Dynamical formation of stable irregular transients in discontinuous map systems

0.73 MB·English

by  Hailin Zou

Dynamical formation of stable irregular transients in discontinuous map systems
Hailin Zou,1 Shuguang Guan,2,3,4 and C.-H. Lai1,4
1Department of Physics and Centre for Computational Science and Engineering,
National University of Singapore, Singapore 117543
2Institute of Theoretical Physics and Department of Physics,
East China Normal University, Shanghai, 200062, P. R. China
3Temasek Laboratories, National University of Singapore, Singapore 117508
4Beijing-Hong Kong-Singapore Joint Center of Nonlinear and Complex systems (Singapore), Singapore 117508
(Dated: January 12, 2010)
0
1 Stable chaos refers to the long irregular transients, with a negative largest Lyapunov exponent,
0 whichisusuallyobservedincertainhigh-dimensionaldynamicalsystems. Themechanismunderlying
2 this phenomenon has not been well studied so far. In this paper, we investigate the dynamical
formation of stable irregular transients in coupled discontinuous map systems. Interestingly, it is
n
foundthatthetransientdynamicshasahiddenpatterninthephasespace: itrepeatedlyapproaches
a
a basin boundary and then jumps from the bundary to a remote region in the phase space. This
J
pattern can be clearly visualized by measuring the distance sequences between the trajectory and
2
thebasinboundary. Thedynamicalformationofstablechaosoriginatesfromtheintersectionpoints
1
of the discontinuous boundaries and their images. We carry out numerical experiments to verify

thismechanism.
]
n
n PACSnumbers: 05.45.-a,87.19.lj
-
s
i
d I. INTRODUCTION chaos was also reported in various types of dynamical
. systems [11–13]. Recently, it was found in the pulse-
t
a coupled oscillators systems which are frequently used to
m Usually, nonlineardynamicalsystems mayhave stable model neuronal activity [14–16]. In all the above works,
attractors such as fixed points, limited cycles, or chaotic
- the stable chaos appears in discontinuous map systems
d attractors. Apart from these, interestingly, it is shown
(ordiscontinuousreturnmaps). Interestingly,it is found
n that irregular long transients can also occur in such sys-
that the stable chaos could also appear in the continu-
o tems. For continuous dynamical systems, these irregular
ous map system where there exists a transition from the
c long transients are usually due to the existence of high-
[ standard chaos to the stable chaos [17].
dimensional chaotic saddles in phase space (see recent

2 review [1] for details). These chaotic saddles often ap-
There are some efforts attempting to illustrate the
v pear after crisis bifurcation [2]. The system may spend
mechanismunderlyingtheformationofstablechaos. For
6
extremely long time in the vicinity ofthe chaotic saddle,
2 example, in Ref. [8], it was conjectured that the stable
and behaves as irregular as chaotic. Due to this reason,
3 chaosisduetothehierarchicalorganizationofsubbasins
thistypeofirregulartransientsisusuallycalledtransient
2 in phase space. The subbasins are subspaces of a basin
. chaos,whichis sensitiveto the initialconditionandthus separated by walls through which an orbit cannot pass
8
haveapositivelargestLyapunovexponent. Forexample,
0 except at portals. The irregular transient is regarded
in Ref. [3], it is found that the development of transient
9 as a sequence of transitions through a hierarchy of sub-
chaosis relatedto the unstable-unstablepair bifurcation
0 basins. However, the formation of these subbasins and
: which involves an unstable periodic orbit in the chaotic portals is still not well understood. In Ref. [18], the
v
attractorandanotherone onthe basinboundary. More-
i stable chaos was attributed to the ordinary chaos in a
X over, another interesting finding along this line is the
continuous system slightly altered from the original dis-
r super-transientwhoseaveragelifetimecouldbeverylong continuous system. One deficiency of this approach is
a even far from the bifurcation point [4, 5]. Such super-
that chaos can exist even in a one-dimensional continu-
transients have also been found in stochastic dynamical
ous map, while stable chaos typically happens in high-
systems [6, 7].
dimensional dynamical systems. In addition, Ref. [10]
However,thereexistsanotherdistincttypeofirregular showed that the alteration could be too large for some
transient that has negative largest Lyapunov exponent, systems. Itwasshownthatthe stablechaosis analogous
usually occurring in discontinuous systems. This phe- to deterministic cellular automata [9]. Along this line,
nomenon was first observed in the coupled map lattice the stable chaos was attributed to the nonlinear propa-
[8]. The complex transients behave irregularly with ex- gation of finite disturbances from the outer regions[10],
ponentialdecayofcorrelationbothintimeandspace[9]. and a stochastic model was presented to understand the
In addition, the transient time usually grows exponen- mechanism of this nonlinear informationflow [12]. How-
tially with system size which makes the attractors un- ever,the direct basic mechanismis still unclear. In spite
reachable in large systems. Due to these properties, the of the works mentioned above, the dynamical formation
transient is termed as stable chaos [9, 10]. Later, stable ofstablechaoshas notbeenwellstudied todate andthe
2
mechanism is still unclear. In particular, how the stable 200,inahyperspherewithcenterxandradiusr. Initially
chaosdevelopsfromthegoverningdynamicalequationsis r is set to be a large value. If all the initial points settle
notfullyunderstood. Onepossiblereasonforthisisthat onto the same attractor as the center x, go to step (3).
these globalbehaviors usually occur in high-dimensional (2) Set the new radius r to be the minimum distance
dynamicalsystems,whichisusuallydifficulttodealwith betweenthe center andthe setofinitial points whohave
mathematically. differentfinalattractorfromthe center. (3)Ifr doesnot
In this paper, we focus on the problem on the dynam- changeinthesequentialm,say5,times,stop. Otherwise
ical formation of stable chaos in the discontinuous sys- repeat the above steps.
tems. For these systems, it is natural to relate the oc- To illustrate this idea, we first consider the following
currence of stable chaos to the discontinuity of the local coupled map lattice with a periodic boundary [8]:
dynamics of the coupled dynamical systems. Motivated
r
by this idea, in this paper, we directly investigate some 1
coupled discontinuous map systems, which are different xi(n+1)= 2r+1 X f(xi+j(n)), (1)
j=−r
from the previous works. Our particular interest is to
reveal how the discontinuity in the local dynamics of a wherethelocaldynamicsisf(x)∼=sx+ω(mod1). When
coupled system can induce stable chaos. To this end,
s is smaller than 1, the system (1) will finally approach
we specifically construct dynamical models whose local
a periodic attractor because of the negative Lyapunov
maps have only contracting pieces with absolute deter-
spectrum. We choose s = 0.9, and ω = 0.118 in this
minantsmallerthan1. Asaconsequence,theoccurrence
paper. We consider nearest-neighbor coupling: r = 1.
of chaos is excluded in such systems, and the generated
ThenumberN ofoscillatorsis28,forwhichtheirregular
long irregular transient is stable chaos by nature.
longtransientisprominent. Fortheindividuallocalmap,
The organization of this papers is as follows. In Sec. there is a discontinuous boundary at x = (1−ω)/s =
II, we show that accompanying stable chaos, a regular 0.98.
patternalwaysexistsinbothcoupleddiscontinuousmaps Fig. 1(a) shows a typical stable irregular transient
and pulse coupled oscillators. In Sec. III, the dynamical in this system. Its corresponding distance sequence to
origin of stable chaos is analyzed and verified by a two- the basin boundaries is shown in Figs. 1(b) and 1(c).
dimensional map. Finally, conclusions are drawn in the To our surprise, we find a regular pattern in the dis-
last section. tance sequence despite of the irregular transient dynam-
ics. As clearly shown in Fig. 1(c), the distance sequence
first gradually approaches the basin boundaries. Then
II. THE REGULAR PATTERN it jumps to some remote regions in the phase space and
ACCOMPANYING STABLE CHAOS begintoapproachthebasinboundariesonceagain. Dur-
ing the whole transient period, such pattern repeatedly
In continuous dynamical systems, the stable mani- occursuntilfinallythe systemsettlesdownonthe trivial
folds of the saddle periodic orbits compose the basin attractors.
boundaries. However, for discontinuous map systems Similarly, the regular pattern in the distance sequence
with only contracting local dynamics, the basin bound- canalsobeobservedintheinhibitorypulse-coupledoscil-
aries could only include the set of points whose dynam- lators. This model describes N oscillators, such as neu-
ics are discontinuous. This set of points comprises the rons, interacting on a direct network by sending and re-
pre-images of the discontinuous boundaries and the dis- ceivingpulses[15,19,22,23]. Thestateofeachoscillator
continuousboundariesthemselves. Intuitively,thestable i is specified by a phase-like variable φi(t) ∈ (−∞,1].
chaos might be related to this set. Normally, we cannot The dynamics of the single oscillator i is given by
obtain this set directly because of the high dimensional-
dφ /dt=1. (2)
ity of the phase space. However, the basin boundaries i
can be easily detected. We can measure the distance
whenφ (t)reachesaphasethreshold1,thisphaseisreset
i
of each point in the trajectory to the basin boundaries.
to zero, φ (t+) = 0, and a pulse is generated. After
Therefore, for each transient trajectory, we will have a i
a delay time τ this pulse is received by all oscillator j
corresponding distance sequence.
havinganin-link fromi. This induces a phase jump in j
The distance of a point x to the basin boundaries B
according to
is defined to be d = minkx−yk , where y ∈ B. This
2
distanced(x)alsoquantifiesthedegreeofstabilityofthe φ (t+τ)+ =min{U−1(U(φ (t+τ))+ǫ ),1}, (3)
j j ji
system under finite perturbation at a given point x. If
theperturbationaddedtothesystemislargerthand(x), wherethe functionU,whichdeterminesthe phasejump,
the system will jump to another attractor. In this sense, is twice continuously differentiable, monotonically in-
the distance is the maximal finite perturbation that the creasing, concave and normalized (U(0)=0 and U(1)=
systemcantoleratewithoutlosingthestability. Weusea 1). Furthermore,thecouplingstrengthisalsonormalized
simple procedure to obtain this distance. The steps are: asǫ =ǫ/k ,wherek representsthenumberofincoming
ij j j
(1) Randomly sprinkle many initial points, for example links of node j (in-degree in graph term). This model is
3
(a)1.0 (a)1.0
0.8
0.6 0.5
1
x
0.4
0.0
0.2
0.0
0 1000 2000 3000 4000 5000 0 500 1000 1500 2000 2500 3000
n n
(b) 0.03 (b) 0.075
0.060
0.02
0.045
d d
0.030
0.01
0.015
0.00 0.000
0 1000 2000 3000 4000 5000 0 500 1000 1500 2000 2500 3000
n n
(c)0.016 (c) 0.05
0.04
0.012
0.03
d0.008 d
0.02
0.004
0.01
0.000 0.00
2600 2800 3000 1000 1100 1200 1300
n n
FIG. 1: (a) The long irregular transients observed in Eq. (1) FIG. 2: (a) The long irregular transients observed for pulse-
for x1. n is the time step. (b) The corresponding distance coupled oscillators for φ2 using oscillator 1 as reference. The
sequence d to the basin boundaries. (c) The enlargement of n represents the numberof times the reference oscillator has
therectangle in (b). beenreset. (b)Thecorrespondingdistancesequencedtothe
basinboundaries. (c)Theenlargementoftherectanglein(b).
equivalenttothestandardleakyintegrate-and-firemodel
with Ui(φ)=γi−1Ii(1−exp(−γiφ)) [15]. and τ = 0.1. The network size is fixed at N = 20. We
Itisconvenienttoinvestigatethedynamicsinareturn observed that long irregular transients exist when dilut-
map by choosing an arbitrary oscillator as reference[22]. ing links from fully coupled networks. Fig. 2(a) shows
Here the oscillator 1 is chosen. When the reference os- one typical transient trajectory for connection probabil-
cillator is reset, the phases are recorded. The number of ity p = 0.2. From the distance sequence to the basin
reset times is used as time step for the return map. The boundaries, as shown in Figs. 2(b) and 2(c), once again,
dynamics can be simulated by an event-by-event based we find the similar pattern as shown in Fig. 1. During
numericalcalculationwhichcanbe appliedtoobtainthe transients, the trajectory repeatedly goes to the basin
solutions for the pulse-coupled network with delay [20]. boundaries and then jumps to some remote regions in
WechoosethesimilarparametervaluesasinRef. [15], the phase space. The existence of the same regular pat-
where the existence of stable irregular transients is ana- terninthe abovetwodifferent systems stronglysuggests
lytically proved, i.e., γ = 1, I = 4.0, b = 1, ǫ = −1.6, that the formation of stable chaos in different discon-
i i
4
tinuous dynamicalsystems actually could havethe same regarded as the generalized map used in Eq. (1) with
dynamical origin. We emphasize that such pattern is more discontinuous boundaries.
uniqueinstablechaos,anddoesnotexistinchaotictran- In the following, we specifically describe the tech-
sients observed in the excitory pulse-coupled oscillators nique to locate the guiding paths accurately for the two-
[21], which should have a different mechanism. dimensionalmapsF whosediscontinuousboundariesare
composed of straight lines. The key procedure of this
techniqueistolocatetheguidingpathswithgivenlength
III. DYNAMICAL FORMATION OF STABLE n andaninitialinterval[A,B]. By varyingnandthe in-
CHAOS terval,wecanlocateallthe guidingpathsofinterest. To
this end, we divide the phase space into squares, or cells
In Sec. II, we observe a regular pattern in the stable bythe discontinuousboundaries. Aninterval[A,B] may
chaosby measuringthe distance to the basinboundaries containstartingpointsoftheguidingpathsifFn(A)and
for each point in the transient trajectory. This regular Fn(B)fallintotwodifferentcells. Considerthemidpoint
pattern displays the internal structure of the stable ir- m= (A+B)/2, we can find a smaller interval [A,m] or
regular transients. During the transient period, the tra- [m,B] whose nth images of the two endpoints fall into
jectory repeatedly goes to different regions on the basin two different cells. We can recursively apply this bisec-
boundaries. This seems to suggest that there is a path tionmethodtoshrinktheintervalandobtainaninterval
thatcanconnecttwodifferentplacesonthebasinbound- [a,b]withgivenaccuracy,say,10−6. Thenonenthimage
aries. Throughextensivenumericalexperiments,wecon- ofthetwoendpointswillbetheendingpointoftheguid-
firmtheexistenceofsuchpaths. Furthermore,itisfound ingpathifitiswithinthedistance10−6 ofdiscontinuous
that the ending point of such a path is an image of the boundaries. After that, we consider the remaining inter-
startingpoint,andboththestartingpointandtheending val[b,B]. Inthis way,we canfind allguiding paths with
point are on the discontinuous boundaries. We thus call length n with starting points belonging to [A,B].
thispathaguidingpath. Thedynamicalformationofthis
path is through the intersection of discontinuous bound-
aries with their images. It can be easily verified that the
1.0
transient structure in the system (1) is formed in this
way where the discontinuous boundaries are x = 0.98.
i 0.8
Thedynamicaloriginofstablechaosliesintheformation
of many guiding pathes in phase space. At the starting
0.6
and the ending points, the dynamics are discontinuous. x)
(
Between these two points, the trajectory usually follows f0.4
the contractingdynamics whichresultsinthe decreasing
distance to the basin boundaries. 0.2
Toverifytheabovemechanismoftheformationofsta-
blechaos,weneedtolocateguidingpathsandthencom- 0.0
0.0 0.2 0.4 0.6 0.8 1.0
pare with the correspondingjumping processesfrom one
region to another region on the basin boundaries. For x
highdimensionalsystems,suchasEq. (1),wecanobtain FIG. 3: A map with only contracting pieces and many dis-
continuousboundaries.
approximately the underlying guiding path by sampling
many initial points, say 106, in a small region where a
A typical transient trajectory in Eq. (4) is shown in
jumping process starts. The length of guiding path n
Fig. 4(a) for e = 0.09. In the distance sequence, as
usually is the same as the jumping process. We can ob-
shown in Fig. 4(b), a regular pattern similar to those in
tain all these trajectories with length n. Then we ap-
Figs. 1(c) and 2(c) is much more evident. Here, since
proximate the guiding path by a trajectory with mini-
our system is only two-dimensional, it is easy for us to
mum quantity D, where D is defined to be the sum of
directly verify our analysis of the mechanism that led to
distance of two endpoints of a trajectory to the discon-
the formation of stable chaos. In Fig. 5, we plot part
tinuousboundaries. Inthisway,wecanverifythatthere
ofthetransienttrajectorywithinaspecifictimewindow,
isaguidingpathgoverningthejumpingprocessfromone
i.e., from n=18 to n=35. As shown in Fig. 4(c), dur-
region of basin boundaries to another region.
ing this time period, the transient trajectory goes from
To be more accurate and more reliable, here we con-
one region of the basin boundary to another region of
struct a two-dimensional map and develop a numerical
the basin boundary. In Fig. 5, we plot both the tran-
techniquetoobtaintheguidingpathswithhighaccuracy.
sient trajectory from n = 18 to n = 35, and a guiding
The dynamical equations of the discontinuous maps are:
path. Remarkably, it is clearly seen that the transient
x = ef(x )+(1−e)f(x ), trajectory exactly follows the guiding path going from
1 1 2
one point of the basin boundary to another point on the
x = ef(x )+(1−e)f(x ), (4)
2 2 1
basinboundary. Since there existplenty ofthese guiding
where the local map f is shown in Fig. 3, which can be pathes in the phase space, we can expect long irregular
5
transients to occur in such coupled map systems, espe- From the microscopic structure, we can also under-
cially when the dimension of the coupled system is very
high.
1.0
25
0.9 31 21 20
0.8
35
0.7
(a) 0.6 27 26
0.9 32
20.5
x
0.4 33 18 19
0.6 0.3 28 23 22 34
1 0.2 29
x
0.1 30 24
0.3 0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
1
0.0
FIG.5: Partoftransienttrajectoryshowninplusandaguid-
0 50 100 150 200
ingpathshownincirclefromn=18ton=35. Thetwolines
n represent two discontinuous boundaries. The number repre-
sents thethetime step n.
0.025
(b)
0.020 stand why stable chaos is unstable against finite small
perturbation(stableagainstinfinitesimallysmallpertur-
0.015
bation)[16]. Duringthelongtransienttime,stablechaos
d
takesplaces near manyguiding paths whose startingand
0.010
ending points belong to the basin boundaries.
0.005
0.000
IV. CONCLUSION
0 20 40 60 80 100 120 140 160
n
In this paper, we investigate the dynamical formation
FIG. 4: A typical transient trajectory (a) for Eq. (2) and its
of long irregular transients with negative largest Lya-
corresponding distance sequence(b).
punov exponent, i.e., the stable chaos, directly based on
the dynamical equations. We show that these irregular
It is interesting to compare stable chaos to the tran- transients actually have certain structure which can be
sient chaos usually occurring in continuous dynamical illustrated by the distance sequence to the basin bound-
systems. The skeleton for transient chaos is the infinite aries. The transients repeatedly approaches the basin
number of unstable periodic orbits (UPOs) embedded in boundaries and then jumps from the boundaries to a
the chaotic saddle. While for the stable irregular tran- remote region in the phase space. Through numerical
sient or stable chaos, the underlying microscopic struc- simulations, it is shown that there exists a guiding path
ture is the guiding path with both starting and ending whoseending pointis animage ofthe startingpointand
points on the discontinuous boundaries. A guiding path both of them are on the discontinuous boundaries. It
is not a cyclic structure, i.e. the starting and ending is these guiding pathes that connect different points on
points are not the same, which is the major difference the basin boundaries, making the dynamics of the sys-
withanUPO.Theregularpatternsoccurringinthehigh tem exhibit long irregular behavior before it goes to the
dimensionalsystemsEq. (1)andEq. (2)impliy thatthe final stable attractor. Thus the present work reveals a
guiding paths areclusteredinphase space,whichis simi- mechanism for the formation of stable chaos in coupled
lar to the dense UPOs. Here,the clusteredguiding paths discontinuous map systems.
are generally associated with the large number of dis-
continuous boundaries in the high dimensional systems.
For Eq. (1), the number of discontinuous boundaries is
N+ N + N +···. Itisexpectedthatthisfastgrowing
(cid:0)2(cid:1) (cid:0)3(cid:1)
Acknowledgments
of the number of discontinuous boundaries with dimen-
sion will make the guiding paths more clustered in high
dimensional systems, somewhat similar to the attractor This work is supported by the National University of
crowding effect [24]. In turn, it will make stable chaos Singapore,andDSTAofSingaporeunderProjectAgree-
more easily observed in high dimensional systems. ment POD0613356.
6
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