long tailed maps as a representation of mixed mode oscillatory systems

Physica D 211 (2005) 74–87

Long tailed maps as a representation of mixed modeoscillatory systems

Rajesh Raghavanb, G. Ananthakrishnaa,b,∗a Materials Research Center, Indian Institute of Science, Bangalore 560012, India

b Center for Condensed Matter Theory, Indian Institute of Science, Bangalore 560012, India

Received 10 December 2004; received in revised form 19 July 2005; accepted 8 August 2005Available online 9 September 2005

Communicated by U. Frisch

Abstract

Mixed mode oscillatory (MMO) systems are known to exhibit generic features such as the reversal of period doubling sequencesand crossover to period adding sequences as bifurcation parameters are varied. In addition, they exhibit a nearly one dimensionalunimodal Poincare map with a long tail. The numerical results of a map with a unique critical point (map-L) show that thesedynamical features are reproduced. We show that a few generic conditions extracted from the map-L are adequate to explain thereversal of period doubling sequences and crossover to period adding sequences. We derive scaling relations that determine theparameter widths of the dominant windows of periodic orbits sandwiched between two successive states of RLk sequence andverify the same with the map-L. As the conditions used to derive the scaling relations do not depend on the form of map, wes s.©

P

K

1

tcal

f

deplexodicodels

s

s:ersalary

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uggest that the analysis is applicable to a family of two parameter one dimensional maps that satisfy these conditon2005 Published by Elsevier B.V.

ACS: 82.40Bj; 05.45Ac

eywords: Multiple time scales; Mixed mode oscillations; One dimensional maps

. Introduction

Dynamical systems with disparate time scales forhe participating modes often exhibit periodic statesharacterized by a combination of relatively largemplitude and nearly harmonic small amplitude oscil-

ations. Such periodic states are called the mixed mode

∗ Corresponding author. Tel.: +91 803 942780;ax: +91 803 600683.

E-mail address: [email protected] (G. Ananthakrishna).

oscillations (MMOs) conventionally denoted byLs

whereL ands correspond to large and small amplituoscillations, respectively. The associated combifurcation sequences consist of alternate perichaotic sequences which have been observed in mand experiments in the area of chemical kinetics[1–5],electrochemical reactions[6–8], biological system[9], and in many physical systems[10–12]. TheseMMO systems typically exhibit the following feature(a) period doubling (PD) sequences and their revin multi-parameter space with respect to a prim

167-2789/$ – see front matter © 2005 Published by Elsevier B.V.oi:10.1016/j.physd.2005.08.004

R. Raghavan, G. Ananthakrishna / Physica D 211 (2005) 74–87 75

Fig. 1. Bifurcation diagrams for the AK model for a plastic instability with primary bifurcation parameter,e and secondary bifurcation parameter,m. (a) Cascading period doubling bifurcations and their reversals forming a bubble structure form = 2.16, and (b) period adding sequences form = 1.2. Chaotic regions exists in vanishingly small parameter regions sandwiched between successive 1s periodic states.

bifurcation parameter keeping other parameters fixed,and (b) crossover to bifurcation sequences of alternateperiodic-chaotic windows when one of the secondarybifurcation parameters is varied wherein dominantwindows of periodicity increase in an arithmetical or-der which we refer to as period adding (PA) sequences[2,4,6–8,12,13]. Usually, MMO systems with thesetwo features are also systems with multiple time scalesof evolution.

As an illustration of the generic features of MMOsystems, we collect some relevant results from our ear-lier study[12,14]on the Ananthakrishna’s model (AKmodel) for a type of plastic instability[15]. As in otherMMO systems, this model also involves disparate timescales. The bifurcation portraits (with respect to a pri-mary bifurcation parameter,e) of the model show pe-riod doubling sequences and their reversal, graduallychanging over to period adding sequences as the sec-ondary bifurcation parameter,m, is decreased. (SeeFig. 1a and b.) As can be seen inFig. 1b, the dom-inant periodic orbits of 1s kind constitute the periodadding sequence. In the parameter space, wherein thereversal of period doubling sequences occurs, periodadding sequences are finite. Periodic orbits of 1s typelose their stability in a period doubling bifurcation as

the (primary) control parameter is increased, restabi-lize through a reverse period doubling bifurcation, andare eventually annihilated in a fold bifurcation[12]. Wehave also studied the next maximal amplitude (NMA)maps[16] obtained by plotting one maximum of theevolution of the fast variable with the next[14]. (TheNMA maps can be regarded as a specific form of thePoincare maps.) These NMA maps show a near one di-mensional unimodal nature with features of sharp max-imum (i.e., the ratio of the height to the width beinglarge)1 and with a long tail as seen in the maps shownin Fig. 2a and b. (Also, see[14,17].) Indeed, severalother MMO systems also exhibit features stated above(and also displayed inFigs. 1 and 2) [5,13,18].

A well studied example of MMO systems is theBelusov Zhabotinsky (BZ) reaction system. Exhaustivetheoretical/experimental studies for the BZ systems intwo parameter space have shown that the NMA mapsof these systems have a unimodal structure with a longtail and show a similar trend as a function of the controlparameters[5,13,19]as in the case of AK model. Other

1 For a one dimensional unimodal map, an acceptable definitionwould bep − f−1(p) is much less thanf (c) − p wheref−1(p) is theonly one preimage ofp distinct from itself. ReferFig. 3for notations.

76 R. Raghavan, G. Ananthakrishna / Physica D 211 (2005) 74–87

Fig. 2. Poincare maps from the AK model form = 1.8 and (a)e = 190.6, and (b)e = 202.7. (Multiple folds arise as a result of finite dissipation.)

MMO systems which display similar features includelasers with a saturable absorber, autocatalytic systemsand a number of oscillating chemical reaction systems[4,11,13,18].

The periodic states representative of MMO systemshave been shown to be associated with systems ex-hibiting global bifurcations in the form of approach tohomoclinicity[20–23]. Although, a variety of tools areavailable for the study of bifurcations in dynamical sys-tems, understanding these features of MMOs has notbeen easy mainly due to the lack of adequate analyt-ical tools to handle the global nature of bifurcationsunderlying the MMOs. However, since information re-garding the nature and properties of periodic orbitsof continuous flow systems are known to be embed-ded in Poincare maps, they have been used effectivelyto understand the underlying bifurcation mechanisms[16,20,24,25]. For instance, considerable insight hasbeen obtained through the Poincare maps derived fromapproach to homoclinicity[20–23]. Poincare mapshave also been used to study the slow manifold struc-ture and hence the origin of MMOs[26]. Since, most ofthe MMO systems exhibit high dissipation, there havebeen attempts to understand the dynamical behavior ofMMO system (specifically BZ system) by modelingits NMA maps as one dimensional discrete dynamicalsystems[19,27–30].

The two distinct bifurcation features, namely, the re-versal of bifurcation sequences and the period addingsequences as seen in MMO systems have been mod-eled separately using one dimensional maps. For in-stance, one of the model suggested for the reversal ofperiod doubling sequencesalone, are maps having asingle critical point along with non-monotonous de-pendence of the control parameter[31]. Another one isa unimodal map having a negative Schwarzian deriva-tive [32]. Yet another one involves a competition be-tween more than one critical point in the dynamics ofthe map[33–35]. Similar efforts to reproduce periodadding sequences have shown that the transmutationof the U-sequences to the Farey sequences or specifi-cally, its subset, the period adding sequences can arisein maps withmore than one critical point with multi-ple parameters[36]. However, experimental and modelMMO systems display Poincare maps with a singlecritical point and yet exhibit a smooth crossover fromreversal of period doubling sequences to period addingsequences for monotonous change of parameter. To thebest of author’s knowledge, there has been no modelmap which capture above described dynamic behaviornor any explanation for the crossover.

The objective of this work is to abstract commonfeatures from the Poincare maps of MMO systems,and impose them in the form of some general con-


straints on one dimensional maps in an effort to modelthe complex bifurcation sequences of higher dimen-sional MMO system. Towards this end, we construct adiscrete example map (hereafter referred to as map-L)that has a shape similar to the Poincare maps of MMOsystems and show that the map exhibits dynamical fea-tures similar to the higher dimensional MMO systemsfor monotonic change in the parameters. Using the de-pendence of the structure of the map-L on the controlparameters (see Section2 for explanation), we extracta few conditions that characterize the Poincare mapsof MMO systems and then analyze the basic mecha-nism responsible for the reversal of periodic sequencesof RLk type. (Admittedly, this is a much less difficultissue than the origin of reversal of all the bifurcation se-quences in these maps.) We also show that the windowsof symbolic sequence RLk are the dominant windowsin the parameter space and derive scaling relations forthe onset of windows of periodic orbits sandwichedbetween successive windows of RLk sequence. The soderived relations connect the slopes near the criticalpoint and unstable fixed point, and are different fromthe usual scaling relations derived in terms of bifur-cation parameters. Investigation of the reversal of PDsequences in turn allows us to understand the mecha-nism for the onset of the period adding sequences aswell. We verify these scaling relations using the map-L.Finally, we discuss the results and the correspondencewith the continuous flow dynamical systems having afinite dissipation.

2

m

(( lly,

( .

U thea bi-fp en-s er-

alization of the Bountis map[31] given by

xn+1 = f (xn; µ, ξ)

f (x; µ, ξ) = a + (ξ − µ) + r(x − c)l

s1 + s2µ(x − c)m(1)

with m ≥ l anda > 0. We refer to this map given byEq.(1) as map-L. We shall use only two parametersµ

andξ of the several parametersl, m, a, r, s1, s2 andc.The other parameters of the mapf (x; µ, ξ) are tunedsuch that the model map has features listed above andthe map looks similar (in structure) to one dimensionalPoincare maps of the MMO systems (Fig. 2). For ournumerical study, we use the following values for themap parameters :l = m = 4.0, a = 3.0, r = 0.6, s1 =1.5, s2 = 0.12 andc = 7.0. We shall useµ andξ asprimary and secondary bifurcation parameters, respec-tively.

The bifurcation diagrams are constructed usingµ asa control parameter for fixed value ofξ. Fig. 3shows atypical shape of the map defined by Eq.(1). This maphas a unique smooth maximum with inflection pointson either side of the maximum. It can be easily verifiedthat the map has a negative Schwarzian derivative ev-erywhere on the positive real line. Sincea > 0, for anappropriate choice ofa, there are no fixed points in theinterval [0, c] for small values ofξ. For a range of val-

F forµ intc

. Example map-L

The features shown inFig. 2 typical of the NMAaps of the MMO systems are :

1) non zero origin2) a symmetric maximum positioned asymmetrica

and3) asymptotic long tail to the right of the maximum

sing a simple two parameter map that mimicsbove features, we first attempt to reproduce the

urcation features of the MMO systems[4,7,11,37], inarticular that of the AK model where we have extive results[12,14,17,38]. Here, we consider a gen

ig. 3. Typical structure of the map-L described in appendix= 4.0 andξ = 18.0. The period one fixed point(p), critical poand the first iteratef (c) are also indicated.


Fig. 4. Bifurcation diagrams for the map-L. (a) Cascading period doubling with a bubble structure forξ = 11.415, and (b) Bubble structure withdominant period three window forξ = 11.43.

ues of the parameterξ < ξ∗, the map has only one fixedpoint, p, which is in the interval [c, f (c)]. For ξ > ξ∗,two more fixed points,p2 andp3, located in the inter-val [0, c], are born in a fold bifurcation atµ = µ∗(ξ).(Both p andp3 are unstable, andp2 is stable.)

In the following, we briefly describe the bifurcationsequences of the mapf (x; µ, ξ) with respect to the pri-mary bifurcation parameterµ for various values ofξ.For small but fixed value ofξ, an increase in the valueof µ leads to a decrease in the value ofp. Forξ < 8.0,the unique fixed pointp is stable for all values ofµ.Beyondξ = 8.0, the fixed pointp loses stability in aperiod doubling bifurcation asµ is increased which isregained in a period undoubling bifurcation thus form-ing a bubble structure (Fig. 4a). For further increasein ξ, the number of nested bubbles grows as 2n withn = 1, 2, 3 . . ..

For ξ ∼ 11.9, a period three window opens up inthe bifurcation diagram separating the period doublingsequences from the reverse period doubling sequences(Fig. 4b). Further increase in the value ofξ unveilsperiod adding sequences of arithmetically increasingperiodic windows coexisting with the bubble structure(Fig. 5a). This sequence of large parameter windowsof stable periodic orbits (inµ) is found to be of RLk

type sequences in the two letter symbolic dynamics

language. Forξ = 12.8, we find that the reversal ofthe dominant bifurcation sequences occur beyondµ ∼4.05.

At ξ = 18.0 (ξ∗ ∼ 13.0), as we increaseµ, atµ =µ∗ = 4.8, the region of the map for small values ofxmakes contact with the bisector giving rise to a fold bi-furcation with the creation of a pair of stable and unsta-ble fixed points,p2 andp3. (The bifurcations diagrambeyond inFig. 5b beyondµ = 4.8 reflects this.) Keep-ing ξ ≥ ξ∗, as the parameterµ is increased towardµ∗,the channel between the bisector and the map gets pro-gressively narrow, thereby stabilizing periodic orbits ofhigher periodicity (SeeFig. 3and alsoFigs. 6a and 8a).The positions of the iterates of higher periodic cycleschange marginally to accommodate the next higher pe-riodic cycle as additional iterates are squeezed in thisintermittent channel. Thus, the intermittency or the foldbifurcation point (µ∗) is an accumulation point for thearithmetically increasing period adding sequences. Be-yond the accumulation point, no further bifurcationsare observed for the stable periodic point (p3).

We note that all the interesting dynamics arises inthe region between the first period doubling bifurca-tion resulting from destabilization of the fixed pointand the re-stabilization of the fixed point, a featureseen in many MMO models as also in the AK model


Fig. 5. The bifurcation diagrams for map-L. (a) period adding sequence coexisting with reversal of period doubling sequences atξ = 12.8(dashed line indicates the unstable monoperiodic orbit), and (b) period adding sequence with fold bifurcation at the accumulation point of RLn

periodic windows forξ = 18.0. Note the similarity withFig. 1b.

[4,7,11,12,14,37]. One other feature of MMO systemsis that the complex bifurcation sequences occur in theregion of the parameter where the amplitude of the prin-cipal periodic orbit (the periodic orbit born out of aHopf bifurcation, which in the case of the map refersto the fixed point), decreases and approaches the re-verse Hopf bifurcation point[4,7,11,12]. This featureis also captured as is clear fromFig. 5a where the un-stable monoperiodic orbitp(µ) is shown by the dashedline.

3. Dynamics of the long tailed maps

Thus, our first objective, namely to construct a maphaving a unique critical point that captures the dynam-ical features of MMO systems for a monotonic changein the parameters is realized. Therefore, one expectsto recover these features on a basis of a few generalconditions that we extract from map-L. One can easilyverify that map-L satisfies the following conditions.

∂f

∂µ(x) < 0, (2)

∣∣∣∣ ∂f∂µ (c)

∣∣∣∣ <∣∣∣∣ ∂f∂µ (p)

∣∣∣∣ (3)

wherec is the critical point andp the fixed point definedby f (p) = p. We consider a general class of two pa-rameter family of mapsf (x; µ, ξ) which is monotonicon either side of a unique critical pointc satisfyingthese two conditions. These two conditions describethe dependence the maps on the parameterµ whichidentify as the primary bifurcation parameter. Geomet-rical meaning of Eq.(3) is that the rate of approachof the fixed point (p) is faster than that of the criticalpoint (c) and thus, the magnitude of the slope atp in-creases withµ. It is intuitively clear that Eq.(3) leadsto a region where the slope is small for largex, whichwe shall refer to as long tail nature (as inFig. 2). Toensure this, we shall assume thatf (x, µ, ξ) decreasesat a slow rate with a formf (x) ∼ x−α, for largex, withα > 0 and small. As in map-L, the map moves down asa function ofµ (Eq. (2)), we assume an intermittencychannel to left ofc becomes operative. Such a channelis actually observed in many systems[4,7,8,12,13]. Weshall, henceforth, refer to the general class of maps de-termined by these few conditions as long tailed maps.

To fix up the preliminaries, we start with some nota-tions and definition. For the sake of brevity, wherevernecessary, we will usef (x) in placef (x; µ, ξ). Thenth iterate ofx is denoted byfn(x) = f (fn−1(x)) =f ◦ fn−1(x). The iterates off (x; µ, ξ) are given by


θ : {x0, x1, x2, . . . , xn, . . .} with the respective neigh-borhoods given by{I0, I1, I2, . . . , In, . . .}, wherexn ∈In. A point q is k-periodic if f k(q; µ, ξ) = q, whilef i(q; µ, ξ) �= q for 0 < i < k − 1. In particular, we de-note the period one fixed point byp. The eigen value

of a k-period cycleq is ∂f k

∂x(q) = λ. If |λ| < 1, then

q is stable. Further, we will always choose the initialpoint,x0, to be the iterate that is closest to the criticalpoint in the stable periodic window. Finally, we willdeal with the nontrivial dynamics of the map in the in-terval I = [c1, c0], wheref (c) = c0 andf 2(c) = c1.Hereafter, we refer to the parameter windows of sta-ble periodic orbits using the corresponding symbolicdynamics notation.

3.1. Reversal of bifurcation sequences

Before we proceed further, we give below somederivatives which will be used to establish subsequentresults. Using the chain rule for iterates of maps, wehave

∂fn

∂x(x0) =

n−1∏i=0

∂f

∂x(f i(x0)). (4)

Using Eq.(4), we can write

∂2fn

∂x2 (x0) = ∂

∂x

(∂fn

∂x(x0)

),

( )

S leadt

al ofp odall plerp thed

argue that these type of periodic orbits are the dominantones.)

To pin down the ideas, consider the map-L whichsatisfies the requirement of the long tailed maps. Thedominance of the RLk periodic orbits are well visual-ized in the bifurcation diagrams of map-L (Fig. 5a).A prominent feature of the bifurcation diagram is thepresence of relatively dark bands of iterates (corre-sponding to superstable orbits) that run across the bi-furcation diagram connecting successive RLk periodicwindows. (Similar features are routinely observed inbifurcation diagrams of MMO systems also.) Thesebands arise as a consequence of the increased stabil-ity of the iterates closer to the critical point of the mapand are known as supertracks in the literature[3]. It isclear that these bands share a similar sequential struc-ture of iterates as the RLk sequence, specifically thesuperstable point of the periodic orbit lies on the enve-lope of these bands. In the following analysis and later,we utilize this contiguous nature of envelope of bandsto determine the parametric dependence of iterates.

Consider then iterates ofk + 2 periodic cycle whichconsists of one visit to the right of the critical point,c,andk visits to the monotonically increasing arm of themap. We examine the reversal of the sign of Eq.(6), asµ is increased for periodic orbits of the form RLk as the

reversal of the periodic sequence is assured if∂f 2

∂µ(x0)

changes sign atµ = µr2. To do this, first consider thesecond iterate (n = 2) of Eq.(6). Usingx0 = c, we get

E q.( nce

e

t ei all

µ ,

L

m

t fµ e

c t

µ

=n−1∑k=0

(∂2f/∂x2)(f k(x0))

(∂f/∂x)(f k(x0))

×n−1∏j=0

∂f

∂x(f j(x0)). (5)

imilarly, the parameter dependence of the iterateso

dfn

dµ(x) = ∂f

∂µ(fn−1(x)) + ∂f

∂x(fn−1(x))

∂fn−1

∂µ(x).

(6)

In order to understand the mechanism of reverseriod doubling sequences exhibited by the unim

ong tailed maps, we restrict our discussion to a simroblem namely, the reversal of period doubling ofominant sequences of the type RLk. (We will soon

∂f 2

∂µ(c) = ∂f

∂µ(f (c)) + ∂f

∂x(f (c))

∂f

∂µ(c). (7)

q. (2) implies that the first term on the RHS of E7) is negative while the second term is positive si∂f∂x

(f (c)) and ∂f∂µ

(f (c)) (Eq. (2)) are negative. As th

ail of map goes asx−α, the magnitude of the slops small for smallα. Hence, there is a region of sm

for which∣∣∣ ∂f∂µ (f (c))

∣∣∣ > ∣∣∣ ∂f∂x (f (c)) ∂f∂µ

(c)∣∣∣ and thus

HS of Eq.(7) is negative. Asµ is increased (asf (c)

oves up towardc),∣∣∣ ∂f∂x (f (c))

∣∣∣ increases and∂f2

∂µ(c)

urns positive, say atµ = µr2. (Note that the value or2 is in principle a function ofξ. For example, in th

ase of map-L, forξ = 12.8, ∂f 2

∂µ(c) changes sign a

∼ 3.4.) Beyondµ = µr2,∂f2

∂µ(c) increases withµ.


Forn > 2, from Eq.(6),

∂f 3

∂µ(c) = ∂f

∂µ(f 2(c)) + ∂f

∂x(f 2(c))

∂f 2

∂µ(c). (8)

For periodic orbits of type RLk, ∂f∂x

(f 2(c)) > 0, and

from Eq.(2), ∂f∂µ

(f 2(c)) < 0. Further, from the previous

discussion, since∂f2

∂µ(c) is zero atµ = µr2, LHS of Eq.

(8) turns negative at this value. Asµ is increased,∂f2

∂µ(c)

turns positive and the second term on the RHS of Eq.(8) is an increasing function ofµ. Thus, the LHS whichis negative forµ < µr2 becomes positive at some valueµ = µr3 > µr2. Similar arguments can be used to showthat∂f

n

∂µ(c), which is negative forµ < µrn−1 will change

sign atµ = µrn .An useful result of this simple analysis is that the re-

versal of successive periodic orbits occur at increasingvalues of the parameter given by

µr2 < µr3 < µr4 < . . . (9)

Later we will use this result to show the order of for-mation of isolated bifurcation structures. These resultscan be verified from the bifurcation portraits of map-Lshown inFig. 5a. Changes in sign of∂f

n

∂µ(c), manifest

as changes in the sign of the slope of the envelope ofsuperstable iterates with parameter. It is clear that loweroutermost envelope corresponding to the second iterateof critical point changes the sign of slope first, signalingt

odico s ort lasug n topi

W

o the

left of c, the sign of∂2f k

∂x2 (c) is positive for all allowed

values ofµ. This positive curvature of the map and its

parametric dependence,∂f k

∂µ(c) < 0 for µ < µrk , im-

plies that ak-periodic orbit is created due to a tangentbifurcation asµ is increased. For further increase inµ, f k(c) comes down, triggering a period doubling bi-furcation making the periodic orbit unstable. Increas-ing µ beyondµrk reverses the period doubling since∂f k

∂µ(c) becomes positive, i.e., the region aroundc of

f k(x) moves up eventually restabilizing thekth peri-odic orbit. Further increase inµ destroys the stableperiodic orbit in another tangent bifurcation (seeFig.5a). The creation of a periodic orbit, its destabilizationand re-stabilization, followed by its destruction con-stitutes an isolated bifurcation curve (isola). This hap-pens for each of the allowed periodic orbit of the formRLk.

From the above arguments and Eq.(9), it is clearthat the isola corresponding to the smallest allowedperiod orbit contains all isolas of the allowed peri-odic orbits of largerk. In the case of map-L, it isclear that the value at which different iterates show

reversal in sign∂fn

∂µ(c) are different as can be seen

from Fig. 5a and the reversal of all the allowed pe-riodic orbits of RLk type occurs beyondµ ∼ 4.05(ξ = 8.0) and the maximum allowed periodic sequenceperiod five (k = 3). Indeed, similar isola features arew Os

meo am-pain es.H theaw ts.E sesw earsd

ea-t m-eU -

he onset of reversal of the periodic sequence.A natural consequence of the reversal of peri

rbits is the presence of isolated bifurcation curvehe isolas. In the following, we trace the origin of isosing the first two constraints (Eqs.(2) and (3)) on theeneral class of maps. Again, we restrict our attentioeriodic orbits of the form RLk. Consider usingx0 = c

n Eq.(5) for an allowedk-periodic RLk−2, we get

∂2f k

∂x2 (c) = ∂2f

∂x2 (c)k−1∏n=1

∂f

∂x(xn)

= ∂2f

∂x2 (c) · ∂f

∂x(f (c)) ·

k−1∏n=2

∂f

∂x(xn) (10)

e first note that excepting∂f

∂x(f (c)) and

∂2f

∂x2 (c), all

ther terms are positive. Thus, as the visits are to

ell documented in bifurcation diagrams of MMystems[4,12].

It is possible to extend these arguments to sother sequences of periodic orbits as well. For exle, consider periodic orbits of RLkR2m type. Similarrguments as before show that (∂2f k+2(m+1)/∂x2)(c)

s positive and (∂f k+2(m+1)/∂µ)(c) changes sign fromegative to positive, hence forming isola structurowever, there are other kinds of periodic orbits inllowed symbolic sequence (for example, RLkR2m+1),hose reversal is interrupted by the homoclinic poinxtending the above type of arguments to such caill depend on the nature of the sequences and appifficult.

As discussed in the introduction, one important fure of MMO systems is the dominance of the parater windows of stable periodic orbits of RLk type.sing Eq. (10), we show that the width of the pa


rameter windows of these stable periodic orbits de-pend critically on the structure of the map as deter-mined by Eqs.(2) and (3). Consider Eq.(10) for kandk + 1 stable periodic orbits. In comparison withthe kth periodic orbit, there is an extra term in theproduct arising from an iterate falling on the mono-tonically increasing arm. Since theµ value of the(k + 1)th periodic orbit is larger than that forkth one,which in turn impliesf (c) has moved towardc, wehave

k−1∏n=2

∂f

∂x(fn(c)) <

k∏n=2

∂f

∂x(fn(c)). (11)

Further, sincef (c) moves toward the critical point,∣∣∣∣∂f∂x (f (c))

∣∣∣∣ increases. Noting that∂2f k

∂x2 (c) is negative,

whose magnitude increases withk (or µ), it impliesthat f k+1(x) exhibits a sharper minimum aroundcthan that for thekth periodic orbit. Assuming that theextent of the neighborhood aroundf k(c) can be ap-

proximated byFk ∼ ∂2f k

∂x2 (c)(x − c)2/2, the window

width of the kth periodic orbit can be approximated

by the product ofFk and∂f k

∂µ(c). Since|Fk| < |Fk+1|

a

∣∣∣∂f k∣∣∣ ∣∣∣∂f k+1

∣∣∣w ato chtc me-t nge

i

( ere

ca -e er ofs byt them oww ss

3.2. Relations between the stability of periodicorbits and structure of the map

Having argued for the predominance of the periodicorbits of sequence of RLk type in the parameter space,we show that periodic windows contained in the chaoticregion bounded by the windows of periodic orbits de-noted by RLk and RLk+1 have much smaller parameterwidth compared to that of RLk and RLk+1. Numericalresults on the map-L are also presented in support ofthe analytical result.

The PA sequence also manifest in NS-unimodalmaps[39] as a part of MSS sequence[40]. In the MSSsequence, between any successive RLk−1 and RLk win-dows, there exist windows of allowed sequences of thetype RLk[S], where [S] is a sequence consistent withthe allowed symbol sequences[41], i.e.,

RLk−1 ≺ . . . ≺ RLk[S] . . . ≺ RLk.

Considering only the increasing behavior of long tailedmaps (prior to reversal sequences), we assume that theMSS ordering corresponding to the periodic orbits ofNS-unimodal maps may be applied for these types ofmaps also.

Let the bifurcation values ˜µk correspond to the lastgenerationk-period orbit in the MSS sequence, i.e.,for µ > µ̃k no more cycles of periodk are present.Thus, the parameter values are ordered in the followingway:

µ

w ion[t

theb odicwa[ -r olics -d ,t nce)p ertp e-

nd ∣∣ ∂µ (c)∣∣ < ∣∣ ∂µ(c)∣∣, it clear that the window

idth of the kth periodic orbit is larger than thf (k + 1) periodic orbit. Since the value at whi

he reversal of PD sequence of the type RLk oc-urs depend on the secondary bifurcation paraer ξ, it is possible that there may not be a cha

n the sign of∂f k

∂µ(c) beyond a certain value ofk.

Note that if∂f k

∂µdoes not change sign, then, th

annot be a change in sign form > k. ) Thus, onlyfew periodic sates of RLk would be seen. How

ver, for the period adding sequences, the numbtates of RLk can go unbounded, which is ensuredhe existence of small channel near the origin ofap. Then, for RLk periodic sequences, the windidths of periodic orbits of the type RLk decreaselowly.

3 = µ̃3 < µ̃h3 < µ̃4 < µ̃h

4 < µ̃5 < . . .

hereµ̃hk refers to degenerate homoclinic bifurcat

42] (whereinf k(c) = p) of the periodic pointp be-weenkth and (k + 1)th periodic orbits.

Using a degenerate homoclinic bifurcation asoundary, the parameter space for all the periindows between RLk−1 and RLk (of period k + 1nd k + 2) can be divided into two regions, asRI :µ̃k+1, µ̃

hk+1] andRII : [µ̃h

k+1, µ̃k+2]. Consider the peiodic window corresponding to the simplest symbequence in the regions RI where stable periodic winows are of the type RLkR[S]. Within this sequence

he most dominant (in the sense of the MSS sequeeriodic sequence is RLkR which has one period high

han the basic sequence RLk. Using Eq.(4), for a ktheriodic orbit (f k(x0) = x0), the eigen value of the p


riodic orbit,λ, is given by,

λ =n=k−1∏n=0

∂f

∂x(xn) (12)

and for stability|λ| ≤ 1.0. (For the period one fixed

point, we haveλp =∣∣∣∣∂f∂x (p)

∣∣∣∣ < 1.0 for µ ≤ µ2.)

Within the first order, we approximate all iteratesfalling in the neighborhoodU of p as having the sameslope as atp. For the sake of clarity, consider the se-quence RLkR which is part of the sequence RLkR[S] inRI . Clearly, the structure of the map ensures that oneiterate of this sequence belongs toU (see for exam-ple map-L inFig. 6a ). In the case of RLkR sequence,f k+2(x0) falls in neighborhoodU of p. Using Eq.(3)for stability of the periodic orbit,∣∣∣∣∣∂f∂x (x0)

∂f

∂x(xk+2)

k+1∏n=1

df

dx(xn)

∣∣∣∣∣ ≤ 1 (13)

wherex0 is an initial point chosen to be in the neigh-borhood of the critical point. (See below Eq.(14) forjustification for the choice ofx0.) Moreover, structureof long tailed maps as derived from Eqs.(2) and (3)indicate that a dominant number of iterates are trappedin a intermittency channel like region near the originof the map. (This point is best illustrated by consider-

ing the concrete example of map-L shown inFig. 6a,where RL3 and RL4R are shown.) Thus, for largek,noting that iterates other thanx0 andxk+2 change onlymarginally, for stability of the periodic orbit, we have

C

∣∣∣∣∂f∂x (x0)∂f

∂x(xk+2)

∣∣∣∣ ≤ 1

whereC(=∏k+1

n=1 f ′(xn))

is a constant factor. Since

xk+2 falls in the neighborhood ofp ∈ U,

∣∣∣ ∂f∂x (xk+2)∣∣∣ ∼

|λp|( 1). Hence,∣∣∣∣∂f∂x (x0)

∣∣∣∣ ≤ [C|λp|]−1. (14)

Recall that for any periodic orbit, one iterate is alwayslocated in the neighborhood of the critical point in theinterval [xSN

0 , xPD0 ], wherexSN

0 andxPD0 correspond to

the onset of the periodic orbit through a saddle node bi-furcation atµSN

n and the destabilization of the periodicorbit through a period doubling bifurcation atµPD

n . Asthe parameter is increased fromµ = µSN

n , the iteratex0 traverses fromxSN

0 to xPD0 throughc. Then, Eq.(14)

implies that for the stability of the periodic orbit RLkR,the iterate (x0) is restricted to a smaller extent of theneighborhood of the critical point than that correspond-ing to RLk in order to compensate for the increase of

F µ = 3.

s ined fr fit.
ig. 6. (a) Map-L and the periodic orbit of form RL3 and RL4R forcaling of ∂f
∂x(xSN

0 ) with |λp|−1 for RLnR periodic sequences obta
7831 andµ = 3.99, respectively (ξ = 18.0). (b) Verification of theom the example map-L. The dashed line is drawn for a linear


∣∣∣∣∂f∂x (xk+2)

∣∣∣∣ ∼ |λp|. Assuming a uniform change inx0

with respect to the parameterµ as it traverses fromxSN0

to xPD0 and using the fact thatλp 1.0, the parameter

windows of RLkR is smaller by a factorλp than thatfor RLk. Similar arguments can be used to show thatany sequence with 2m + 1 extra visits to the right thanRLk would have a window width smaller by a factorλ−(2m+1)

p . (The assumption here is that these visits to

the right are close top.) Since any RLkR[S] sequenceconsists of 2m + 1 right visits over and above the ba-sic sequence RLk, the width of this periodic window isvanishingly small for largem. Thus,∣∣∣∣∂f∂x (x0)

∣∣∣∣ ≤ C−1|λ−(2m+1)p |. (15)

More specifically, Eq.(15) allows us to get a scal-ing relation between the onset of the periodic windowsof type RLkR2m+1 which can be verified by using themap-L. For this we fix the control parameter,µ at thefold bifurcation points of the periodic windows so thatthe equality holds in Eq.(15). Under this condition,

Eq. (15) (for m = 0) implies that∣∣∣ ∂f∂x (xSN

0 )∣∣∣ λp = C−1

which is almost independent ofk in RLkR. To verify

this, we have plotted∣∣∣ ∂f∂x (xSN

0 )∣∣∣ as a function of|λp|−1

which exhibits linear scaling behavior for the onset val-ues of RLnR windows using the map-L (Fig. 6b). In asimilar fashion, we have shown a plot of∂f (xSN

0 ) asa -sN d,t6

rem tt dingR ico oil ht tiallya pesa e atp fortb

Fig. 7. Verification of the scaling of∂f∂x

(xSN0 ) with |λp|−(2m+1) for

RL2R2m+1 periodic sequences obtained from example map-L.

seen to increase by a factorλ2p compared to RLk orbit.

(The approximation gets better with increasing valueof λp.) Hence, we have

∣∣∣∣∂f∂x (x0)

∣∣∣∣ ≤ C−1 · |λp|−2. (16)

Following arguments presented forRI , we see thatthe iteratex0 is restricted to a smaller extent aroundthe critical point by a factorλ−2

p which in turn leads to

the width of RLkR2 being smaller by the same factorcompared to RLk. Using the equality, the above relationhas been verified numerically for the map-L with thecontrol parameter,µ, kept at the fold bifurcation point.Fig. 8b shows the scaling for the onset of the periodicorbit RLkR2 with λ−2

p which confirms the analyticalresult.

While we have derived the scaling relations for a par-ticular sequence of periodic windows, namely RLkRm,stable windows of other symbolic sequences also existin the chaotic region between RLk and RLk+1. How-ever, as the structure of the map ensures that any peri-odic orbit with periodicity higher than that of RLkR andRLkR2 in the regionsRI andRII , respectively, typicallyhave more number of iterates falling into the neighbor-hoodU of p ensures that the other window widths ofthese periodic orbits are increasingly small.

∂x

function of|λp|−(2m+1) for variousm values correponding to periodic orbits of RL2R2m+1 type (Fig. 7).ote that, even though largek approximation is use

he scaling relations work well whenk is not large (Figs.b and 7).

In the regionRII , the allowed periodic orbits aade up of the sequence RLkR2[S] which has at leas

wo more iterates to the right than the corresponLk orbit. (SeeFig. 8a.) The smallest allowed periodrbit in this region is RLkR2 which has exactly tw

terates more than the periodic orbit RLk both of whichie in the neighborhoodU of p. (Note that even thoughe actual distance of these iterates may be substanway fromp, due to the structure of map, the slot these points can be approximated by the slop.) Thus, approximating the derivatives of the maphese two iterates falling in the neighborhoodU of p toeλp, the eigen value of the periodic orbit RLkR2 is


Fig. 8. (a) Periodic orbit sequence of RL4R (dotted line) and RL4R2 (continuous line) forµ = 3.990345 andµ = 4.001927 forξ = 18.0 forthe map-L (the maps are indistinguishable from each other on this scale). (b) Verifications of the scaling relation∂f

∂x(xSN

0 ) with |λp|−2 for RLnR2

periodic sequences obtained from map-L.

4. Concluding comments

In summary, the numerical study of the map-L hav-ing a unique critical point shows that it captures thetypical dynamical features of MMO systems, in par-ticular that of the AK model, such as the reversal ofperiod doubling sequences and the crossover to periodadding sequence. We extract a few general conditionsfrom the map-L (Eqs.(2) and (3), the long tail prop-erty and an intermittency channel) that define a generalclass of long tailed maps. Using these conditions, wehave demonstrated the reversal of period doubling se-quence and crossover to the period adding sequencesas a part of the complex alternate periodic-chaotic se-quences in MMO systems. Using these two conditions,the dominance (width) of the periodic orbits of the typeRLkRm are shown to be controlled byscaling relationsthat connect the slope near the critical point to that atthe fixed point. We stress that these scaling relationsare very different from those usually derived where thewidths of the periodic orbits are evaluated as a func-tion of the parameters. Again, map-L has been used toverify these scaling relations.

Since, many maps of experimental MMO systemsexhibit generic features similar to those used here,the scaling relations derived for the onset of the pe-

riodic orbits of the form RLkRm can be taken to serveas a check for the correctness of the analysis. How-ever, while earlier experiments do show the domi-nance of the period adding sequences[30], the ex-isting results are not accurate enough to verify thescaling relations derived here. We believe that care-ful experiments, particularly in laser systems wherethe accuracy of control is high, should bear out ourresults.

The long tailed nature of the maps has been noted inexperimental NMA maps of BZ reaction systems andits importance in modeling the bifurcations sequencehas also been recognized[19]. The intended trapezoidalmap used in the study by Coffman et al.[19] to repro-duce specific alternate periodic chaotic bifurcation se-quence includes typical long tail map features namely,(a) the long tail part with magnitude of slope far lessthan unity (b) large magnitude of slope for period oneorbit, and (c) the re-injection is close to the origin ofthe map. Even though, these are minimal features re-quired for the map to model the complex bifurcationsequences, the exact form of the maps are not required.Indeed, our effort here has been directed at demonstrat-ing that the conditions are general enough to reproducethe dominant bifurcation sequences of the MMO sys-tems.


Here it is worth while to comment on connection be-tween the long tail nature of the NMA maps of MMOsystems and dissipation, in view of the fact, they con-stitute the dimensionally reduced form of the contin-uous time systems. In these class of maps, apart fromthe neighborhood of the critical point, there exists alarge tail region wherein the slope is small. By def-inition, these regions are related to high dissipation,since the dispersion of the iterates in its next iterationis suppressed (i.e., given a neighborhoodI1 in the tailregion,I2 = f (I1; µ, ξ) � I1). One type of sequenceof periodic orbits which include visits to the regionsof high dissipation with no visits to the neighborhoodcontainingp are the RLk sequences. High dissipationfavors periodic orbits, hence the stability of these pe-riodic sequences are enhanced at the expense of thechaotic regions[43]. In the same spirit, the windowsof periodic orbits occurring within the chaotic regionbetween any two successive RLk sequences, involvingat least one iterate in the neighborhood of the unstablefixed point and favoring negative dissipation (local ex-pansion) have window widths smaller by a factor ofλp.For a maps under consideration, the eigen value of theperiod one fixed point is large and consequently smallerwindows widths for the periodic orbits contained inthe chaotic region sandwiched between RLk−1 andRLk.

Since the dissipation involved in the higher dimen-sional continuous time system is related to the sepa-ration of time scales operating in the system, a largers ionaln ap.S ntiali thepfi heo r thek nals l crit-i dy-n it isc l pa-r andp y at-t on-s n bes s inh

Acknowledgments

This work was completed while R.R. was vistingCenter for Condensed Matter Theory (CCMT), IndianInstitute of Science. R.R. thanks the Institute for kindhospitality and Department of Science and Technologyfor financial support.

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long tailed maps as a representation of mixed mode oscillatory systems

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