locally and globally riddled basins in two coupled piecewise-linear maps · 2013-06-06 · locally...

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Locally and globally riddled basins in two coupled piecewise-linear maps Yu. Maistrenko, 1 T. Kapitaniak 2 and P. Szuminski 2 1 Institute of Mathematics, Academy of Sciences of Ukraine, 3 Tereshchenkivska St., Kiev 252601, Ukraine 2 Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland ~Received 1 May 1997; revised manuscript received 21 July 1997! The chaos synchronization and riddled basins phenomena are discussed for a family of two-dimensional piecewise linear endomorphisms that consist of two linearly coupled one-dimensional maps. Rigorous condi- tions for the occurrence of both phenomena are given. Different scenarios for the transition from locally to globally riddled basins and blowout bifurcation have been identified and described. @S1063-651X~97!09511-1# PACS number~s!: 05.45.1b I. INTRODUCTION Two identical chaotic systems x n 11 5 f ( x n ) and y n 11 5 f ( y n ), x , y PR, evolving on an asymptotically stable cha- otic attractor A , when one-to-one coupling x n 11 5 f ~ x n ! 1d 1 ~ y n 2x n ! , y n 11 5 f ~ y n ! 1d 2 ~ x n 2 y n ! ~1! is introduced, can be synchronized for some ranges of d 1,2 PR, i.e., u x n 2 y n u 0 as n ‘@1–13#. In the synchronized regime the dynamics of the coupled system ~1! is restricted to one-dimensional invariant sub- space x n 5 y n , so the problem of synchronization of chaotic systems can be understood as a problem of stability of the one-dimensional chaotic attractor A in two-dimensional phase space @14,15#. The basin of attraction b ( A ) is the set of points whose v-limit set is contained in A . In Milnor’s definition @16# of an attractor the basin of attraction need not include the whole neighborhood of the attractor, i.e., we say that A is a weak Milnor attractor if b ( A ) has a positive Lebesgue measure. For example, a riddled basin @14,15,17–20#, which has re- cently been found to be typical for a certain class of dynami- cal systems with one-dimensional invariant subspace @such as x n 5 y n in the example ~1!#, has positive Lebesgue mea- sure but does not contain any neighborhood of the attractor. In this case the basin of attraction b ( A ) may be a fat fractal, so that any neighborhood of the attractor intersects the basin with positive measure, but may also intersect the basin of another attractor with positive measure. The dynamics of the system ~1! is described by two Lyapunov exponents. One of them describes the evolution on the invariant manifold x 5 y and is always positive. The sec- ond exponent characterizes the evolution transverse to this manifold and is called transversal. If the transversal Lyapunov exponent is negative, the set A is an attractor, at least in the weak Milnor sense. When the transversal Lyapunov exponent is negative and there exist trajectories in the attractor A , which are transver- sally repelling, A is a weak Milnor attractor with a locally riddled basin, i.e., there is a neighborhood U of A such that in any neighborhood V of any point in A , there is a set of points in V øU of positive measure that leave U in a finite time. The trajectories that leave neighborhood U can either go to the other attractor ~attractors! or after a finite number of iterations be diverted back to A . If there is a neighborhood U of A such that in any neighborhood V of any point in U there is a set of points of positive measure that leave U and go to the other attractor ~attractors!, then the basin of A is globally riddled. In this paper we identify and describe different ways of transition from locally to globally riddled basins and discuss the conditions for the basin of attraction to be one or the other. We consider the dynamics of a four-parameter family of a two-dimensional piecewise linear noninvertible map F 5 H f l , p ~ x n ! 1d ~ y n 2x n ! : x n 11 5 px n 1 l 2 S 1 2 p l D S U x n 1 1 l U 2 U x n 2 1 l U D 1d 1 ~ y n 2x n ! f l , p ~ y n ! 1d ~ x n 2 y n ! : y n 11 5 py n 1 l 2 S 1 2 p l D S U y n 1 1 l U 2 U y n 2 1 l U D 1d 2 ~ x n 2 y n ! , ~2! where l , p , d 1,2 PR, which consists of two identical linearly coupled one-dimensional maps being the generalization of the skew tent map. Chaotic attractors of the skew tent map have been considered in @21–27#. In comparison with the maps studied in @14,15,17,18#, our map ~2! has the advantage that, as we show below, conditions for the occurrence of riddled basins can be given analytically. We use the map ~2! as a test model of coupled chaotic systems. The outline of this paper is as follows. In Sec. II we recall the definitions of different types of attractor stability and PHYSICAL REVIEW E DECEMBER 1997 VOLUME 56, NUMBER 6 56 1063-651X/97/56~6!/6393~7!/$10.00 6393 © 1997 The American Physical Society

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Page 1: Locally and globally riddled basins in two coupled piecewise-linear maps · 2013-06-06 · Locally and globally riddled basins in two coupled piecewise-linear maps Yu. Maistrenko,1

PHYSICAL REVIEW E DECEMBER 1997VOLUME 56, NUMBER 6

Locally and globally riddled basins in two coupled piecewise-linear maps

Yu. Maistrenko,1 T. Kapitaniak2 and P. Szuminski2

1Institute of Mathematics, Academy of Sciences of Ukraine, 3 Tereshchenkivska St., Kiev 252601, Ukraine2Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland

~Received 1 May 1997; revised manuscript received 21 July 1997!

The chaos synchronization and riddled basins phenomena are discussed for a family of two-dimensionalpiecewise linear endomorphisms that consist of two linearly coupled one-dimensional maps. Rigorous condi-tions for the occurrence of both phenomena are given. Different scenarios for the transition from locally toglobally riddled basins and blowout bifurcation have been identified and described.@S1063-651X~97!09511-1#

PACS number~s!: 05.45.1b

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I. INTRODUCTION

Two identical chaotic systemsxn115 f (xn) and yn115 f (yn), x,yPR, evolving on an asymptotically stable chotic attractorA, when one-to-one coupling

xn115 f ~xn!1d1~yn2xn!,

yn115 f ~yn!1d2~xn2yn! ~1!

is introduced, can be synchronized for some ranges ofd1,2PR, i.e., uxn2ynu→0 asn→` @1–13#.

In the synchronized regime the dynamics of the coupsystem ~1! is restricted to one-dimensional invariant suspacexn5yn , so the problem of synchronization of chaotsystems can be understood as a problem of stability ofone-dimensional chaotic attractorA in two-dimensionalphase space@14,15#.

The basin of attractionb(A) is the set of points whosev-limit set is contained inA. In Milnor’s definition @16# ofan attractor the basin of attraction need not include the whneighborhood of the attractor, i.e., we say thatA is a weakMilnor attractor if b(A) has a positive Lebesgue measuFor example, a riddled basin@14,15,17–20#, which has re-cently been found to be typical for a certain class of dynacal systems with one-dimensional invariant subspace@suchas xn5yn in the example~1!#, has positive Lebesgue measure but does not contain any neighborhood of the attracIn this case the basin of attractionb(A) may be a fat fractal,

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so that any neighborhood of the attractor intersects the bwith positive measure, but may also intersect the basinanother attractor with positive measure.

The dynamics of the system~1! is described by twoLyapunov exponents. One of them describes the evolutionthe invariant manifoldx5y and is always positive. The second exponent characterizes the evolution transverse tomanifold and is called transversal. If the transverLyapunov exponent is negative, the setA is an attractor, atleast in the weak Milnor sense.

When the transversal Lyapunov exponent is negativethere exist trajectories in the attractorA, which are transver-sally repelling,A is a weak Milnor attractor with a locallyriddled basin, i.e., there is a neighborhoodU of A such thatin any neighborhoodV of any point inA, there is a set ofpoints inVùU of positive measure that leaveU in a finitetime. The trajectories that leave neighborhoodU can eithergo to the other attractor~attractors! or after a finite number ofiterations be diverted back toA. If there is a neighborhoodUof A such that in any neighborhoodV of any point inU thereis a set of points of positive measure that leaveU and go tothe other attractor~attractors!, then the basin ofA is globallyriddled.

In this paper we identify and describe different waystransition from locally to globally riddled basins and discuthe conditions for the basin of attraction to be one orother. We consider the dynamics of a four-parameter famof a two-dimensional piecewise linear noninvertible map

F5H f l ,p~xn!1d~yn2xn!: xn115pxn1l

2 S 12p

l D S Uxn11

l U2Uxn21

l U D1d1~yn2xn!

f l ,p~yn!1d~xn2yn!: yn115pyn1l

2 S 12p

l D S Uyn11

l U2Uyn21

l U D1d2~xn2yn!,~2!

of

alld

where l ,p,d1,2PR, which consists of two identical linearlycoupled one-dimensional maps being the generalizationthe skew tent map. Chaotic attractors of the skew tent mhave been considered in@21–27#. In comparison with themaps studied in@14,15,17,18#, our map~2! has the advantag

ofp

that, as we show below, conditions for the occurrenceriddled basins can be given analytically. We use the map~2!as a test model of coupled chaotic systems.

The outline of this paper is as follows. In Sec. II we recthe definitions of different types of attractor stability an

6393 © 1997 The American Physical Society

Page 2: Locally and globally riddled basins in two coupled piecewise-linear maps · 2013-06-06 · Locally and globally riddled basins in two coupled piecewise-linear maps Yu. Maistrenko,1

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6394 56YU. MAISTRENKO, T. KAPITANIAK, AND P. SZUMINSKI

give rigorous conditions for asymptotic and weak Milnstability of the synchronized chaotic attractor of map~2!.These analytical conditions allow us to obtain a twdimensional bifurcation diagram of coupled maps~2!. Twodifferent types of transitions from locally to globally riddlinbasins and their connection with global bifurcations of tbasins of attraction that we identify are studied in Sec.Section IV describes recently identified features ofblowoutbifurcation from locally riddled and globally riddled basinsFinally, we summarize our results in Sec. V.

II. BIFURCATION DIAGRAM FOR STABILITY OFATTRACTORS OF CHAOTIC SYNCHRONIZATION

Consider the two-dimensional map of the plane (x,y) intoitself ~2!. When

~ l ,p!PP5H l .1,2l

l 21,p<21J ,

the one-dimensional mapf l ,p has two symmetrical attractorG (2),@21,0# andG (1),@0,1#, which are cycles of 2m cha-otic intervals ~so-called 2m-piece chaotic attractors!. De-pending on parametersl andp, m can be any positive integer. DenotePm as a subregion ofP, whereGm

(6) is a period-2m cycle of chaotic intervals. Bifurcation curves for the trasition Gm

(6)→Gm11(6) can be found in@28#.

For the mapFl ,p , each set

A5Am~6 !5$x5yPGm

~6 !%

is a one-dimensional chaotic invariant set that may or mnot be an attractor in the plane (x,y). We distinguish threetypes of attraction. The various notions of attractors invotwo properties:~i! that it attracts nearby trajectories and~ii !that it cannot be decomposed into smaller attractors.shall concentrate on the first property sinceA has every-where dense trajectories. Thus the definitions given hshould be completed by some minimality condition in ordto be generally valid.

Definition 1. The setA is anasymptotically stable attractor if for any sufficiently small its neighborhoodU(A)there exists its neighborhoodV(A) such that if (x,y)PV(A) then Fn(x,y)PU(A) for any nPZ1 and distancer„Fn(x,y);A…→0, n→`.

Definition 2. The setA is a weak Milnor attractorif itsbasin of attractionb(A) has a positive Lebeague measureR2.

Note that in@28#, an asymptotically stable attractor is rferred to as an attractor with the property of strong stabiand a weak Milnor attractor is referred to as an attractor wthe property of weak stability. Among weak Milnor attrators ~which are not asymptotically stable! we can distinguishtwo classes depending on whether or not the basinb(A) hasa full measure in a neighborhoodU(A). In the first case, i.e.when measure@b(A)ùU#5measureU, the attractorA canbe referred to as aMilnor attractor.

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A. Asymptotic stability of the attractor

In our previous study@28# of map ~2! some preliminary

analytical conditions for thed5def

d11d2 parameter regions inwhich Am5Am

(6) , mPZ1, is ~i! an asymptotically stable attractor and~ii ! a weak Milnor attractor were obtained. In thfollowing we generalize these conditions, being both necsary and sufficient.

Theorem.The attractorAm is asymptotically stable if andonly if ( l ,p)PPm

(k),P ~m50,1, . . . ; k52,3, . . .! and theconditions

u l 2puamup2duam11,1,

u l 2dukam1~21!m~k21!up2dukam111~21!m11~k21!,0 ~3!

are fulfilled, wheream is a sequence of integer numbers dfined asa050, a151, and

am5am2112am22 ~m52,3, . . .!

and subregionsPm(k) are given as

Pm~2!5H ~ l ,p!PP: 2

l m11

l m,pm,2

11A114l m2

2l mJ ,

Pm~k!5H ~ l ,p!PP: 2

l mk 21

~ l m21!l mk21,pm,2

l mk2121

~ l m21!l mk22J ,

k53,4, . . . , ~4!

where

l m5 l am1~21!mpam111~21!m11

,

pm5 l ampam11. ~5!

In Fig. 1 regionsA1 of asymptotic stability of attractorAare indicated in black; the boundaries ofA1 are obtainedfrom relations~3!, in which inequality signs were replaceby equality. Numerical calculations have been done forp52 l , i.e., for the tent mapf l ,2 l , when formulas~4! and~5!are transformed to

FIG. 1. Regions of asymptoticA1 and weak MilnorA2 stabilityfor map ~2!; p52 l . RegionsA1 andA2 are shown in black andgray, respectively.

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56 6395LOCALLY AND GLOBALLY RIDDLED BASINS IN TW O . . .

Pm~2!5H 1, l 52p<2: &, l m,

l m11

l mJ ,

Pm~k!5H 1, l 52p<2:

l mk2121

~ l m21!l mk22, l m,

l mk 21

~ l m21!l mk21J

~k53,4, . . .!, ~48!

where

l m52pm5 l 2m. ~58!

B. Weak Milnor stability of the attractor

Conditions for the chaotic invariant setAm5Am(6) to be a

weak Milnor attractor were obtained using an invariant msurem5m l ,p of the mapf l ,p and can be given in the form

l'5@am1~21!mm# lnu l 2du

1@am111~21!m11m!] lnup2du,0, ~6!

where

m5m l m ,pmH uxu,1

l J .

RegionsA2 of the weak Milnor stability of the attractorA5Am

(6) are shown in Fig. 1 in gray; boundaries ofA2 areobtained from condition~6! by replacing the inequality by anequality.

Unfortunately, generally we do not have analytical epression for the densityr of the measurem as it can beexplicitly found only in exceptional cases. For example,the moment of the first homoclinic bifurcation of the fixepoint of f l ,p @i.e., whenl 5p/(12p2)# the density functionr5r(x) is piecewise constant with two break pointsx561/l such thatm$uxu,1/l %51/p2.

Conditions~6! shows the negativeness of the transverLyapunov exponentl' of the typical trajectory in the attractor A. In numerical calculations shown in Fig. 1, we used texpression forl' from Birkhoff’s ergodic theorem

l'5 limN→`

1

N (n50

`

lnu f 8~xn!2du,

where$xn5 f n(x0)% is a typical trajectory of the mapf l ,p inthe attractorA.

C. Locally and globally riddled basins

Suppose that the transversal Lyapunov exponentl' of atypical trajectory on the attractorA5Am

(6) is negative, butstill there exist trajectories onA, which are transversallyrepelling, i.e., we are in the gray regionA2 in Fig. 1. Thenone can simply check that the attractorA is not stable in aLyapunov sense as there exists a neighborhoodU such thatany neighborhoodV contains a positive Lebesgue measuset of points that leaveU under the iterations ofFl ,p . In thiscase the basin of attraction ofA is locally riddled@2#.

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e

Definition 3. A set A is an attractor with a locallyriddled basin of attraction if there is a neighborhoodU of Asuch that in any neighborhoodV of any points inA there isa set of points inVùU of positive Lebesgue measure thleavesU in a finite time.

This riddling property has apparently a local charactertakes place in a sufficiently small neighborhoodU5U(A)and gives no information about the further behavior of ttrajectories after their leave fromU. In the model underconsideration, different situations related to this global dnamics property take place. Two of the most spread of thare the following~note that the mapFl ,p is noninvertible!.

(i) Locally riddled basin. After leaving neighborhoodU(A), almost all ~in a measure sense! points return toU.Then some of them, after a finite number of iterations, leaU again and so on. The dynamics of such trajectories psents nonregular temporal ‘‘bursting’’: The trajectory spensome time~usually long enough! near attractorA until itgoes away; then, after some other time~usually shortenough! it returns to the neighborhood. This behavior dscribes a sort ofon-off intermitencywide spread in practiceIf l' is negative but not too close to zero, the bursts wusually stop: Finally, the trajectory will be attracted by thattractorA. If l' is negative and sufficiently close to zerthe bursts can never stop.

(ii) Globally riddled basin.After leavingU(A), a positivemeasure set of points goes to another attractor~s! or to infin-ity. This another attractor may be, for instance, an attractpoint cycle or attracting cycle of chaotic two-dimension~2D! sets.

Only in the globally riddled case can the basin of attration b(A) have a riddled structure of a fat fractal as a subof R2, which means the following: The neighborhood of apoint (x,y)Pb(A) is filled by a positive Lebesgue measuset of points (x,y) that are attracted by another attractor~orother attractors!.

In Figs. 2~a! and 2~b! we show the examples of locallriddled basins forl 51.3, p522, and d15d250.6 @Fig.2~a!# and globally riddled basins forl 51.3, p522, andd1

5d250.725 @Fig. 2~b!#. The attractorA(1) of Fig. 2~a! at-tracts almost all points from its neighborhood, but it is nLyapunov stable. In Fig. 2~b! the attractorA(1) is not asymp-totically stable as in any its neighborhood there exists a ptive measure set of points that goes to another attractorA(2)

or to infinity. These properties are clearly visible at the elargements shown in Figs. 2~c!–2~e!. In Fig. 2~c! we noticethat all points from the neighborhood (0.5,1)3(0.5,1) go tothe attractorA1. However, some of these points temporaleave this neighborhood. In Fig. 2~d! we presented the pointin the neighborhood ofA(1) that under iterations of map~2!leave the neighborhoodU5(0.4,1.1)3(0.4,1.1)~white area!and points~gray area! that do not leave this neighborhoodFinally, almost all points from both areas converge toA(1),but those from the white region have to follow a longer paNote that the locally riddled basin contains also a zemeasure set of points~including unstable periodic ones! thatare not attracted byA(1) ~see@29–31#!. But when a com-puter simulation is processed, points not attracted byA(1)

cannot usually be seen on the screen and one can arrive awrong conclusion that a locally~but not globally! riddled

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6396 56YU. MAISTRENKO, T. KAPITANIAK, AND P. SZUMINSKI

FIG. 2. AttractorsA1 andA2 of map~2!; l 51.3 andp522. ~a! d15d250.6, ~b! d15d250.725, and~c! and~d! enlargements of~a!and~b!, respectively. Points that under iterations of map~2!, temporally leave the neighborhood (0.4,1.1)3(0.4,1.1) are indicated in whitein ~c!. Basins of attraction of attractorsA1 andA2 are shown, respectively, in darker and lighter gray and the basin of attraction at inis indicated in white.

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s

s

ew

i-

basin can represent a set open inR2 that includes a neighborhood of the attractorA(1). Finally, in Fig. 2~e! we canobserve a typical case of global riddling. The basin of attrtor A(1) ~darker gray! is riddled by the basin of attractorA2

~lighter gray!.

III. TRANSITION FROM LOCALLYTO GLOBALLY RIDDLED BASINS

For map~2! we observed two types of bifurcation leadinto the transition from locally to globally riddled basins~l -gbifurcation!. The examples of these bifurcations are shownFigs. 3~a!–3~f!. The first type ofl -g bifurcation is shown inFigs. 3~a!–3~c! for l 51.95 andp521.95. Before the bifur-cation ford15d2521 @Fig. 3~a!# the basins of both attractors A(1) andA(2) contain a neighborhood of attractors ecept of a zero measure set. After the bifurcation ford15d2

-

n

520.9 basins of attractorsA(1) andA(2) are riddled by thebasin of the attractor at infinity as shown in Figs. 3~b! and3~c! @particularly in the closeup in Fig. 3~c!#. The similartype of l -g bifurcation occurs in the case shown in Figs. 2~a!and 2~b!, where basins of both attractorsA(1) andA(2) areriddled by each other.

A different type ofl -g bifurcation is shown in Figs. 3~d!–3~f! for l 52p5&. As in the first type before bifurcation@Fig. 3~d!, d15d2520.94#, the basins of both attractorA(1) andA(2) are locally but not globally riddled~only at-tractorA(1) is shown!. After the bifurcation, new attractorA1 and A2 form @Fig. 3~e! and the closeup in Fig. 3~f!, d1

5d250.935# in the neighborhood ofA(1) and the basin ofA(1) becomes globally riddled by the basins of these nattractors.

These types ofl -g bifurcations were observed to be typ

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ay

56 6397LOCALLY AND GLOBALLY RIDDLED BASINS IN TW O . . .

FIG. 3. l -g bifurcations of map~2!: ~a!–~c! bifurcations of the first type and~d!–~f! bifurcations of the second type.~a!–~c! l 51.95 andp521.95: ~a! d15d2521, ~b! d15d2520.9, and~c! enlargement of~b!. ~d!–~f! l 5& and p52&: ~d! d15d2520.94, ~e! d1

5d2520.935, and~f! enlargement of~e!. Basins of attraction of attractorsA1 andA2 are shown, respectively, in darker and lighter grand the basin of attraction at infinity is indicated in white.

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cal for map~2!. We can summarize their properties in thfollowing definitions.

Definition 4.The l -g bifurcation is of the first~or inner!type if in the local and global riddling the basins of the saattractors are involved.

Definition 5. The l -g bifurcation is of the second~orouter! type if in the local and global riddling the basinsdifferent attractors are involved.

IV. BLOWOUT BIFURCATION

The bifurcation of losing the weak Milnor stability habeen called a blowout bifurcation@15,17#. Recently, Ashwin,Buescu, and Stewart@32# suggested to use the notion of critcality for the classification of blowout bifurcations, analgous to that for the bifurcation of fixed points in invariasubspaces. In@32# the blowout bifurcation atn5n0 wascalled subcritical if there exists an unstable invariant setBn ,

e

namely, the boundary dividing the basins of attractorA andthe attractor at infinity forn,n0 , which is destroyed onpassing throughn5n0 . If for n.n0 there exists a family ofattractorsAn that correspond to the on-off intermittent attrators, then the blowout bifurcation is called supercritical; s@32# for examples and further discussion.

From our model, roughly speaking, we can conclude tthe blowout bifurcation is subcritical if in the bifurcatiomoment the basin of attractorA is globally riddled by thebasin of the attractor at infinity. The blowout bifurcationsupercritical if in the bifurcation moment the basin ofA islocally riddled or it is globally riddled by the basins of somattractors among which there are no attractors at infinity.

Note that in the subcritical case the blowout bifurcationreally like an explosion giving rise to the immediate disapearance of attractorA: It cannot be seen in computer simulations, so its basin becomes a zero-measure set. Cardidifferent is the scenario of the bifurcation in the supercritic

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6398 56YU. MAISTRENKO, T. KAPITANIAK, AND P. SZUMINSKI

case. In this case the bifurcation consists in the transifrom the 1D to 2D attractor~s! and it does not resemble aexplosion, as the density of the new 2D attractor~s! changes‘‘slowly’’ when the parameter is in a neighborhood of a trasition point.

The examples of blowout bifurcations for map~2! at l51.3 and p522 are shown in Figs. 4~a! and 4~b!. Ournumerical calculations allow identification of the two typof blowout bifurcation. Figure 4~a! shows the blowout bifur-cation in the case where before the bifurcation attracA(1) andA(2) were locally [email protected], Fig. 2~a!#.After the blowout bifurcation the synchronized statex5y isno longer stable and we observe two hyperchaotic twdimensional attractorsA(1)8 and A(2)8 @d15d250.5, Fig.4~a!#. An unstable invariant set, the boundary between ba

FIG. 4. Attractors after blowout bifurcations:l 51.3 and p522. ~a! d15d250.5, before bifurcation attractorsA1 and A2

are locally riddled as shown in Fig. 2~a!, and~b! d15d250.8, be-fore bifurcation attractorsA1 andA2 are globally riddled as shownin Fig. 2~b!.

,

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of attraction of the attractorsA(1) andA(2) is not destroyed.Shortly after the bifurcation, we observed the intermittenbetween the unstable synchronized attractorsA(1) andA(2)

and hyperchaotic attractorsA(1)8 andA(2)8, respectively, inthe form of a ‘‘burst.’’

In Fig. 4~b! we observe the blowout bifurcation of attrators A(1) and A(2), the basins of which are mutually globally riddled by each other~i.e., they are intermingled according to the definition in@18#! @d15d250.725, Fig. 2~b!#.After the [email protected], Fig. 4~b!# we observed aunique two-dimensional hyperchaotic attractorA. In thiscase the boundary between basins of attraction of the attors A(1) and A(2) has been destroyed before bifurcatio~transition from the locally to the globally riddled basin!, butthe boundary between the sum of basinsb(A(1))ùb(A(2))and the basin of the attractor at infinity is not destroyed y

V. CONCLUSION

Depending on the coupling parameters, the synchronistate of map~2! is characterized by different types of stabity. In the case of weak synchronization, the attractor ofsynchronized state is not asymptotically stable and twoferent states of riddling are possible. Conditions for weMilnor and asymptotic stability of the synchronized chaoattractor of map~2! are given analytically in a rigorous form

We showed that thel -g bifurcation, i.e., the transitionfrom locally to globally riddled basins, can occur in twrecently identified different way. In the first~inner! type ofbifurcation in local and global riddling the same attractoare involved. When at the transition of the bifurcation poinnew attractor~s! is ~are! formed and the basins of this~these!new attractor~s! riddle the basins of the initial attractor~s! wehave the second type ofl -g bifurcation.

The blowout bifurcation, i.e., the transition from weastability to weak instability, can be sub- or supercritical.before bifurcation attractors are globally riddled by thetractor at infinity the bifurcation is subcritical; otherwise itsupercritical. The observed properties ofl -g and blowoutbifurcations seem to be typical for a class of system withlower-dimensional invariant manifold and important fstudies of chaos synchronization problems.

ACKNOWLEDGMENT

This work was supported by the KBN~Poland! underProject No. 7TO7A 039 10.

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