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About Two Mechanisms of Reunion of Chaotic Attractors
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Chaos\ Solitons + Fractals Vol[ 8\ No[ 7\ pp[ 0262Ð0289\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9859Ð9668:87 ,08[99¦9[99
Pergamon
PII] S9859!9668"87#99969!7
About Two Mechanisms of Reunion of Chaotic Attractors YURI MAISTRENKO and IRINA SUSHKO Institute of Mathematics National Academy of Sciences of Ukraine\ Kiev\ Ukraine and
LAURA GARDINI$ Dipartimento di Metodi Quantitativi\ University of Brescia\ and Istituto di Scienze Economiche\ University of Urbino\ 50918 Urbino\ Italy "Accepted 13 February 0887#
Abstract*Considering a family of two!dimensional piecewise linear maps\ we discuss two di}erent mech! anisms of reunion of two "or more# pieces of cyclic chaotic attractors into a one!piece attracting set\ observed in several models[ It is shown that\ in the case of so!called {contact bifurcation of the 1nd kind|\ the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor[ In the case of so!called {contact bifurcation of the 0st kind|\ the reunion is a result of a contact of the attractor with its basin boundary which is fractal\ including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ INTRODUCTION
Our purpose is to describe some mechanisms of reunion of two or more pieces of a cyclic chaotic attractor in R1 into a one!piece chaotic attracting set[ When observing such a reunion in several nonlinear models\ one can often classify it in one of the following two possibilities] 0[ It is due to a so!called {contact bifurcation of the 1nd kind| ð0Ð3Ł\ giving rise to a direct reunion of the pieces of the attractor by pairs\ and the resulting shape of the attractor is a junction\ two by two\ of the pieces existing before separately "Fig[ 0#[ 1[ Another type of reunion is due to a so!called {contact bifurcation of the 0st kind| "see the same references as above#] two or more pieces of some cyclic chaotic attractor\ being situated far enough from each other\ under small parameter variation\ can suddenly give rise to the appearance of bursts*rare points of the trajectories in some region of the phase space including the old pieces of attractors ðFig[ 1"a#Ł[ The number of the rare points increases as the iterations are continued\ and they belong to a new chaotic attractor ðFig[ 1"b#Ł[ In some sense\ chaotic attractors begin to {feel each other| when the distance between them is still _nite[ When calculating trajectories in a system with {rare points|\ their appearance seems rather unusual as we cannot see\ in phase space\ any other attracting set except for the two
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Fig[ 0[ An example of the {{contact bifurcation of the 1nd kind|| when the reunion of two pieces of an attractor occurs due to the _rst homoclinic bifurcation of a saddle cycle belonging to the basin boundary of the attractor[
"or more# pieces of attractor under consideration[ Due to this\ it can be supposed that\ in the region between the attractors\ a repeller should exist[ Contact bifurcation with it causes the rare points[ Moreover\ such repeller have a complicated topological structure\ so that an observer cannot usually see any order in the appearance of such rare points and this visual randomness has both phase space and temporal nature[ Really\ as described in ð3Ł "ch[ 4#\ besides the two types recalled above\ there are several kinds of contact bifurcations between chaotic attractors and their basins\ with di}erent dynamical e}ects[ However\ we shall restrict our interest to the two mentioned above\ because these are the two kinds of contact bifurcations involved in the piece!wise linear maps which interest us in this work[ The family F of maps that we consider is the family of two!dimensional piecewise linear continuous maps\ as a function of four parameters\ given by] F GF 0 ] F j J GF1] f
0 1 0 10 1 0 1 0 10 1 0 1 x l 9 Ł y 9 a
x \ y
x l−b b Ł y a−d d
y²x¦0\ "0#
x −b ¦ \ yrx¦0\ y a−d
where both the above mentioned scenarios can be realized\ depending on the parameters of the family[ Note that the map F can be considered in some sense as a 1Dim analog of the skew tent map[ We shall consider the case in which F has two saddle _xed points\ O and P\ which are at the boundary of a closed bounded trapping region\ Z\ of the phase plane "where trapping means mapped into itself] F"Z#W ¹ Z#[
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Fig[ 1[ An example of {{contact bifurcation of the 0st kind|| when the reunion of two pieces of an attractor is via appearance of rare points of the trajectories in the regions between pieces\ when they are still far from each other[ The reunion occurs due to a boundary crisis] each chaotic piece has a contact with its fractal basin boundary[
It should be noted that the case in which a dynamical system has two saddle _xed points occurs quite often in the modern theory of nonlinear dynamics[
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We shall see that\ in the _rst case\ the contact bifurcation "without rare points# is a result of the _rst homoclinic bifurcation of the saddle P[ In the second case\ the reunion bifurcation "with rare points# is caused by the contact of the attractors with the boundary of its basin of attraction\ which includes the stable set of the saddle P after its _rst homoclinic bifurcation[ Therefore\ two di}erent situations can occur in our model] in the _rst case\ the _rst homoclinic bifurcation of P gives rise to immediate reunion of the attractors[ In the second one\ the homoclinic bifurcation of the saddle P creates an hyperbolic invariant set L "on the boundary of the basins# without reunion of attractors[ The stable set W s"L# of the set L has Cantor like structure[ It plays the role of {separator| of the basins of the di}erent pieces of the cyclic attractor[ Under parameter variation\ a bifurcation occurs when the chaotic attractors have a contact with W s"L#[ At this bifurcation value\ the basin of each piece of the cyclic attractor comes to be intermingled\ assuming that the preimages of each contact point are dense everywhere on the {attractor|[ Soon after the contact bifurcation\ the old attractor disappears due to the in_nitely many intersections with W s"L#\ and a new attractor arises[ The structure of the paper is as follows[ In Section 1\ we describe the map F[ A so!called {saddleÐsaddle| area is de_ned in the parameter space when both the _xed points of the map F are saddles[ Correspondingly\ a bounded invariant area is found in the phase space[ Section 2 is devoted to the study of di}erent kinds of attractors of the map F[ Depending on the parameters\ the attractor can be either a cycle of any period or a cyclic chaotic attractor of any period[ We give exact formulae for the boundaries of the regions of existence of the attracting cycles of di}erent periods[ A conjecture is formulated about the order of appearance of the attractors in the system under parameter variation[ It is shown that the case of stability loosing of any point cycle due to the ~ip bifurcation "besides the cycle of period 1# results in the appearance of a cyclic chaotic attractor of doubling period[ In the case in which the cycle of period 1 becomes unstable\ a cyclic chaotic attractor of period 1i can appear\ where i depends on the parameters[ The bifurcations which may then occur consists of the pairwise reunion of the pieces of the attractor to give only a one!piece chaotic set[ The main problem concerns the bifurcation causing the reunion[ It is connected with the _rst homoclinic bifurcation of the saddle P "Section 3# as well as with an invariant hyperbolic set analogous to the Smale horseshoe which appears after the homoclinic bifurcation[ We give a construction of this set in Section 5[ Di}erent mechanisms of reunion of pieces of cyclic chaotic attractors are explained in Section 4\ considering the sequences of bifurcations of the 3!\ 1! and 0!piece chaotic attractors\ as well as those of the 5!\ 2! and 0!piece chaotic attractors[ 1[ DESCRIPTION OF THE MAP[ {SADDLEÐSADDLE AREA| IN THE PARAMETER SPACE[ BOUNDED INVARIANT REGION IN THE PHASE SPACE
Consider the four!parameter family F of two!dimensional piecewise linear continuous maps given in "0#[ The map F consists of two linear maps F0 and F1 de_ned below and above\ respectively\ the straight line LC−0 ""x\y#]y x¦0# which is a line of discontinuity of the Jacobian of the map\ F[ The map\ F\ is one!to!one if d×ab:l and is noninvertible otherwise[ Indeed\ if d³ab:l\ the map\ F\ is of "Z9−Z1# type\ i[e[ each point "x\y#$R1 has two preimages\ provided that y³ax:l¦a^ has zero preimages\ provided that y×ax:l¦a and has a unique preimage if y ax:l¦a[ Thus\ the straight line\ LC ""x\y#]y ax:l¦a#\ is the locus of {critical values| of F\ i[e[ the so!called critical line of rank!0 ð4\ 1\ 3Ł[ Due to these properties\ the map F can be considered as a two!dimensional analog of the skew tent map[ Denote the _xed point of F0 by O "9\9#[ The eigenvalues of F0 are m0 a and m1 l with corresponding eigendirections
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Fig[ 2[ The triangle of stability of the _xed point P in the "b\ d#!parameter plane\ a × 0\ 9 ³ l ³ 0^ D is the region in which P is saddle[
u0 "9\0# and u1 "0\9#[ Let us _x the parameters a and l so that the _xed point O is a saddle] a × 0\ 9 ³ l ³ 0[ The local stable "unstable# manifold of O is denoted by w9"O# "a9"O## which is obviously a segment of the straight line "y 9# ""x 9##[ The _xed point of the map F1 is denoted by P "x9\y9#\ where x9
b"0−a# "0−l#"d−a# ^ y9 [ b"a−0#−"0−d#"0−l# b"a−0#−"0−d#"0−l#
We are interested in a so!called {saddleÐsaddle| area D in the parameter space in which both the _xed points O and P are saddles and\ moreover\ a bounded trapping region Z exists in the phase plane "x\y#[ For any _xed a×0 and 9³l³0\ the region of stability of P "x9\y9# in the "b\d# parameter plane is the triangle P0 bounded by three segments of the following straight lines d −0¦b"0¦a#:"0¦l#\ "d0# d 0¦b"0−a#:"0−l#\ "d1# d "ab¦0#:l\ "d2# The _xed point P becomes a saddle if the parameter point "b\d# intersects either "d0# "the eigenvalue of P crosses through −0# or "d1# "the eigenvalue crosses through 0#[ In the last case\ obviously a bounded invariant area in the phase plane does not exist[ Thus\ the region D\ in the parameters| space we are interested in is de_ned as follows] D ""a\b\d\l#] a×0\ d ³d³minb$R0""d0#\"d1#\"d2##\9³l³0# where d denotes a bifurcation curve of destruction of the trapping region Z which will be de_ned below "Fig[ 2#[ We will restrict ourself to the condition b×9 "at b 9\ the map F is a triangular map and its dynamics are described in ð7Ł#[
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Fig[ 3[ Trapping region Z of the map F\ in the phase plane "x\ y#[
The local stable "unstable# manifold of P is denoted by w9"P# "a9"P## which is a segment of the straight line y Qsx¦Rs denoted as "ys# for short\ "y Qux¦Ru\ denoted as "yu# for short# where Qs
l0¦b−l s d"0−l#−"l0¦b#"0−a#¦l−a ^R ^ b ba−dl¦d¦l−b−0
Qu
l1¦b−l u d"0−l#−"l1¦b#"0−a#¦l−a ^R ^ b ba−dl¦d¦l−b−0
and 9³l0³0\ l1³−0 are eigenvalues of F1] l0 "l−b¦d¦z"b−l−d#1−3"dl−ba##:1\ l1 "l−b¦d−z"b−l−d#1−3"dl−ba##:1[ Assuming now that the parameters belong to the region D\ we shall _nd a bounded region in the phase plane "x\y# which is trapping for F[ Consider the region Z\ bounded by the segments of the four straight lines] "x 9#\ "y 9# "containing\ respectively\ a9"O# and w9"O##\ LC and of the straight line "yu# "containing a9"P##[ The region Z is a quadrilateral with corner points O "9\9#\ A0 LC+"yu#\ B0 "9\a# and C0 "−Ru:Qu\9# "Fig[ 3#[ It is easy to see that if "a\b\d\l#$D and b×9 then the region Z is trapping "i[e[ mapped into itself] F"Z#UZ#[ The bifurcation destroying the invariance of Z takes place when F1"A0# C0[ This bifurcation deter! mines the appearance of the _rst heteroclinic trajectory from the saddle P to the saddle O and also the homoclinic connection for O "see ð1\ 3Ł#[ The curve d in the parameter space corresponds to this last condition[ 2[ ATTRACTORS] ATTRACTING CYCLES AND CYCLIC CHAOTIC ATTRACTORS
Let "a\b\d\l#$D\ b×9 and\ consequently\ the bounded invariant region Z exists in the phase plane[ We shall see that an attractor of the map F\ necessarily belonging to Z\ is either an attracting cycle gn of period n or a cyclic chaotic attractor An\m of period m\ where n\ m can assume any natural value\ depending on the parameters[
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Fig[ 4[ The regions Pn of existence of attracting cycles gn in the "b?\ d#!parameter plane where b? 1b\ at a 0¦09−4 and l 9[2 for n 1\ [ [ [ \06[
We shall _nd the regions Pn$D\ nr1\ such that if parameters belong to Pn\ then an attracting cycle of period n of the map F exists[ Let gn ""x0\y0#\"x1\y1#[\[[\ "xn\yn## be a cycle of period n such that one of its points "let us assume the point "x0\y0## belongs to the region x¦0³y³ax:l¦a\ i[e[\ x0¦0³y0³ax0:l¦a^ and the other points belong to the region 9³y³x¦0\ i[e[\ 9³yi³xi¦0\ i 1[\[[\ n[ The appearance of gn is a result of saddleÐnode bifurcation and it is de_ned by the condition "x0\y0#$LC[ From "x0\y0# F0n−0"F1"x0\y0## at y0 ax0:l¦a we get the locus\ in the parameter space\ of the appearance of gn] dn\e
bl n−1"0−l#"an−0#¦"l n−0#"an−0−0# \ an−1"a−0#"0−ln#
which is a straight line in the "b\d# parameter plane at _xed values for a\ l and n[ Note that the cycle g1 arises at the stability loosing of the _xed point P\ that is d1\e "d0#[ The curve of stability loosing of gn can be found as a result of ~ip bifurcation\ when one of the cycle multiplicators becomes equal to −0[ Also\ this curve is a straight line in the "b\d# parameter plane at _xed a\ l and n] dn\s −
bln−0"an¦0# ¦ [ an−0 an−0"ln¦0# 0
Thus\ the regions Pn$D of existence of attracting cycles gn are] Pn ""a\b\d\l#$D]dn\s³d³dn\e#[ The regions Pn are shown in Fig[ 4 in the "b?\d# parameter plane where b? 1b at a 0¦09−4\ l 9[2 where n 1\[[[\ 06[ Note that\ at _xed values\ a×0\ 9³l³0\ the number of the regions Pn is _nite\ that is n 1\[[[\ n \ where n is a number which depends on a and l[ Moreover\ if al×0\ there is not any attracting cycle of the map F at "a\b\d\l# e D[ But\ if a:0 and l:0\ then n :[
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When the parameter point "b\d# crosses one of the regions Pn through the curve dn\s\ a ~ip bifurcation for gn occurs[ What is the result for the dynamics of the map F< Computer simulations show that this bifurcation gives rise to the appearance of a cyclic chaotic attractor\ of doubling period 1n when nr2 and of period 1k for n 1\ k depending on the parameters[ Further variation of the parameters can give rise to the bifurcation of the pairwise reunion of the pieces of the attractor and\ _nally\ all pieces merge into a one!piece attracting set[ By An\m\ we denote a cyclic chaotic attractor of period m\ arising after stability loosing of the cycle of period n[ Using analytical results and computer simulations\ we can deduce the following consequences for the bifurcations under parameters variation] 0[ Let n 1] g1cA1\1kcA1\1k−0c [ [ [ cA1\1cA1\0\ where k can be any positive integer depending on the parameters\ moreover\ k:\ as a:0\ b:9\ d:−0 and l:9[ 1[ Let nr2] gncAn\1ncAn\ncAn\0[ As was mentioned above\ the bifurcations g1cA1\1k and gncAn\1n are ~ip bifurcations for the corresponding cycles "g1 and gn#[ Our purpose is to describe the bifurcations causing the reunion of the pieces of the cyclic chaotic attractors\ i[e[ bifurcations An\1ncAn\n\ An\ncAn\0\ A1\1icA1\1i−0\ i 1\[[[\ k[
3[ STRUCTURE OF STABLE AND UNSTABLE SETS OF P[ FIRST HOMOCLINIC BIFURCATION OF P
In order to describe the _rst homoclinic bifurcation of the saddle _xed point P\ we consider the structure of its stable and unstable sets[ The stable set of P has the form] i0\ [ [ [ \in W s"P# w9"P#* k F−n"w9"P## w9"P#* k w−n \ i0 0^ i1\ [ [ [ \in 0\1\ n−0
n−0
where w9"P# is the local stable set of P\ that is\ a hal~ine of equation "ys# with the end point W9 LC+"ys#[ The global stable set W s"P# is given by the union of all the preimages of w9"P#] −0 0 F−0"w9# F−0 0 "w9#*F1 "w9# w9*w−0
where the set w9*w0−0 consists of two half!lines issuing from the point W−0 "see Fig[ 5a#\ 0 −0 0 00 01 F−1"w9# F−0 0 "w−0#*F1 "w−0# w−1*w−1\ −0 000 001 011 010 F−2"w9# F−0"w00 "w01 −1#*F −1# w−2*w−2*w−2*w−2\ −0 −0 −0 "w001 "w011 "w010 F−3"w9# F−0"w000 −2#*F −2#*F −2#*F −2 # 0001 0010 0011 0110 0111 0101 0100 w0000 −3 *w−3 *w−3 *w−3 *w−3 *w−3 *w−3 *w−3 \ 011 and so on[ By main beak we shall call the beak formed by the hal~ines w010 −2 and w−2 with the end −0 −0 0 point C−2 where C−2 F1 F0 "C−0#\C−0 w−0+LC0 ðsee Fig[ 5"a#Ł[ The unstable set of P is given by]
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Fig[ 5[ The structure of the stable set W s"P# of the saddle _xed point P before "a# and in the moment "b# of the _rst homoclinic bifurcation of P[
W u"P# a9"P#* k Fn"a9"P##\ n−0
where a9"P# is the local unstable manifold of P\ given by the segment ðA0\A1Ł\ where A0 F"A9#\ A9 LC−0+"yu#\ A1 F1"A0#[ The _rst homoclinic bifurcation of P occurs when C−2 A9 "which corresponds also to the merging of C−1 with A0 and C−0 with A1#\ i[e[ when the main beak "and thus the stable set of P# has a _rst contact with Wu"P# and\ consequently\ in_nitely many preimages of the main beak also have the contacts with Wu"P# ðsee Fig[ 5"b#Ł[ The curve H in the parameter plane "b\d# corresponding to the _rst homoclinic bifurcation of P is shown in Fig[ 6\ as well as the curves H1i\ i 0[\[[ 3\ corresponding to the _rst homoclinic bifurcations of saddle cycles of period 1i\
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Fig[ 6[ The curve H\ in the parameter plane "b\ d#\ of the _rst homoclinic bifurcation of P[ The curves H1i \ i 0\ [ [ [ \ 3 of the _rst homoclinic bifurcations of period!1i saddle cycles such that a half of the cycle points lie above LC−0 and the others are below[ Each curve H1i correspond to the bifurcation of pairwise reunion of the pieces of the cyclic chaotic attractor A1\1i¦0 N A1\1i [
such that half of the cycle points lie above LC−0 and the others are below[ The curves H1i separate the regions of existence of the cyclic chaotic attractors A1\1i¦0 of period 1i¦0 and A1\1i of period 1i\ i[e[ corresponding to the bifurcation of pairwise reunion of the pieces of the cyclic chaotic attractor] A1\1i¦0cA1\1i[ While for the curve H\ as we will see\ for some parameter values\ it also corresponds to the reunion of two pieces of the chaotic attractor into a one!piece attractor\ while for other parameter values\ the homoclinic bifurcation of P does not cause such bifurcation in the attractor[ In order to simplify the exposition\ let us study in detail these two di}erent cases considering the bifurcations A1\3cA1\1 and A1\1cA1\0[ 4[ TWO DIFFERENT MECHANISMS FOR THE REUNION OF TWO PIECES OF THE 1!CYCLIC CHAOTIC ATTRACTOR INTO A ONE!PIECE ATTRACTOR AND OF THREE PIECES OF THE 2!CYCLIC CHAOTIC INTO A ONE!PIECE ATTRACTOR
Let us _x the parameter values a 0[4\ l 9[0\ b 9[34 and vary the parameter d starting from d0 −9[6[ As can be seen from Fig[ 6\ at such parameter values\ the dynamical system generated by F has a cyclic chaotic attractor A1\3 of period 3] A1\3 "A01\3\A11\3\A21\3\A31\3#^ i 3 0 # Ai¦0 F"A1\3 1\3 \ i 0\1\2\ F"A1\3# A1\3 "Fig[ 7#[ Using the theory of critical lines ð4Ł\ we can _nd the boundary of the basin of attraction of i each element A1\3 WA1\3 with respect to the fourth iteration of F "see ð1\ 3Ł for details#[ Indeed\ the immediate basin of attraction of A01\3 is bounded by the segments of the critical line LC\ the local unstable manifold a9"P# of P\ as well as the local stable and unstable manifolds of the point p0 which is one of the points of the saddle cycle g1 "p0\p1# of period 1[ The _rst homoclinic i bifurcation of g1 occurs at d −9[607[[[\ giving rise to the pairwise reunion of A1\3 "A01\3 with 2 1 3 A1\3 and A1\3 with A1\3#[ The result of the reunion is the cyclic chaotic attractor A1\1 of period 1] A1\1 "A01\1\A11\1# "Fig[ 8#[ Note that all the bifurcations A1\1kcA1\1k−0\ k 2\[[[\ are
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Fig[ 7[ A 3!cyclic chaotic attractor\ A1\3 "A01\3\ A11\3\ A21\3\ A31\3# of the map F[
Fig[ 8[ A 1!cyclic chaotic attractor A1\1 "A01\1\ A11\1# of the map F[
analogous to the one described above\ being the result of homoclinic bifurcation of the period −1k−0 saddle cycle[ Let us consider the last bifurcation A1\1cA1\0[ The parameter point "b\d# intersects the curve H at d −9[724[[[\ and the _rst homoclinic bifurcation of P takes place "Fig[ 09#[ As in the previous cases\ the bifurcation causes the reunion of the pieces of the attractor A01\1 and A11\1 giving rise to the one!piece attracting set A1\0[ Let us now consider a di}erent crossing of the curve H\ in another point] d −9[42[[[\ b 9[7[ As can be seen from numerical experiments\
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Fig[ 09[ Contact bifurcation of the 1nd kind corresponding to the reunion of two pieces of the attractor A1\1 into a one! piece chaotic attractor A1\0[
in this case the homoclinic bifurcation of P does not cause the reunion of A01\1 with A11\1[ After this bifurcation\ as we shall see in Section 5\ an invariant hyperbolic set L appears in the neighborhood of P[ The stable set Ws"L# of this set has a Cantor like structure and is a limit set of the stable set of P[ It plays the role of {separator| of the basins of the two disjoint attractors of the map F1[ Thus the basin boundary becomes a fractal set at this homoclinic bifurcation of P[ In Fig[ 00\ the 1!piece attractor A1\1 and some branches of W s"P# are shown at d −9[44\ b 0\ at parameter values just beyond the _rst homoclinic bifurcation of P[ At d −9[46692[[[ the attractor A1\1 has a _rst contact with W s"L#[ It is clear that such a contact shall cause the destruction of the closed invariant chaotic set\ assuming that the preimages of the contact point are dense everywhere on the attractor[ Just after this bifurcation\ numerical evidence shows the existence of an attractor with a high density of iterated points in the places previously occupied "i[e[ before the contact# by the old two pieces of the attractor and rare iterated points between them "Fig[ 01#[ These rare points belong to the trajectories which escape through the gaps appearing after the intersection of the attractor with W s"L#[ They spend a relatively short time all over the attractor\ following the unstable set of P\ then return again inside the old position of the two!piece attractor and remain there for a su.ciently long time[ And so on[ Indeed\ we already get a one!piece attractor A1\0[ If we continue to iterate\ the new shape of the attractor comes to be close to the unstable set W u"P# ðsee Fig[ 1"b#Ł[ When considering the curve H\ one can _nd a point h separating two di}erent kinds of mechanisms of reunion of two pieces of A1\1 as described above] a crossing of H to the left of h gives rise to the immediate reunion at the moment of homoclinic bifurcation of P^ to the right side of h\ the homoclinic bifurcation results in basin fractalization only and the real curve of merging of the two pieces of the attractor\ starting from the h\ lies below H[ It is a curve characterized by the contact of the attractor with its basin boundary[ Above\ we have considered the consequence of bifurcations A3\1cA1\1cA1\0\ i[e[ the reunion bifurcations for the pieces of the 1i!cyclic chaotic attractor\ i 0\[[[\ k[ We assume that\ in the general case\ nr2 the bifurcation An\1ncAn\n is always of the 1nd kind\ i[e[ the result of the
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Fig[ 00[ An example of the 1!cyclic chaotic attractor A1\1 "A01\1\ A11\1# of the map F and some brunches of the stable set of the saddle P\ after the _rst homoclinic bifurcation of P[
Fig[ 01[ Contact bifurcation of the 0st kind corresponding to the reunion of two pieces of the attractor A1\1 into a one! piece chaotic attractor A1\0 via appearance of rare points of the trajectories in the regions between the pieces[
homoclinic bifurcation of the saddle cycle of period n on the basin boundary\ while the bifurcation An\ncAn\0 is always of the 0st kind[ As an example\ let us consider the sequence of bifurcations A2\5cA2\2 and A2\2cA2\0[ In Fig[ 02"a#\ the 5!cyclic chaotic attractor\ A2\5\ is shown at parameter values a 0[94\ b 3\ d −9[16 and l 9[2[ This attractor appears after the ~ip bifurcation of the attracting cycle
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Fig[ 02[ A 5!cyclic chaotic attractor A2\5 "A02\5\ A12\5\ [ [ [ \ A52\5# of the map F "a# which bifurcates into the 2!cyclic chaotic attractor A2\2 "A02\2\ A12\2\ A22\2# "b# due to the contact bifurcation of 1nd kind[ The one!piece chaotic attractor A2\0 "c# is obtained as a result of the contact bifurcation of 0st kind[
g2 "p02\p12\p22#\ which becomes a saddle[ At d −9[18[[[ the attractor A2\5 bifurcates into the 2!cyclic chaotic attractor A2\2 ðFig[ 02"b#Ł due to the contact bifurcation of the 1nd kind\ which corresponds to the _rst homoclinic bifurcation of the saddle cycle g2[ At d −9[204[[[\ the one!piece chaotic attractor\ A2\0 is obtained ðFig[ 02"c#Ł as a result of the contact bifurcation of the 0st kind\ that is\ the reunion of the three pieces of the attractor occurs due to the contact of each piece of the attractor with its basin boundary\ which is already fractal[
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Fig[ 02*continued[
5[ DESCRIPTION OF THE HORSESHOE
In this section\ we construct an invariant hyperbolic set analogous to the Smale horseshoe\ which appears after the _rst homoclinic bifurcation of P[ Let us consider the parameter values soon after this bifurcation of P "described by the curve H in Fig[ 6#\ for example\ by taking a lower value of d[ Then the beaks belonging to the stable set of P will cross the unstable set of P\ creating in_nitely many homoclinic points of P[ Our purpose is to show that\ also in this case "our map F is not a di}eomorphism but a two!dimensional piecewise linear map with a non unique inverse#\ the existence of homoclinic points of P implies the existence of an hyperbolic Cantor set L\ with chaotic dynamics\ near P\ for a suitable power of F\ say Fm\ and corre! spondingly\ this gives an invariant chaotic set for the map F\ called L in the previous sections[ To see this\ let us recall the {folding| action of the critical curve LC[ If we consider a strip S crossing LC−0\ as in Fig[ 03"a# and one side along the unstable set Wu"P# for convenience\ then F"S# is {folded| along LC[ Figure 03"a# also schematically shows the relative positions of the critical segments of interest with the stable and unstable sets of P[ Note that S0 F"S# S"0#*S"1# F0"S+R0#*F1"S+R1# and\ for the non invertibility of F\ we have "0# −0 "1# F−0"S0#wS[ However\ we can recover S as follows] F−0 0 "S #*F1 "S # S[ Note also that if we take the strip S crossing not only LC−0\ but also the two branches of v−2 issuing from the point C−2 of LC−0 "see Fig[ 03"a##\ then F"S# intersects v−1[ Moreover\ we can always take S in such a way that F"S# intersects v−1 in two disjoint segments\ as in Fig[ 03"a#[ In fact\ we can proceed in reverse order\ choosing S"0#*S"1# _rst\ as in Fig[ 03"a#\ with disjoint intersections with v−1\ and having two portions labelled p0 and p1 below v−1[ Note that both p0 and p1 have one side bounded by a segment of W s"P#\ one side bounded by a segment of W u"P# and p0+p1 f[ Then −0 F−0 0 "p0# and F1 "p1# are two disjoint strips on opposite sides with respect to LC−0\ and de_ning −j −0 H0 F1 ) F0 "p0#\ H1 F−"j¦0# "p1#\ we get two disjoint {horizontal| strips\ as close to P as we 1 want\ by _xing suitable values of the integer j[ Let us assume a _xed value of j[ Then F j¦0"H0*H1# p0*p1 and it is easy to see that\ in a _nite number of iterations by F\ we shall get F m"H0*H1# V0*V1\ where V0 and V1 are disjoint strips which intersect H0*H1 as in Fig[ 03[
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Fig[ 03[ Construction of Smale horseshoe "a# and its enlargements "b\ c#\ after the _rst homoclinic bifurcation of saddle _xed point P "for the details see text#[
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Fig[ 03*continued[
In fact\ after "j¦0# applications of F\ we get p0*p1 and then F"p0*p1# F1"p0*p1#\ F 1"p0*p1# F0 ) F1"p0*p1# and after i applications of F F1\ for a suitable integer i\ we shall be su.ciently near P to get disjoint intersections with H0 and H1\ as in Fig[ 03"a#[ It is now clear that the {standard horseshoe mechanism| can be instaured\ proving the existence of an invariant Cantor set\ LWH0*H1[ In fact\ V0*V1 T"H0*H1# where T F m and m i¦2¦j[ Moreover V0 T0"H0#\ V1 T1"H1# where T0 F 1j ) F0 ) F1 ) F0 ) F 1i
and T1 F 1j ) F0 ) F i¦1 1 [
Thus we can consider T]"H0*H1#:"V0*V1# as a map with a unique inverse by de_ning T−0 as −0 follows] let x$V0*V1\ then T−0"x# T−0 "x# T−0 0 "x# if x$V0\ T 1 "x# if x$V1where −i −0 −0 −0 −j T −0 0 F1 ) F0 ) F1 ) F0 ) F1
−"i¦1# −j and T −0 ) F−0 1 F1 0 ) F1 [
For the map T]"H0*H1#:"V0*V1# and its inverse T−0\ the standard procedure used for invertible maps works well[ Let us repeat a few steps[ De_ne Li j Hi+Vj "see Fig[ 03"b##[ Consider L10 and L00 as subsets of V0[ Then they are the image by T of two subsets of H0[ Note that one side of L10 and L00\ the right one in our _gure\ belongs to the stable set W s"P#[ Thus\ these sides must be the image by T of two disjoint pieces on the right side of H0\ bounded by W s"P#[ The opposite sides of L10 and L00 belong to the side of V0 which is the image by T of the side of H0 opposite to W s"P#[ It follows that L10 and L00 must be the image by T of two disjoint horizontal strips included in H0\ say H10 and H00\ T"H00# L00 and T"H10# L10\ or equivalently\ T−0"L00# H00 and T−0"L10# H10[ Analogous reasonings lead to the de_nition of two disjoint strips inside H1] T−0"L11# H11WH1 and T−0"L01# H01WH1\ H11+H01 f[
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Now\ let us take the images[ Consider the two subsets L10 and L11 of H1[ As these are subsets of H1\ their images by T belong to V1[ The low boundary of such subsets belong to the stable set W s"P#\ thus\ their images by T are two disjoint pieces on the side of V1\ belonging to W s"P# "see Fig[ 03"b##[ The opposite sides of these subsets belong to the side of H1 opposite to W s"P# and thus\ their images by T belong to the opposite sides of V1[ That is\ T"L10# V10 and T"L11# V11 are two disjoint {vertical| strips belonging to V1[ In a similar way T"L00# V00 and T"L01# V01 are two disjoint {vertical| strips belonging to V0\ and so on[ The mechanism is standard "see e[g[ ð5\ 6Ł#[ We shall _nd a Cantor set LWH0*H1 which is both forward and backward invariant by T\ i[e[ T"L# L and T−0"L# L[ Now\ regarding hyperbolicity we note that\ by construction\ we started with two horizontal strips H0 and H1\ having one side on W s"P# and one side on W u"P#[ However\ from the process of construction of the subsets Hi j and Vi j "which also have one side on W s"P# and one side on W u"P##\ we can deduce that\ really\ in a neighbourhood of P\ the structure of the stable and unstable sets of P is such that W s"P# includes in_nitely many {segments parallel| to v9 "i[e[ the local stable set W sloc"P## and the same holds for W u"P#] it includes in_nitely many {segments parallel| to W uloc"P# "see Fig[ 03"c##[ Thus\ we can presume to start the process with two horizontal strips\ H0 and H1\ having two opposite sides on W s"P# and the other two opposite sides on W u"P#[ Then the Cantor set L is hyperbolic[ REFERENCES 0[ Fournier!Prunaret\ D[\ Mira\ C[ and Gardini\ L[ Some contact bifurcations in two!dimensional examples[ In Pro! ceedin`s ECIT |83[ World Scienti_c "in press#[ 1[ Mira\ C[\ Gardini\ L[\ Barugola\ A[ and Cathala\ J[ C[ Chaotic Dynamics in Two!Dimensional Noninvertible Maps[ World!Scienti_c\ Singapore[\ 0885[ 2[ Abraham\ R[\ Gardini\ L[ and Mira\ C[\ Chaos in Discrete Dynamical Systems "A Visual Introduction in two Dimensions#[ Telos\ Springer[\ 0886[ 3[ Mira\ C[\ Rauzy\ C[\ Maistrenko\ Y[\ Sushko\ I[\ Some properties of a two!dimensional piecewise!linear noninvertible map[ Int[ J[ Bifurcation + Chaos\ 0885\ 5"01A#\ 1188Ð1208[ 4[ Gumowski\ I[ and Mira\ C[\ Dynamique Chaotique[ Cepadues Editions\ Toulouse\ 0879[ 5[ Wiggins\ S[ Global Bifurcations and Chaos[ Springer!Verlag\ N[Y[\ 0877[ 6[ Wiggins\ S[ Introduction to Applied Nonlinear Dynamical Systems and Chaos[ Springer!Verlag\ N[Y[\ 0889[ 7[ Maistrenko\ Yu[ L[ and Sushko\ I[ M[\ Bifurcation phenomena for two!dimensional piecewise linear maps[ J[ Tech[ Phys[\ 0885\ 26\ 260Ð267[
Chaos\ Solitons + Fractals Vol[ 8\ No[ 7\ pp[ 0262Ð0289\ 0887 Þ 0887 Elsevier Science Ltd[ All rights reserved Printed in Great Britain 9859Ð9668:87 ,08[99¦9[99
Pergamon
PII] S9859!9668"87#99969!7
About Two Mechanisms of Reunion of Chaotic Attractors YURI MAISTRENKO and IRINA SUSHKO Institute of Mathematics National Academy of Sciences of Ukraine\ Kiev\ Ukraine and
LAURA GARDINI$ Dipartimento di Metodi Quantitativi\ University of Brescia\ and Istituto di Scienze Economiche\ University of Urbino\ 50918 Urbino\ Italy "Accepted 13 February 0887#
Abstract*Considering a family of two!dimensional piecewise linear maps\ we discuss two di}erent mech! anisms of reunion of two "or more# pieces of cyclic chaotic attractors into a one!piece attracting set\ observed in several models[ It is shown that\ in the case of so!called {contact bifurcation of the 1nd kind|\ the reunion occurs immediately due to homoclinic bifurcation of some saddle cycle belonging to the basin boundary of the attractor[ In the case of so!called {contact bifurcation of the 0st kind|\ the reunion is a result of a contact of the attractor with its basin boundary which is fractal\ including the stable set of a chaotic invariant hyperbolic set appeared after the homoclinic bifurcation of a saddle cycle on the basin boundary[ Þ 0887 Elsevier Science Ltd[ All rights reserved[
0[ INTRODUCTION
Our purpose is to describe some mechanisms of reunion of two or more pieces of a cyclic chaotic attractor in R1 into a one!piece chaotic attracting set[ When observing such a reunion in several nonlinear models\ one can often classify it in one of the following two possibilities] 0[ It is due to a so!called {contact bifurcation of the 1nd kind| ð0Ð3Ł\ giving rise to a direct reunion of the pieces of the attractor by pairs\ and the resulting shape of the attractor is a junction\ two by two\ of the pieces existing before separately "Fig[ 0#[ 1[ Another type of reunion is due to a so!called {contact bifurcation of the 0st kind| "see the same references as above#] two or more pieces of some cyclic chaotic attractor\ being situated far enough from each other\ under small parameter variation\ can suddenly give rise to the appearance of bursts*rare points of the trajectories in some region of the phase space including the old pieces of attractors ðFig[ 1"a#Ł[ The number of the rare points increases as the iterations are continued\ and they belong to a new chaotic attractor ðFig[ 1"b#Ł[ In some sense\ chaotic attractors begin to {feel each other| when the distance between them is still _nite[ When calculating trajectories in a system with {rare points|\ their appearance seems rather unusual as we cannot see\ in phase space\ any other attracting set except for the two
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Fig[ 0[ An example of the {{contact bifurcation of the 1nd kind|| when the reunion of two pieces of an attractor occurs due to the _rst homoclinic bifurcation of a saddle cycle belonging to the basin boundary of the attractor[
"or more# pieces of attractor under consideration[ Due to this\ it can be supposed that\ in the region between the attractors\ a repeller should exist[ Contact bifurcation with it causes the rare points[ Moreover\ such repeller have a complicated topological structure\ so that an observer cannot usually see any order in the appearance of such rare points and this visual randomness has both phase space and temporal nature[ Really\ as described in ð3Ł "ch[ 4#\ besides the two types recalled above\ there are several kinds of contact bifurcations between chaotic attractors and their basins\ with di}erent dynamical e}ects[ However\ we shall restrict our interest to the two mentioned above\ because these are the two kinds of contact bifurcations involved in the piece!wise linear maps which interest us in this work[ The family F of maps that we consider is the family of two!dimensional piecewise linear continuous maps\ as a function of four parameters\ given by] F GF 0 ] F j J GF1] f
0 1 0 10 1 0 1 0 10 1 0 1 x l 9 Ł y 9 a
x \ y
x l−b b Ł y a−d d
y²x¦0\ "0#
x −b ¦ \ yrx¦0\ y a−d
where both the above mentioned scenarios can be realized\ depending on the parameters of the family[ Note that the map F can be considered in some sense as a 1Dim analog of the skew tent map[ We shall consider the case in which F has two saddle _xed points\ O and P\ which are at the boundary of a closed bounded trapping region\ Z\ of the phase plane "where trapping means mapped into itself] F"Z#W ¹ Z#[
About Two Mechanisms of Reunion of Chaotic Attractors
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Fig[ 1[ An example of {{contact bifurcation of the 0st kind|| when the reunion of two pieces of an attractor is via appearance of rare points of the trajectories in the regions between pieces\ when they are still far from each other[ The reunion occurs due to a boundary crisis] each chaotic piece has a contact with its fractal basin boundary[
It should be noted that the case in which a dynamical system has two saddle _xed points occurs quite often in the modern theory of nonlinear dynamics[
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We shall see that\ in the _rst case\ the contact bifurcation "without rare points# is a result of the _rst homoclinic bifurcation of the saddle P[ In the second case\ the reunion bifurcation "with rare points# is caused by the contact of the attractors with the boundary of its basin of attraction\ which includes the stable set of the saddle P after its _rst homoclinic bifurcation[ Therefore\ two di}erent situations can occur in our model] in the _rst case\ the _rst homoclinic bifurcation of P gives rise to immediate reunion of the attractors[ In the second one\ the homoclinic bifurcation of the saddle P creates an hyperbolic invariant set L "on the boundary of the basins# without reunion of attractors[ The stable set W s"L# of the set L has Cantor like structure[ It plays the role of {separator| of the basins of the di}erent pieces of the cyclic attractor[ Under parameter variation\ a bifurcation occurs when the chaotic attractors have a contact with W s"L#[ At this bifurcation value\ the basin of each piece of the cyclic attractor comes to be intermingled\ assuming that the preimages of each contact point are dense everywhere on the {attractor|[ Soon after the contact bifurcation\ the old attractor disappears due to the in_nitely many intersections with W s"L#\ and a new attractor arises[ The structure of the paper is as follows[ In Section 1\ we describe the map F[ A so!called {saddleÐsaddle| area is de_ned in the parameter space when both the _xed points of the map F are saddles[ Correspondingly\ a bounded invariant area is found in the phase space[ Section 2 is devoted to the study of di}erent kinds of attractors of the map F[ Depending on the parameters\ the attractor can be either a cycle of any period or a cyclic chaotic attractor of any period[ We give exact formulae for the boundaries of the regions of existence of the attracting cycles of di}erent periods[ A conjecture is formulated about the order of appearance of the attractors in the system under parameter variation[ It is shown that the case of stability loosing of any point cycle due to the ~ip bifurcation "besides the cycle of period 1# results in the appearance of a cyclic chaotic attractor of doubling period[ In the case in which the cycle of period 1 becomes unstable\ a cyclic chaotic attractor of period 1i can appear\ where i depends on the parameters[ The bifurcations which may then occur consists of the pairwise reunion of the pieces of the attractor to give only a one!piece chaotic set[ The main problem concerns the bifurcation causing the reunion[ It is connected with the _rst homoclinic bifurcation of the saddle P "Section 3# as well as with an invariant hyperbolic set analogous to the Smale horseshoe which appears after the homoclinic bifurcation[ We give a construction of this set in Section 5[ Di}erent mechanisms of reunion of pieces of cyclic chaotic attractors are explained in Section 4\ considering the sequences of bifurcations of the 3!\ 1! and 0!piece chaotic attractors\ as well as those of the 5!\ 2! and 0!piece chaotic attractors[ 1[ DESCRIPTION OF THE MAP[ {SADDLEÐSADDLE AREA| IN THE PARAMETER SPACE[ BOUNDED INVARIANT REGION IN THE PHASE SPACE
Consider the four!parameter family F of two!dimensional piecewise linear continuous maps given in "0#[ The map F consists of two linear maps F0 and F1 de_ned below and above\ respectively\ the straight line LC−0 ""x\y#]y x¦0# which is a line of discontinuity of the Jacobian of the map\ F[ The map\ F\ is one!to!one if d×ab:l and is noninvertible otherwise[ Indeed\ if d³ab:l\ the map\ F\ is of "Z9−Z1# type\ i[e[ each point "x\y#$R1 has two preimages\ provided that y³ax:l¦a^ has zero preimages\ provided that y×ax:l¦a and has a unique preimage if y ax:l¦a[ Thus\ the straight line\ LC ""x\y#]y ax:l¦a#\ is the locus of {critical values| of F\ i[e[ the so!called critical line of rank!0 ð4\ 1\ 3Ł[ Due to these properties\ the map F can be considered as a two!dimensional analog of the skew tent map[ Denote the _xed point of F0 by O "9\9#[ The eigenvalues of F0 are m0 a and m1 l with corresponding eigendirections
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Fig[ 2[ The triangle of stability of the _xed point P in the "b\ d#!parameter plane\ a × 0\ 9 ³ l ³ 0^ D is the region in which P is saddle[
u0 "9\0# and u1 "0\9#[ Let us _x the parameters a and l so that the _xed point O is a saddle] a × 0\ 9 ³ l ³ 0[ The local stable "unstable# manifold of O is denoted by w9"O# "a9"O## which is obviously a segment of the straight line "y 9# ""x 9##[ The _xed point of the map F1 is denoted by P "x9\y9#\ where x9
b"0−a# "0−l#"d−a# ^ y9 [ b"a−0#−"0−d#"0−l# b"a−0#−"0−d#"0−l#
We are interested in a so!called {saddleÐsaddle| area D in the parameter space in which both the _xed points O and P are saddles and\ moreover\ a bounded trapping region Z exists in the phase plane "x\y#[ For any _xed a×0 and 9³l³0\ the region of stability of P "x9\y9# in the "b\d# parameter plane is the triangle P0 bounded by three segments of the following straight lines d −0¦b"0¦a#:"0¦l#\ "d0# d 0¦b"0−a#:"0−l#\ "d1# d "ab¦0#:l\ "d2# The _xed point P becomes a saddle if the parameter point "b\d# intersects either "d0# "the eigenvalue of P crosses through −0# or "d1# "the eigenvalue crosses through 0#[ In the last case\ obviously a bounded invariant area in the phase plane does not exist[ Thus\ the region D\ in the parameters| space we are interested in is de_ned as follows] D ""a\b\d\l#] a×0\ d ³d³minb$R0""d0#\"d1#\"d2##\9³l³0# where d denotes a bifurcation curve of destruction of the trapping region Z which will be de_ned below "Fig[ 2#[ We will restrict ourself to the condition b×9 "at b 9\ the map F is a triangular map and its dynamics are described in ð7Ł#[
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Fig[ 3[ Trapping region Z of the map F\ in the phase plane "x\ y#[
The local stable "unstable# manifold of P is denoted by w9"P# "a9"P## which is a segment of the straight line y Qsx¦Rs denoted as "ys# for short\ "y Qux¦Ru\ denoted as "yu# for short# where Qs
l0¦b−l s d"0−l#−"l0¦b#"0−a#¦l−a ^R ^ b ba−dl¦d¦l−b−0
Qu
l1¦b−l u d"0−l#−"l1¦b#"0−a#¦l−a ^R ^ b ba−dl¦d¦l−b−0
and 9³l0³0\ l1³−0 are eigenvalues of F1] l0 "l−b¦d¦z"b−l−d#1−3"dl−ba##:1\ l1 "l−b¦d−z"b−l−d#1−3"dl−ba##:1[ Assuming now that the parameters belong to the region D\ we shall _nd a bounded region in the phase plane "x\y# which is trapping for F[ Consider the region Z\ bounded by the segments of the four straight lines] "x 9#\ "y 9# "containing\ respectively\ a9"O# and w9"O##\ LC and of the straight line "yu# "containing a9"P##[ The region Z is a quadrilateral with corner points O "9\9#\ A0 LC+"yu#\ B0 "9\a# and C0 "−Ru:Qu\9# "Fig[ 3#[ It is easy to see that if "a\b\d\l#$D and b×9 then the region Z is trapping "i[e[ mapped into itself] F"Z#UZ#[ The bifurcation destroying the invariance of Z takes place when F1"A0# C0[ This bifurcation deter! mines the appearance of the _rst heteroclinic trajectory from the saddle P to the saddle O and also the homoclinic connection for O "see ð1\ 3Ł#[ The curve d in the parameter space corresponds to this last condition[ 2[ ATTRACTORS] ATTRACTING CYCLES AND CYCLIC CHAOTIC ATTRACTORS
Let "a\b\d\l#$D\ b×9 and\ consequently\ the bounded invariant region Z exists in the phase plane[ We shall see that an attractor of the map F\ necessarily belonging to Z\ is either an attracting cycle gn of period n or a cyclic chaotic attractor An\m of period m\ where n\ m can assume any natural value\ depending on the parameters[
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Fig[ 4[ The regions Pn of existence of attracting cycles gn in the "b?\ d#!parameter plane where b? 1b\ at a 0¦09−4 and l 9[2 for n 1\ [ [ [ \06[
We shall _nd the regions Pn$D\ nr1\ such that if parameters belong to Pn\ then an attracting cycle of period n of the map F exists[ Let gn ""x0\y0#\"x1\y1#[\[[\ "xn\yn## be a cycle of period n such that one of its points "let us assume the point "x0\y0## belongs to the region x¦0³y³ax:l¦a\ i[e[\ x0¦0³y0³ax0:l¦a^ and the other points belong to the region 9³y³x¦0\ i[e[\ 9³yi³xi¦0\ i 1[\[[\ n[ The appearance of gn is a result of saddleÐnode bifurcation and it is de_ned by the condition "x0\y0#$LC[ From "x0\y0# F0n−0"F1"x0\y0## at y0 ax0:l¦a we get the locus\ in the parameter space\ of the appearance of gn] dn\e
bl n−1"0−l#"an−0#¦"l n−0#"an−0−0# \ an−1"a−0#"0−ln#
which is a straight line in the "b\d# parameter plane at _xed values for a\ l and n[ Note that the cycle g1 arises at the stability loosing of the _xed point P\ that is d1\e "d0#[ The curve of stability loosing of gn can be found as a result of ~ip bifurcation\ when one of the cycle multiplicators becomes equal to −0[ Also\ this curve is a straight line in the "b\d# parameter plane at _xed a\ l and n] dn\s −
bln−0"an¦0# ¦ [ an−0 an−0"ln¦0# 0
Thus\ the regions Pn$D of existence of attracting cycles gn are] Pn ""a\b\d\l#$D]dn\s³d³dn\e#[ The regions Pn are shown in Fig[ 4 in the "b?\d# parameter plane where b? 1b at a 0¦09−4\ l 9[2 where n 1\[[[\ 06[ Note that\ at _xed values\ a×0\ 9³l³0\ the number of the regions Pn is _nite\ that is n 1\[[[\ n \ where n is a number which depends on a and l[ Moreover\ if al×0\ there is not any attracting cycle of the map F at "a\b\d\l# e D[ But\ if a:0 and l:0\ then n :[
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When the parameter point "b\d# crosses one of the regions Pn through the curve dn\s\ a ~ip bifurcation for gn occurs[ What is the result for the dynamics of the map F< Computer simulations show that this bifurcation gives rise to the appearance of a cyclic chaotic attractor\ of doubling period 1n when nr2 and of period 1k for n 1\ k depending on the parameters[ Further variation of the parameters can give rise to the bifurcation of the pairwise reunion of the pieces of the attractor and\ _nally\ all pieces merge into a one!piece attracting set[ By An\m\ we denote a cyclic chaotic attractor of period m\ arising after stability loosing of the cycle of period n[ Using analytical results and computer simulations\ we can deduce the following consequences for the bifurcations under parameters variation] 0[ Let n 1] g1cA1\1kcA1\1k−0c [ [ [ cA1\1cA1\0\ where k can be any positive integer depending on the parameters\ moreover\ k:\ as a:0\ b:9\ d:−0 and l:9[ 1[ Let nr2] gncAn\1ncAn\ncAn\0[ As was mentioned above\ the bifurcations g1cA1\1k and gncAn\1n are ~ip bifurcations for the corresponding cycles "g1 and gn#[ Our purpose is to describe the bifurcations causing the reunion of the pieces of the cyclic chaotic attractors\ i[e[ bifurcations An\1ncAn\n\ An\ncAn\0\ A1\1icA1\1i−0\ i 1\[[[\ k[
3[ STRUCTURE OF STABLE AND UNSTABLE SETS OF P[ FIRST HOMOCLINIC BIFURCATION OF P
In order to describe the _rst homoclinic bifurcation of the saddle _xed point P\ we consider the structure of its stable and unstable sets[ The stable set of P has the form] i0\ [ [ [ \in W s"P# w9"P#* k F−n"w9"P## w9"P#* k w−n \ i0 0^ i1\ [ [ [ \in 0\1\ n−0
n−0
where w9"P# is the local stable set of P\ that is\ a hal~ine of equation "ys# with the end point W9 LC+"ys#[ The global stable set W s"P# is given by the union of all the preimages of w9"P#] −0 0 F−0"w9# F−0 0 "w9#*F1 "w9# w9*w−0
where the set w9*w0−0 consists of two half!lines issuing from the point W−0 "see Fig[ 5a#\ 0 −0 0 00 01 F−1"w9# F−0 0 "w−0#*F1 "w−0# w−1*w−1\ −0 000 001 011 010 F−2"w9# F−0"w00 "w01 −1#*F −1# w−2*w−2*w−2*w−2\ −0 −0 −0 "w001 "w011 "w010 F−3"w9# F−0"w000 −2#*F −2#*F −2#*F −2 # 0001 0010 0011 0110 0111 0101 0100 w0000 −3 *w−3 *w−3 *w−3 *w−3 *w−3 *w−3 *w−3 \ 011 and so on[ By main beak we shall call the beak formed by the hal~ines w010 −2 and w−2 with the end −0 −0 0 point C−2 where C−2 F1 F0 "C−0#\C−0 w−0+LC0 ðsee Fig[ 5"a#Ł[ The unstable set of P is given by]
About Two Mechanisms of Reunion of Chaotic Attractors
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Fig[ 5[ The structure of the stable set W s"P# of the saddle _xed point P before "a# and in the moment "b# of the _rst homoclinic bifurcation of P[
W u"P# a9"P#* k Fn"a9"P##\ n−0
where a9"P# is the local unstable manifold of P\ given by the segment ðA0\A1Ł\ where A0 F"A9#\ A9 LC−0+"yu#\ A1 F1"A0#[ The _rst homoclinic bifurcation of P occurs when C−2 A9 "which corresponds also to the merging of C−1 with A0 and C−0 with A1#\ i[e[ when the main beak "and thus the stable set of P# has a _rst contact with Wu"P# and\ consequently\ in_nitely many preimages of the main beak also have the contacts with Wu"P# ðsee Fig[ 5"b#Ł[ The curve H in the parameter plane "b\d# corresponding to the _rst homoclinic bifurcation of P is shown in Fig[ 6\ as well as the curves H1i\ i 0[\[[ 3\ corresponding to the _rst homoclinic bifurcations of saddle cycles of period 1i\
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Fig[ 6[ The curve H\ in the parameter plane "b\ d#\ of the _rst homoclinic bifurcation of P[ The curves H1i \ i 0\ [ [ [ \ 3 of the _rst homoclinic bifurcations of period!1i saddle cycles such that a half of the cycle points lie above LC−0 and the others are below[ Each curve H1i correspond to the bifurcation of pairwise reunion of the pieces of the cyclic chaotic attractor A1\1i¦0 N A1\1i [
such that half of the cycle points lie above LC−0 and the others are below[ The curves H1i separate the regions of existence of the cyclic chaotic attractors A1\1i¦0 of period 1i¦0 and A1\1i of period 1i\ i[e[ corresponding to the bifurcation of pairwise reunion of the pieces of the cyclic chaotic attractor] A1\1i¦0cA1\1i[ While for the curve H\ as we will see\ for some parameter values\ it also corresponds to the reunion of two pieces of the chaotic attractor into a one!piece attractor\ while for other parameter values\ the homoclinic bifurcation of P does not cause such bifurcation in the attractor[ In order to simplify the exposition\ let us study in detail these two di}erent cases considering the bifurcations A1\3cA1\1 and A1\1cA1\0[ 4[ TWO DIFFERENT MECHANISMS FOR THE REUNION OF TWO PIECES OF THE 1!CYCLIC CHAOTIC ATTRACTOR INTO A ONE!PIECE ATTRACTOR AND OF THREE PIECES OF THE 2!CYCLIC CHAOTIC INTO A ONE!PIECE ATTRACTOR
Let us _x the parameter values a 0[4\ l 9[0\ b 9[34 and vary the parameter d starting from d0 −9[6[ As can be seen from Fig[ 6\ at such parameter values\ the dynamical system generated by F has a cyclic chaotic attractor A1\3 of period 3] A1\3 "A01\3\A11\3\A21\3\A31\3#^ i 3 0 # Ai¦0 F"A1\3 1\3 \ i 0\1\2\ F"A1\3# A1\3 "Fig[ 7#[ Using the theory of critical lines ð4Ł\ we can _nd the boundary of the basin of attraction of i each element A1\3 WA1\3 with respect to the fourth iteration of F "see ð1\ 3Ł for details#[ Indeed\ the immediate basin of attraction of A01\3 is bounded by the segments of the critical line LC\ the local unstable manifold a9"P# of P\ as well as the local stable and unstable manifolds of the point p0 which is one of the points of the saddle cycle g1 "p0\p1# of period 1[ The _rst homoclinic i bifurcation of g1 occurs at d −9[607[[[\ giving rise to the pairwise reunion of A1\3 "A01\3 with 2 1 3 A1\3 and A1\3 with A1\3#[ The result of the reunion is the cyclic chaotic attractor A1\1 of period 1] A1\1 "A01\1\A11\1# "Fig[ 8#[ Note that all the bifurcations A1\1kcA1\1k−0\ k 2\[[[\ are
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Fig[ 7[ A 3!cyclic chaotic attractor\ A1\3 "A01\3\ A11\3\ A21\3\ A31\3# of the map F[
Fig[ 8[ A 1!cyclic chaotic attractor A1\1 "A01\1\ A11\1# of the map F[
analogous to the one described above\ being the result of homoclinic bifurcation of the period −1k−0 saddle cycle[ Let us consider the last bifurcation A1\1cA1\0[ The parameter point "b\d# intersects the curve H at d −9[724[[[\ and the _rst homoclinic bifurcation of P takes place "Fig[ 09#[ As in the previous cases\ the bifurcation causes the reunion of the pieces of the attractor A01\1 and A11\1 giving rise to the one!piece attracting set A1\0[ Let us now consider a di}erent crossing of the curve H\ in another point] d −9[42[[[\ b 9[7[ As can be seen from numerical experiments\
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Fig[ 09[ Contact bifurcation of the 1nd kind corresponding to the reunion of two pieces of the attractor A1\1 into a one! piece chaotic attractor A1\0[
in this case the homoclinic bifurcation of P does not cause the reunion of A01\1 with A11\1[ After this bifurcation\ as we shall see in Section 5\ an invariant hyperbolic set L appears in the neighborhood of P[ The stable set Ws"L# of this set has a Cantor like structure and is a limit set of the stable set of P[ It plays the role of {separator| of the basins of the two disjoint attractors of the map F1[ Thus the basin boundary becomes a fractal set at this homoclinic bifurcation of P[ In Fig[ 00\ the 1!piece attractor A1\1 and some branches of W s"P# are shown at d −9[44\ b 0\ at parameter values just beyond the _rst homoclinic bifurcation of P[ At d −9[46692[[[ the attractor A1\1 has a _rst contact with W s"L#[ It is clear that such a contact shall cause the destruction of the closed invariant chaotic set\ assuming that the preimages of the contact point are dense everywhere on the attractor[ Just after this bifurcation\ numerical evidence shows the existence of an attractor with a high density of iterated points in the places previously occupied "i[e[ before the contact# by the old two pieces of the attractor and rare iterated points between them "Fig[ 01#[ These rare points belong to the trajectories which escape through the gaps appearing after the intersection of the attractor with W s"L#[ They spend a relatively short time all over the attractor\ following the unstable set of P\ then return again inside the old position of the two!piece attractor and remain there for a su.ciently long time[ And so on[ Indeed\ we already get a one!piece attractor A1\0[ If we continue to iterate\ the new shape of the attractor comes to be close to the unstable set W u"P# ðsee Fig[ 1"b#Ł[ When considering the curve H\ one can _nd a point h separating two di}erent kinds of mechanisms of reunion of two pieces of A1\1 as described above] a crossing of H to the left of h gives rise to the immediate reunion at the moment of homoclinic bifurcation of P^ to the right side of h\ the homoclinic bifurcation results in basin fractalization only and the real curve of merging of the two pieces of the attractor\ starting from the h\ lies below H[ It is a curve characterized by the contact of the attractor with its basin boundary[ Above\ we have considered the consequence of bifurcations A3\1cA1\1cA1\0\ i[e[ the reunion bifurcations for the pieces of the 1i!cyclic chaotic attractor\ i 0\[[[\ k[ We assume that\ in the general case\ nr2 the bifurcation An\1ncAn\n is always of the 1nd kind\ i[e[ the result of the
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Fig[ 00[ An example of the 1!cyclic chaotic attractor A1\1 "A01\1\ A11\1# of the map F and some brunches of the stable set of the saddle P\ after the _rst homoclinic bifurcation of P[
Fig[ 01[ Contact bifurcation of the 0st kind corresponding to the reunion of two pieces of the attractor A1\1 into a one! piece chaotic attractor A1\0 via appearance of rare points of the trajectories in the regions between the pieces[
homoclinic bifurcation of the saddle cycle of period n on the basin boundary\ while the bifurcation An\ncAn\0 is always of the 0st kind[ As an example\ let us consider the sequence of bifurcations A2\5cA2\2 and A2\2cA2\0[ In Fig[ 02"a#\ the 5!cyclic chaotic attractor\ A2\5\ is shown at parameter values a 0[94\ b 3\ d −9[16 and l 9[2[ This attractor appears after the ~ip bifurcation of the attracting cycle
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Fig[ 02[ A 5!cyclic chaotic attractor A2\5 "A02\5\ A12\5\ [ [ [ \ A52\5# of the map F "a# which bifurcates into the 2!cyclic chaotic attractor A2\2 "A02\2\ A12\2\ A22\2# "b# due to the contact bifurcation of 1nd kind[ The one!piece chaotic attractor A2\0 "c# is obtained as a result of the contact bifurcation of 0st kind[
g2 "p02\p12\p22#\ which becomes a saddle[ At d −9[18[[[ the attractor A2\5 bifurcates into the 2!cyclic chaotic attractor A2\2 ðFig[ 02"b#Ł due to the contact bifurcation of the 1nd kind\ which corresponds to the _rst homoclinic bifurcation of the saddle cycle g2[ At d −9[204[[[\ the one!piece chaotic attractor\ A2\0 is obtained ðFig[ 02"c#Ł as a result of the contact bifurcation of the 0st kind\ that is\ the reunion of the three pieces of the attractor occurs due to the contact of each piece of the attractor with its basin boundary\ which is already fractal[
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Fig[ 02*continued[
5[ DESCRIPTION OF THE HORSESHOE
In this section\ we construct an invariant hyperbolic set analogous to the Smale horseshoe\ which appears after the _rst homoclinic bifurcation of P[ Let us consider the parameter values soon after this bifurcation of P "described by the curve H in Fig[ 6#\ for example\ by taking a lower value of d[ Then the beaks belonging to the stable set of P will cross the unstable set of P\ creating in_nitely many homoclinic points of P[ Our purpose is to show that\ also in this case "our map F is not a di}eomorphism but a two!dimensional piecewise linear map with a non unique inverse#\ the existence of homoclinic points of P implies the existence of an hyperbolic Cantor set L\ with chaotic dynamics\ near P\ for a suitable power of F\ say Fm\ and corre! spondingly\ this gives an invariant chaotic set for the map F\ called L in the previous sections[ To see this\ let us recall the {folding| action of the critical curve LC[ If we consider a strip S crossing LC−0\ as in Fig[ 03"a# and one side along the unstable set Wu"P# for convenience\ then F"S# is {folded| along LC[ Figure 03"a# also schematically shows the relative positions of the critical segments of interest with the stable and unstable sets of P[ Note that S0 F"S# S"0#*S"1# F0"S+R0#*F1"S+R1# and\ for the non invertibility of F\ we have "0# −0 "1# F−0"S0#wS[ However\ we can recover S as follows] F−0 0 "S #*F1 "S # S[ Note also that if we take the strip S crossing not only LC−0\ but also the two branches of v−2 issuing from the point C−2 of LC−0 "see Fig[ 03"a##\ then F"S# intersects v−1[ Moreover\ we can always take S in such a way that F"S# intersects v−1 in two disjoint segments\ as in Fig[ 03"a#[ In fact\ we can proceed in reverse order\ choosing S"0#*S"1# _rst\ as in Fig[ 03"a#\ with disjoint intersections with v−1\ and having two portions labelled p0 and p1 below v−1[ Note that both p0 and p1 have one side bounded by a segment of W s"P#\ one side bounded by a segment of W u"P# and p0+p1 f[ Then −0 F−0 0 "p0# and F1 "p1# are two disjoint strips on opposite sides with respect to LC−0\ and de_ning −j −0 H0 F1 ) F0 "p0#\ H1 F−"j¦0# "p1#\ we get two disjoint {horizontal| strips\ as close to P as we 1 want\ by _xing suitable values of the integer j[ Let us assume a _xed value of j[ Then F j¦0"H0*H1# p0*p1 and it is easy to see that\ in a _nite number of iterations by F\ we shall get F m"H0*H1# V0*V1\ where V0 and V1 are disjoint strips which intersect H0*H1 as in Fig[ 03[
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Fig[ 03[ Construction of Smale horseshoe "a# and its enlargements "b\ c#\ after the _rst homoclinic bifurcation of saddle _xed point P "for the details see text#[
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Fig[ 03*continued[
In fact\ after "j¦0# applications of F\ we get p0*p1 and then F"p0*p1# F1"p0*p1#\ F 1"p0*p1# F0 ) F1"p0*p1# and after i applications of F F1\ for a suitable integer i\ we shall be su.ciently near P to get disjoint intersections with H0 and H1\ as in Fig[ 03"a#[ It is now clear that the {standard horseshoe mechanism| can be instaured\ proving the existence of an invariant Cantor set\ LWH0*H1[ In fact\ V0*V1 T"H0*H1# where T F m and m i¦2¦j[ Moreover V0 T0"H0#\ V1 T1"H1# where T0 F 1j ) F0 ) F1 ) F0 ) F 1i
and T1 F 1j ) F0 ) F i¦1 1 [
Thus we can consider T]"H0*H1#:"V0*V1# as a map with a unique inverse by de_ning T−0 as −0 follows] let x$V0*V1\ then T−0"x# T−0 "x# T−0 0 "x# if x$V0\ T 1 "x# if x$V1where −i −0 −0 −0 −j T −0 0 F1 ) F0 ) F1 ) F0 ) F1
−"i¦1# −j and T −0 ) F−0 1 F1 0 ) F1 [
For the map T]"H0*H1#:"V0*V1# and its inverse T−0\ the standard procedure used for invertible maps works well[ Let us repeat a few steps[ De_ne Li j Hi+Vj "see Fig[ 03"b##[ Consider L10 and L00 as subsets of V0[ Then they are the image by T of two subsets of H0[ Note that one side of L10 and L00\ the right one in our _gure\ belongs to the stable set W s"P#[ Thus\ these sides must be the image by T of two disjoint pieces on the right side of H0\ bounded by W s"P#[ The opposite sides of L10 and L00 belong to the side of V0 which is the image by T of the side of H0 opposite to W s"P#[ It follows that L10 and L00 must be the image by T of two disjoint horizontal strips included in H0\ say H10 and H00\ T"H00# L00 and T"H10# L10\ or equivalently\ T−0"L00# H00 and T−0"L10# H10[ Analogous reasonings lead to the de_nition of two disjoint strips inside H1] T−0"L11# H11WH1 and T−0"L01# H01WH1\ H11+H01 f[
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Now\ let us take the images[ Consider the two subsets L10 and L11 of H1[ As these are subsets of H1\ their images by T belong to V1[ The low boundary of such subsets belong to the stable set W s"P#\ thus\ their images by T are two disjoint pieces on the side of V1\ belonging to W s"P# "see Fig[ 03"b##[ The opposite sides of these subsets belong to the side of H1 opposite to W s"P# and thus\ their images by T belong to the opposite sides of V1[ That is\ T"L10# V10 and T"L11# V11 are two disjoint {vertical| strips belonging to V1[ In a similar way T"L00# V00 and T"L01# V01 are two disjoint {vertical| strips belonging to V0\ and so on[ The mechanism is standard "see e[g[ ð5\ 6Ł#[ We shall _nd a Cantor set LWH0*H1 which is both forward and backward invariant by T\ i[e[ T"L# L and T−0"L# L[ Now\ regarding hyperbolicity we note that\ by construction\ we started with two horizontal strips H0 and H1\ having one side on W s"P# and one side on W u"P#[ However\ from the process of construction of the subsets Hi j and Vi j "which also have one side on W s"P# and one side on W u"P##\ we can deduce that\ really\ in a neighbourhood of P\ the structure of the stable and unstable sets of P is such that W s"P# includes in_nitely many {segments parallel| to v9 "i[e[ the local stable set W sloc"P## and the same holds for W u"P#] it includes in_nitely many {segments parallel| to W uloc"P# "see Fig[ 03"c##[ Thus\ we can presume to start the process with two horizontal strips\ H0 and H1\ having two opposite sides on W s"P# and the other two opposite sides on W u"P#[ Then the Cantor set L is hyperbolic[ REFERENCES 0[ Fournier!Prunaret\ D[\ Mira\ C[ and Gardini\ L[ Some contact bifurcations in two!dimensional examples[ In Pro! ceedin`s ECIT |83[ World Scienti_c "in press#[ 1[ Mira\ C[\ Gardini\ L[\ Barugola\ A[ and Cathala\ J[ C[ Chaotic Dynamics in Two!Dimensional Noninvertible Maps[ World!Scienti_c\ Singapore[\ 0885[ 2[ Abraham\ R[\ Gardini\ L[ and Mira\ C[\ Chaos in Discrete Dynamical Systems "A Visual Introduction in two Dimensions#[ Telos\ Springer[\ 0886[ 3[ Mira\ C[\ Rauzy\ C[\ Maistrenko\ Y[\ Sushko\ I[\ Some properties of a two!dimensional piecewise!linear noninvertible map[ Int[ J[ Bifurcation + Chaos\ 0885\ 5"01A#\ 1188Ð1208[ 4[ Gumowski\ I[ and Mira\ C[\ Dynamique Chaotique[ Cepadues Editions\ Toulouse\ 0879[ 5[ Wiggins\ S[ Global Bifurcations and Chaos[ Springer!Verlag\ N[Y[\ 0877[ 6[ Wiggins\ S[ Introduction to Applied Nonlinear Dynamical Systems and Chaos[ Springer!Verlag\ N[Y[\ 0889[ 7[ Maistrenko\ Yu[ L[ and Sushko\ I[ M[\ Bifurcation phenomena for two!dimensional piecewise linear maps[ J[ Tech[ Phys[\ 0885\ 26\ 260Ð267[