < x′ < x′

1h. If λ > 0, Eq. (5) has twosolutions if x′ > x′

2h or if x′ < x′

1h, and no solutionswhen x′

1h < x′ < x′

2h.

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Thus the coordinates of the points that markthe frontier between the different Zi-zones on the di-agonal are: x′

1d ≈ 0.65λ and x′

2d ≈ −0.1λ, x′

1h = 4λand x′

2h ≈ 0.44λ. For example, the origin p0 isalways in the Z5-zone. It is located into the in-terval that determines x′

1d and x′

2d as extremes,then p0 has three preimages on the diagonal. Alsoit is outside of the interval determined by the ex-tremes x′

1h and x′

2h, then p0 has two preimages onthe hyperbola (4). In fact, its preimages are (1, 1),(−1/3, −1/3) and p0 itself on the diagonal, and(1, 0), (0, 1) on the hyperbola. Taking and connect-ing the results of former paragraphs (a) and (b), thenumber of preimages of rank-one of a point (x′, x′)on the diagonal can be summarized in the followingtables.

λ > 0:

Interval Number of Preimages

x′ < x′

2d 3

x′

2d < x′ < x′

2h 5

x′

2h < x′ < x′

1d 3

x′

1d < x′ < x′

1h 1

x′ > x′

1h 3

λ < 0:

Interval Number of Preimages

x′ < x′

1h 3

x′

1h < x′ < x′

1d 1

x′

1d < x′ < x′

2h 3

x′

2h < x′ < x′

2d 5

x′ > x′

2d 3

To summarize, the many different zones Zi sep-arated by branches of the critical curve LC canbe classified, according to the nomenclature es-tablished in [Mira et al., 1996], in two differentschemes. For λ > 0, the map (1) is of the typeZ3 − Z5 Z3 − Z1 ≺ Z3 and for λ < 0, the map isof the type Z3 Z1 − Z3 ≺ Z5 − Z3.

Observe in Figs. 4(b) and 4(c) that the branchesof the critical curve LC are not smooth. They ex-hibit two cusp points that coincide with x′

1h andx′

2h. A cusp point can be considered as a singularpoint of LC where three first rank preimages coin-cide. These points give information on the sheetstructure of the map foliation. In the case of the

cusp point x′

1h the two preimages on the hyperbola(4) and the preimage on the diagonal are mixedup at the point (−1, −1). Equivalently, x′

2h hastwo preimages, (0.91, 0.91) and (−0.52, −0.52) onthe diagonal and the other three preimages coin-cide with the point (1/3, 1/3). For the map T thereis no bifurcation involving the cusp points and as-sociated with the qualitative change of the sheetorganization. The only bifurcation occurs in λ = 0,and gives rise to an inversion in the plane of thesheet structure [Figs. 4(b) and 4(c)].

5. Basin Fractalization

5.1. General properties

The set D of initial conditions that converge to-wards an attractor at finite distance when the num-ber of iterations of T tends toward infinity is thebasin of the attracting set at finite distance. Whenonly one attractor exists at finite distance, D is thebasin of this attractor. When several attractors atfinite distances exist, D is the union of the basins ofeach attractor. The set D is invariant under back-ward iteration T−1 but not necessarily invariant byT : T−1(D) = D and T (D) ⊆ D. A basin may beconnected or nonconnected. A connected basin maybe simply connected or multiply connected, whichmeans connected with holes. A nonconnected basinconsists of a finite or infinite number of connectedcomponents, which may be simply or multiply con-nected [Mira et al., 1994].

The closure of D includes also the points ofthe boundary ∂D, whose sequences of images arealso bounded and lay on the boundary itself. Ifwe consider the points at infinite distance as an at-tractor, its basin D∞ is the complement of the clo-sure of D. When D is multiply connected, D∞ isnonconnected, the holes (called lakes) of D beingthe nonconnected parts (islands) of D∞. Inversely,nonconnected parts (islands) of D are holes of D∞

[Mira et al., 1996].In Sec. 3, we explained that the map (1) may

possess one, two or three attractors at a finite dis-tance. The points at infinity constitute the fourthattractor of T . Thus, if a different color for eachdifferent basin is given we obtain a colored pat-tern with a maximum of four colors. In the presentcase, the phenomena of basin boundary fractaliza-tion have their origin in the competition betweenthe attractor at infinity (whose basin is D∞) and

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the attractors at finite distance (whose basin isD). The competition between the attractors atfinite distance can also fractalize the set D (seeSec. 5.3.2). When a bifurcation of D takes place,some important changes appear in the colored fig-ure, and, although the dynamical causes cannot beclear, the colored pattern becomes an important vi-sual tool to analyze the behavior of basins.

We study in detail, and with the help of graph-ical representations, the mechanisms of basin frac-talization for the map T . The changes of the shapeand evolution of D are given as a function of thesign of the parameter λ. Two scenarios are distin-guished: Scenario I for λ > 0 and Scenario II forλ < 0.

5.2. Scenario I, λ > 0

When 0 < λ < 1.084, we observe important modifi-cations of the area of D, the basin of the attractingset at finite distance. Its size decreases when λ in-creases. Two main contact bifurcation phenomenagiving rise to a fractal basin boundary take placeinto this interval: fractal islands disaggregation if0.39 < λ < 0.61 and infinitely many converging se-quences of lakes if 1.08 < λ < 1.084. We proceedto explain the role played by critical curves in thebifurcations giving rise to these phenomena.

5.2.1. Fractal islands disaggregation,0.39 < λ < 0.61

When λ increases from λ ≈ 0.39 to λ ≈ 0.61,D undergoes two bifurcations connected – noncon-nected and nonconnected – connected. When Dis nonconnected, it is made up of the immediatebasin D0 containing the single attractor (p0) andinfinitely many small regions without connection(islands). This disaggregation is the result of in-finitely many contact bifurcations, which are ex-plained in the next paragraph. Such phenomenoncan be also found in some quadratic Z0 − Z2

maps and has been explained in [Mira & Rauzy,1995]. We explain for decreasing λ the correspond-ing mechanisms [Figs. 6(a)–6(h)].

When λ ≈ 0.61 [Figs. 6(e)–6(h)], a first rankisland is created due to the intersection of LC (1)

with the two symmetric narrow “tongues” of D lo-cated in the third quadrant, and having the linesx = −1/3 and y = −1/3 as asymptotes. Thosepoints of D crossing LC (1) from the below Z3-zoneto the Z5-zone acquire two new preimages of rank-one. These preimages appear as shaped islands in-

tersecting LC(1)−1 in the middle Z3-zone. Higher rank

preimages of these seminal islands give rise to newsmaller islands. The same mechanism of creationof new islands comes from the intersection of the

xn

yn

λλλλ = 0.3

D

(∆∆∆∆)

xn = 0

yn = 0

P2

P1

(a)

xn

yn

λλλλ = 0.3

xn = 0

(b)

Fig. 5. (a) Basin D for λ = 0.3. Observe the “tongues” having as asymptotes the lines: x = −1/3, y = −1/3, x = 0 andy = 0. (b) Details of the fractal structure of the tongues.

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existing islands with LC arcs [Figs. 6(c) and 6(d)].Thus islands crossing LC (3) from the middle Z3-zone to the Z5-zone undergo a contact bifurcationthat creates a new pair of rank-one islands inter-

secting LC(3)−1.

Then, [Figs. 6(a) and 6(b)] islands evolving inthe plane from the middle Z3-zone to the Z1-zoneacross LC (2) give place to the aggregation of a pair

of rank-one islands located on both sides of LC(2)−1.

This cascade of bifurcations, considered when λ

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(a)

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op/q'r s tvu

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(b)

xy

z

|~

U

')

') U'

¡¢£'¤)¥ ¦§

¨© ª«d¬f ®D¯F°

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³O´

µ¶·¸

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(c)

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(d)

Fig. 6. (a) Collective islands disaggregation of basin D for λ = 0.396. (b) Observe the fractal pattern of islands on both

sides of the three like-axes of symmetry: LC(1)−1, LC

(2)−1 and LC

(3)−1 for λ = 0.45. (c–f) First rank islands generated from the

intersection of LC (1) with basin “tongues”, for λ = 0.6, (g–h) λ = 0.7.

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xn

yn

LC(1)

LC(2)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

LC-1(4)

λλλλ=0.6

Z1

Z3

Z3 Z5

LC(4)

(e)

xn

yn

λλλλ=0.6

LC(1)

Z3

Z5

(f)

xn

yn

λλλλ=0.7

LC(1)

LC(2)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

LC-1(4)

(g)

xn

yn LC

(1)

λλλλ=0.7

Z5 Z3

(h)

Fig. 6. (Continued )

increases, generates a fractal pattern of noncon-nected islands in the vicinity of the immediate basinD0 and on both sides of its three like-axes of sym-

metry: LC(1)−1, LC

(2)−1 and LC

(3)−1 [Fig. 6(c)]. When

λ ≈ 0.396, the two headlands of D∞ enclosed be-tween LC (1) and the two former “tongues” disap-pear and an infinitely complex aggregation occurs:the frontier of every island contacts with the bound-

ary ∂D0 simultaneously and a connected basin D isobtained finally [see contact at point A in Fig. 6(a)].

Observe that D has now a nonsmooth bound-ary [Figs. 5(a) and 5(b)]. The stable set of theperiod-two saddle (p1, p2) and its preimages formpart of it. To find the analytical expression ofthe boundary is a difficult task but in this particu-lar case, we can obtain by direct visual inspection

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of the figures, some partial and approximated in-formation. The coordinates axes are T 2-invariant:T reduces on the axes to the double logistic mapxn+1 = f2(xn) with f(x) = λx(1 − x), so that itgives rise to a known dynamics on the axes for the

map T . Two apparently unbounded “tongues” es-caping toward infinity are in the asymptotical direc-tion of the lines x = −1/3 and y = −1/3 as can beinspected in Fig. 5(a). These two lines are the rank-one preimages of the coordinate axes, which are also

xn

yn

LC(1)

LC(2)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

LC-1(4)

λλλλ = 0.9

Z1

Z3

Z3

Z5

Z1

LC(1)

(a)

xn

yn

LC(1)

LC(2)

LC(3)

LC-1(2) LC-1

(1)

LC-1(3)

LC-1(4)

D0

D∞∞∞∞

λλλλ = 1

Z1

Z3

Z3 Z5

(b)

xn

yn

(∆∆∆∆)

λλλλ = 1.

(c)

xn

yn

LC(1)

LC(2)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

LC-1(4)

D0

D∞∞∞∞

λλλλ = 1.03

Z1

Z3

Z3

Z5 S1

-1

S2-1

S1

S2

(d)

Fig. 7. (a) Basin D formed by the square D0 ≡ [1, 0]×[0, 1], which contains the attractors, and four small like-triangle regionslinked to the square by four narrow arms for λ = 0.9. (b) D has five disconnected components for λ = 1. (c) Enlargment of(b). (d) Observe the two semicircular shaped regions of D∞, S−1

1 and S−12 , intersecting D0 for λ = 1.03.

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asymptotes of other two similar “tongues”. Then ananalytical approximation of the frontier of the for-mer “tongues” can be obtained by backward itera-tion of the boundary of the latter “tongues”. Aftersome straightforward calculation, it can be foundthat the map T 2 generates stable dynamics on theaxes when the initial conditions (x0, 0) or (0, y0)are lying on the intervals, 1 − 1/λ < x0 < 1/λ or1 − 1/λ < y0 < 1/λ, for λ < 1. We call theseintervals δ and δ′, respectively. Preimages of differ-ent ranks higher than 1 of points in the vicinity ofδ and δ′ generate a set of curves that come closeto or can give us a rough idea of the fractal struc-ture of the basin boundary at the “tongues”. Forinstance, the piece of curve, x = −1/3 + 1/(9λy2)with 0 < y < λ−3, close to the asymptote x = −1/3,is a preimage of rank-two of points in the vicinityof the positive part of δ′ [Fig. 5(b)].

5.2.2. Finite disaggregation andinfinitely disconnected basin,0.7 < λ < 1.032

When 0.7 < λ < 1, D seems to be formed bythe square D0 ≡ [1, 0] × [0, 1], which contains the

xn

yn

λλλλ = 1

D1

D2

C2

C1

P6

P5

Fig. 8. D0 for λ = 1. The two colors correspond to the non-connected basins of two attractive closed invariant curves.

attracting set at finite distance, and four small like-triangled regions linked to the square by four nar-row arms [Fig. 7(a)]. These arms shrink when λapproaches 1, and disappear for λ = 1 when the

xn

yn LC

(2)

LC(3)

LC-1(2)

LC-1(3)

λλλλ = 1.0803

Z1

Z3 Z5

H01(1)

H02-1

H01

H02(1)

(a)

xn

yn LC

(2)

LC(3)

LC-1(2)

LC-1(3)

λλλλ = 1.082

Z1

Z3

Z5

H01(1)

H01(21)

H02(1)

H02(21)

P5

P6

(b)

Fig. 9. (a) D0 for λ = 1.0803: first rank holes H(1)01 and H

(1)02 (and higher rank preimages holes) of the bays H01 and H02,

respectively. (b–c) New arborescent sequence of holes created from the crossing of H(21)01 and H

(21)02 with LC (2). (d) Chaotic

attractor and its basin. (e) Strange repulsor for λ = 1.084.

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xn

yn

λλλλ = 1.082

V

LC(1)

LC(3)

LC-1(1)

LC-1(3)

LC-1(2) LC

(2)

(c)

xn

yn

λλλλ = 1.083

LC(2)

LC(3)

LC-1(2)

LC-1(3)

(d)

xn

yn

λλλλ = 1.084

LC(2)

LC(3)

LC-1(2)

LC-1(3)

(e)


origin p0 undergoes a transcritical bifurcation. Themain part of D is then a disconnected pattern offive components: the square D0, a triangle-shapedcomponent located in a Z3 neighborhood of the ver-tex point (−1/3, −1/3) (preimage of rank-one ofthe point p0), and the three triangle-shaped regions

that are preimages of rank-one of the latter compo-nent [Fig. 7(b)]. As there is no region Z0, an infinitesequence of rank-one preimages exists and gives risein this case to the appearance of much smaller is-lands of preimages. They can be observed by theenlargement of figures [see some in Fig. 7(c)].

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Points (1, 0) and (0, 1) cross through LC (2)

when λ = 1. When λ > 1, two regions appearS1

1 and S12 , which are part of D∞ and are located in

a Z3 zone. The rank-one preimages of S11 and S1

2 ,respectively S−1

1 and S−12 are two new semicircular

regions and intersect LC(2)−1. They are located in the

vicinity of points (1, 0.5) and (0.5, 1), preimages of(1, 0) and (0, 1) [Fig. 7(d)].

When 0.866 < λ < 1.032, T possesses two orthree attractors (see Sec. 3). If the basin of eachattractor is colored in a different way, we observethat each basin included in D0 is nonconnected andis made up of infinitely many components. Theboundary of each connected component belongs tothe stable set of unstable points [Gardini et al.,1994]. Figures 7(c) and 8 show D0 for λ = 1. Inthis case, two closed invariant curves attract the dy-namics and two different colors define the infinitelydisconnected basin pattern.

5.2.3. Infinitely many convergingsequences of lakes, 1.08 < λ < 1.084

When λ increases, the two semicircular shapedzones of D∞, S−1

1 and S−12 , located in the imme-

diate basin D0 ≡ [1, 0] × [0, 1] in the neighbor-hood of points (1, 0.5) and (0.5, 1), grow in size.For λ ≈ 1.0801, the basin undergoes a contactbifurcation. D∞ crosses through LC (2) and twobays (headlands of D∞), H01 and H02, are cre-

ated. Their rank-one preimages, H(1)01 and H

(1)02 ,

are holes (lakes) intersecting LC(2)−1 into the middle

Z3-region [Fig. 9(a)]. Rank-one preimages of the

latter holes generate four new lakes in D0, H(21)0i

and H(22)0i , i = 1, 2 [Fig. 9(b)]. Preimages with in-

creasing rank give rise to an arborescent sequenceof lakes. Inside D0, the accumulation points of thisinfinite sequence of holes are the two unstable focip5,6 and their rank-one preimages inside the imme-diate basin. Outside D0, the other two rank-onepreimages of p5,6 are limit points of the arborescentsequence of holes generated on the bigger triangleisland with vertex point (−1/3, −1/3) [Fig. 9(c)].

When λ ≈ 1.0806, H(21)01 and H

(21)02 cross

through LC (2) [Fig. 9(b)]. This new contact bi-furcation is the germ of a new arborescent andspiralling sequence of lakes converging towards thesame accumulation points. When λ increases val-ues, new holes intersect LC (2) and give rise to new

holes crossing through LC(2)−1 and new sequences of

lakes converging towards the unstable foci p5,6 and

their preimages. Due to the fact that the preimageshave a finite number of accumulation points, thestructure is not fractal [Fig. 9(b)]. A similar phe-nomenology has been found and studied in Z0 −Z2

maps [Mira et al., 1994].When λ increases (λ ≈ 1.0835), the chaotic at-

tractor, which is limited by arcs of LCn curves, isdestroyed by a contact bifurcation with its basinboundary. A new dynamical state arises. The in-finite number of unstable cycles and their rank-nimages belonging to the existing chaotic area be-fore the bifurcation define a strange repulsor whichmanifests itself by chaotic transients [Fig. 9(e)]. Forλ ≈ 1.085 the basin pattern disappears definitively.

5.3. Scenario II, λ < 0

When −1.545 < λ < 0, the size of the domain ofbounded iterated sequences decreases when λ de-creases. Two different qualitative phenomena arisein this case for decreasing λ: countable self-similardisaggregation if −1.2 < λ < −0.85 and sharpboundary fractalization if −1.545 < λ < −1.45.We analyze the patterns generated in the followingparagraphs.

5.3.1. Countable self-similardisaggregation, −1.2 < λ < −0.85

In the interval −1 < λ < 0, the origin p0 is the onlyattractor. The basin D is connected. It presentsseveral “tongues”: unbounded segments escapingapparently towards infinity and having the verti-cal lines x = −1/3 and x = 1, and the horizontalstraight lines y = −1/3 and y = 1 as asymptotes(see Sec. 5.2.1). When λ ≈ −0.85, LC (2) crossesthrough the boundary ∂D, creating a small head-land H0 of D∞ trapped between LC (2) itself and∂D [Figs. 10(a)–10(c)]. A rank-one preimage of thissmall region creates a big lake H−1

0 in D intersect-

ing LC(2)−1. Other lakes of smaller size appear on

the tongues as preimages of H−10 . The sequence of

rank-n preimages of these holes generates an infi-nite sequence of lakes along the diagonal and on thetongues [Figs. 10(c)]. The basin becomes multiplyconnected.

New headlands of D∞, H1 and H2, are createdbetween LC (2) and ∂D when λ decreases. It givesrise to new arborescent sequences of holes locatedclose to the former ones [Figs. 10(d) and 10(e)].Afterwards, the aggregation of neighbor headlandsof D∞ gives rise to the aggregation of their image

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lakes in the interior of D. Other bays open in thesea, due to the crossing of LC (2) through ∂D, giv-ing rise to the transformation of their image lakesin roadsteads [Figs. 10(f)–10(h)]. For λ ≈ −1.05,the combination of these two phenomena breaks

D in a set of infinitely many components locatedon the diagonal and on the tongues, as a conse-quence of that LC (2) is totally contained in D∞ andthe former headlands of D∞ are open in the sea.The new basin is a set with a countable number of

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self-similar parts. If we make enlargments of someparts, they seem identical to the former in the se-quence [Figs. 10(i)–10(k)]. It means that the basinD can be seen as a collection of arbitrarily smallpieces, each of which is an exact scaled down ver-sion of the entire basin.

If ln is the size of component n of this sequence,the scaling factor r defined as r ≈ limn→∞(ln/ln+1)takes the value r ≈ 7.43.

To calculate this scaling factor we remark thatthe sequence of diagonal preimages xn (with x1 = 1and xn > 1 for all n > 1) of the origin p0(x0 = 0)

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are located one by one in the core of each compo-nent of the self-similar basin. For λ = −1, this se-quence xn can be calculated with the restrictionof recurrence (1) on the diagonal:

xn = 3x3n+1 − 2x2

n+1 − xn+1 , (7)

where x0 = 0, x1 = 1, . . . . The accumulation pointx∞, where the sequence xn converges, is obtainedas the fixed point of the former recurrence:

xn = xn+1 → x∞ =1 +

√7

3≈ 1.21 .

Observe that x∞ is the coordinate of the fixed pointp4 on the diagonal. That is, the sequence xn isan heteroclinic orbit between p0 and p4. The dif-ference between two consecutive terms, xn − xn−1,gives us, approximately, the size ln of the nth com-ponent of the self-similar basin. The scaling factorr is obtained by linearizing Eq. (7) around the fixedpoint x∞. The result is:

r ≈ limn→∞

lnln+1

= limn→∞

xn − xn−1

xn+1 − xn

= 9x2∞

− 4x∞ − 1 ≈ 7.43

Observe that r is the diagonal multiplier of theunstable fixed point p4.

5.3.2. Sharp fractal boundary,−1.545 < λ < −1.45

Two stable period two cycles are created for λ =−1, as a consequence of a codimension-two bifur-cation in p0 (see Sec. 3). These 2-cycles persiststable in the parameter range −1.45 < λ < −1.Afterwards the system undergoes a double cascadeof flip bifurcations that gives rise to a symmet-ric pair of chaotic areas for λ ≈ −1.5131 (seeFig. 3). Then two colors define the basin of attrac-tors at finite distance, D = D1 ∪D2, in the interval−1.545 < λ < −1, being D1 and D2 the basin ofeach attractor [Fig. 10(f) and following).

For λ ≈ −1.45, the critical curve LC (3) inter-sects the boundary separating D1 and D2, creatingheadlands of one basin between LC (3) itself and theboundary of the other basin [Figs. 11(a) and 11(b)].After this contact bifurcation a cascade of islandsof basin D1 into D2, and vice versa, is generated.A fractal pattern is created. Observe that the frac-talization is located in very determined zones of D.Each of these regions is limited apparently by two

xn

yn

λλλλ = -1.46

LC(1)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

Z3

Z3

Z5

H02

H01

(a)

xn

yn

LC(3)

LC-1(1)

LC-1(3)

λλλλ = -1.46

Z3

Z5 H01

(b)

Fig. 11. Sharp boundary fractal of basin D (explanation in the text): (a) λ = −1.46, (b) λ = −1.46 (enlargement),(c) λ = −1.5, (d) λ = −1.5 (enlargement), and, (e) λ = −1.54, (f) λ = −1.54 (enlargement).

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xn

yn

LC(1)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

λλλλ = -1.5

Z3

Z5

Z3

Z3

(c)

xn

yn

LC(3)

LC-1(3)

λλλλ = -1.5

LC-1(1)

Z3

Z5

(d)

xn

yn

LC(1)

LC(3)

LC-1(2)

LC-1(1)

LC-1(3)

λλλλ = -1.54

Z5

Z3

Z3

(e)

xn

yn

λλλλ = -1.54

LC(3)

(f)


smooth curves (whose equations are unknown forus) intersecting on the diagonal. The origin andits preimages are on the frontier of these regionsas accumulation points [Figs. 11(a) and 11(b)]. Al-though islands of D1 are spread out over D2, andvice versa, in the fractal zones, these are not rid-

dled basins in the sense defined in [Alexander et al.,1992]. We can always find on those regions a disk,which intersects one of the basins in a set of positivemeasure but does not intersect the other basin.

A similar behavior is repeated for λ ≈ −1.485but with the basin of infinity. LC (3) crosses through

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the boundary of this basin and a cascade of islandsof D∞ is created in the regions where D1 and D2

are intermingled [Figs. 11(c) and 11(d)]. Observethat now the origin and the sequence of its preim-ages are cusp points located on the frontier of D.For λ decreasing, many different D∞-islands havecontacts with the external boundary of D and theyopen in the sea [Fig. 11(e)]. This arborescent se-quence of roadsteads gives place to a pattern witha sharp fractal boundary.

As in Sec. 5.2.3, for λ ≈ −1.545, a contact be-tween ∂D and the frontier of the chaotic area de-stroys the attractor and a strange repulsor takes itsplace. The dynamics derives towards infinity givingrise to the disappearance of the basin.

6. Conclusion

Basins of a noninvertible two-dimensional mappingformed by two symmetrically coupled logistic mapshave been analyzed. Two approaches have beenused to study the bifurcations leading to their frac-talization. On one hand, the geometrical and col-ored representations of basins as a phase-plane vi-sual instrument allow us a qualitative focus, and,by the other hand, a more precise and analyticalapproach is furnished by the tool of critical curves.

The map is of degree five and a region withfive first rank preimages exists for every value ofthe parameter λ. Also there are regions with oneand three rank-one preimages. The many differ-ent zones Zi separated by branches of the criti-cal curve LC can be classified, according to thenomenclature established in [Mira et al., 1996], inZ3 − Z5 Z3 − Z1 ≺ Z3 type for λ > 0 and,Z3 Z1−Z3 ≺ Z5−Z3 for λ < 0. The symbols “”and “≺” indicate the existence of two cusp pointslocated on the frontiers separating the Z1 and Z5

zones from two of the Z3 zones. We point out thenonexistence of a zone with zero preimage, then ev-ery point on the plane has an infinite sequence ofpreimages.

Although the system has only one parameter,it gives rise to a rich basin behavior, which gener-ates many different complex patterns on the plane.Considering the complexity of all this phenomenol-ogy and being aware of the difficulty of such anattempt, we have organized every different typeof basin fractalization present in this system inthe following way: fractal islands disaggregation

for 0.39 < λ < 0.61, finite disaggregation and in-

finitely disconnected basin for 0.7 < λ < 1.032,infinitely many converging sequences of lakes for1.08 < λ < 1.084, countable self-similar disaggrega-

tion for −1.2 < λ < −0.85 and sharp fractal bound-

ary for −1.545 < λ < −1.45. Each of these frac-talization types have common features with thosepresent in the simplest and well-studied case ofZ0 −Z2 maps. We want to call attention on the setof patterns found in the interval −1.2 < λ < −0.85.It constitutes a new and easily calculable exampleof a self-similar countable set on the plane. It canbe seen as a collection of arbitrarily small pieces,each of which is an exact scaled down version ofthe entire basin. The scaling factor has been calcu-lated by analytical direct inspection of recurrenceequations.

Finally we remark that, in general, basin frac-talization does not imply the existence of a chaoticattractor in the system. In fact, two of the frac-talization types analyzed in our mapping present aperiodic underlying dynamical behavior.

Acknowledgments

We would like to thank Prof. Mira and Dr. Taha fortheir helpful comments. We also acknowledge thesignificant improvement of this article after care-fully including all the referee’s remarks. R. Lopez-Ruiz wishes to thank the group Systemes Dy-namiques at INSA (Toulouse), where most part ofthis work was done, for his kind hospitality, and theProgram Europa of CAI-CONSI+D (Zaragoza) forfinancial support.

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mappings-2003

Documents

connected basin

basin of attraction

invariant subset

theattracting set

chaotic attractor

attracting setis

finite disaggregation

complex patterns onthe