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Physics Letters A 352 (2006) 83–88

www.elsevier.com/locate/pla

Highly incoherent phase dynamics in the SprottE chaotic flow

J.M. González-Miranda

Departamento de Física Fundamental, Universidad de Barcelona, Avenida Diagonal 647, 08028 Barcelona, Spain

Received 28 July 2005; received in revised form 2 November 2005; accepted 8 November 2005

Available online 28 November 2005

Communicated by C.R. Doering

Abstract

Despite its simple mathematical structure, the SprottE flow displays a chaotic dynamics whose complexity increases with the bifurcparameter. This complexity shows up in the phase dynamics, which becomes highly incoherent. Such behavior is interpreted as the suof the dynamics of several different embedded standard maps. 2005 Elsevier B.V. All rights reserved.

PACS: 05.45.-a; 05.45.Ac; 05.45.Pq

Keywords: Chaotic oscillators; Phase coherence; Bifurcation diagram

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Chaotic oscillators are dynamical systems whose timelution is characterized by an intrinsic instability which makthem unpredictable, and a fractal structure that makes tcomplex. One might expect that the mathematical modelsessary to describe such dynamics have to be very complicHowever, it is possible to observe chaotic oscillations in modthat bear a relatively simple mathematical structure. In paular, for systems whose dynamics occurs in a continuous tit has been found[1] that it is possible to have chaotic osclations for a three-dimensional flow described by a vector fiwhich has only five terms, being nonlinear only one of theMoreover, by means of a systematic computational searchthe simplest chaotic flows, Sprott[2] has determined a set onineteen chaotic flows whose fields have five terms includtwo nonlinearities, or six terms including only one nonlinearThese are mean to be among the simplest existing chaoticregarding its mathematical structure.

Research performed in the last three decades has provechaotic oscillators are ubiquitous in science, medicine andgineering. Therefore, besides its intrinsic interest, the studchaotic dynamics has become an issue of fundamental imtance because of the realm of its potential theoretical and prcal applications. In particular, simple mathematical models

E-mail address: jgm@ffn.ub.es(J.M. González-Miranda).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2005.11.057

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display complex dynamical behaviors are of great interestcause they allow easy study of such complexities. In this Lewe will present the results of an study of one of the Spchaotic flows[2], the SprottE flow, which displays a compledynamics with some special features. These include an alcontinuous and steady decrease of the coherence of theof the chaotic oscillations[3,4], which is a signal of an increasin the irregularity of the dynamics. This makes this systemcandidate for further study, specially in the subfield of synchnization and control of chaos[5], which is a very active onewithin the field of nonlinear dynamics and chaos.

We will study a nonlinear oscillator which evolves in a thredimensional space of variables,(x, y, z). The dynamics of thissystem is given by the following set of autonomous ordindifferential equations, written in dimensionless form:

(1)x = y · z,(2)y = x2 − y,

(3)z = 1− α · x.

This oscillator, withα = 4, has been introduced by Sprott[2]as one the four dissipative flows, with only two quadratic nlinear terms out of five, which can display chaos. We hextended here this mathematical model by means of the insion of the variableα as a bifurcation parameter, which allowto explore the variety of dynamical behaviors available to

84 J.M. González-Miranda / Physics Letters A 352 (2006) 83–88

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system. This does not alter the simplicity of the equations,allows to investigate the variety of chaotic attractors thatflow has available.

The divergence of the flow given by Eqs.(1)–(3) is equalto −1, so it contracts volumes in phase space; i.e. it is asipative system for all values ofα. These equations have onone fixed point,xf (α), which is located at(1/α,1/α2,0). Anelementary linear analysis shows that the eigenvalues of thearization atxf areλ1 = −1, andλ2,3 = ±i/

√α. For α < 0,

xF is unstable because the system has three real eigenvtwo of them negative (λ1 = −1, λ2 = −1/

√|α| ), and one positive (λ3 = 1/

√|α| ). For α > 0, xF has a stable manifold odimension one, given by the eigenvectorv1 = (0,1,0) of λ1,and a two-dimensional neutral manifold associated to the cplex conjugate eigenvaluesλ2,3, which have a null real part ana modulus which decreases from infinity accordingly to atential law ofα.

We have studied numerically the solutions of Eqs.(1)–(3)for initial conditions far enough ofxf as to allow the effects othe nonlinearities to show up. A standard fourth order RunKutta algorithm[6] was used for this aim. Different time stepranging fromτ = 0.001 to τ = 0.040, have been used to tethe robustness of the solutions found. Several initial conditrandomly chosen within a sphere of radius 10, centered axf ,have been considered to test the existence of several solufor the sameα. The trajectories that we have analyzed are ththat follow a previous large evolution, always greater than6

time steps from the initial condition chosen, to avoid transitoprevious to the stationary regime. Forα 0, all solutions founddiverge to infinity; therefore, we will concentrate onα > 0 inthe remainder of the Letter.

The nonlinear dynamics of the SprottE flow, given byEqs. (1)–(3) with α = 4.0, is chaotic with a maximal Lyapunov exponentλ1 = 0.078> 0 [2]. Projections of a trajectory of the system ontox–z and they–z planes, displayed in(Fig. 1(a), (b)), show that the dynamics in phase space isaperiodic rotation aroundxF . The projection onto thex–z planedescribes a proper rotation in the sense[7] that the trajectory inphase space can be described as a rotation around a uniquter, xF here, and along a well-defined direction, clockwisethis case. The projection onto they–z plane shows that whilethe phase point rotates around the fixed point in thex–z plane,it moves back and forth along the positive part of they-axis,which is the direction of the stable manifold. This type of ortopology is maintained whenα changes.

A global view of the dynamics given by Eqs.(1)–(3)whenα

is varied is given by the bifurcation diagram and the Lyapuspectrum presented inFig. 1(c) and (d), respectively. We havcomputed bifurcation diagrams from the set of relative maxi(X,Y,Z), obtained from the time series of the system variab[x(t), y(t), z(t)]. These trajectories have been computed foset of 550 evenly spaced values ofα, using the same initial condition for all of them. The results are essentially the samethe three variables, so we present here only the bifurcationgram obtained fromz(t). The Lyapunov spectrum,(λ1, λ2, λ3),has been computed by means of standard techniques defor flows [8].

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Fig. 1. The dynamics of the SprottE flow as illustrated by projections onto (athe x–z and (b) they–z planes of a trajectory computed forα = 4.0. A crossindicates the position of the fixed point,xF . (c) The bifurcation diagram, an(d) the two largest Lyapunov exponents as functions of the bifurcation parter,α.

Fig. 1(c), (d) show a typical transition to chaos through ariod doubling cascade forα small (α 3.36). This is similar tothat of the standard map, and also alike to the one displayethe simplest chaotic flow studied by Sprott[1]. For intermediateand large values ofα we observe a fast increase in the slopethe upper envelope of the bifurcation diagram, while the larLyapunov exponent stays positive and growths steadily. Mover, nearαC1 = 3.5, andαC2 = 4.1, discontinuous changeoccur in the wide of the bifurcation diagram chaotic band.

A detailed study of the bifurcation diagram made forα ∈[3.3,3.6], presented inFig. 2(a), (b), unveils a complexity beyond the typical standard map like behavior. There is indea discontinuous change in the attractor size. Moreover, anomenon of bistability is observed: when the bifurcation dgram is computed by increasingα from 3.3 and using the finastate of a run as the initial condition for the following, resulike those shown inFig. 2(a) are obtained. However, when tbifurcation diagram is computed by the same procedure,decreasingα from 3.6 the results are like those ofFig. 2(b).These figures show that within the intervalα ∈ [3.37,3.51] thesystem shows hysteresis. Discontinuous changes of shapeattractor[9] and hysteresis[10,11] are characteristics of somof a kind of chaos–chaos transitions known in the literatas crisis. Detailed observation of the bifurcation diagramα ∈ [4.0,4.1] and forα ∈ [4.4,4.5] also shows abrupt changin the attractor size, although smaller as the values ofα aregreater (Fig. 2(c), (d)). These crises possibly occur for even

J.M. González-Miranda / Physics Letters A 352 (2006) 83–88 85

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Fig. 2. Bifurcation diagram near the first crisis for: (a) increasing, and (b)creasing bifurcation parameter. The directions for the change ofα in the cal-culation of the diagram are indicated by arrows. Top region of the bifurcadiagram near (c) the second crisis, and (d) the third crisis.

creasing values ofα, but they are difficult to observe as thdiscontinuities become smaller. All this observations tell usfor α 3.36 the system dynamics deserves more detailed s

We have observed that these deviations from the standynamics shown by the SprottE flow for intermediate andlarge values ofα are linked to a special behavior of an inteesting property of a chaotic attractor: its phase coherence[3,4].This property refers to the existence of a predominant frequeof oscillation in the dynamics of the chaotic system. Cohephase oscillators exhibit a relatively regular chaotic dynamwhile incoherent oscillators are more irregular and fluctuatPhase coherence has proven to be a relevant property of chdynamics when issues related to chaos synchronization aresidered[5], specially when phase synchronization[4,12–14],or amplitude envelope synchronization[15] are involved. Sig-natures of phase coherence are the presence of sharpemerging from a broadband structure in the power spectand fluctuations of the phase from those of a uniform rotati

To gain some insight on how the phase of the SprottE flowbehaves we have studied the one-sided power spectral denof the time series[x(t), y(t), z(t)]. For a given time seriess(t),this is defined as[6]

(4)Ps(ω) = 2

∣∣∣∣∣∞∫

0

s(t) · eiωt dt

∣∣∣∣∣2

,

being s ∈ x, y, z in the present case. InFig. 3(a) the resultsobtained fromx(t), for eight values ofα, between 3.2 and 6.0,

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Fig. 3. (a) Change of the power spectral density ofx(t) with the bifurcation pa-rameter for the series of values ofα: 3.2, 3.6, 4.0, 4.1, 4.5, 5.0, 5.5 and 6.0.The plot of each function has been shifted up from its predecessor by muling Px(ω) by 10i−1 with i its number in the series. The power spectral denat α = 3.20 (thick line) and atα = 6.00 (thin line) computed from (b)y(t) and(c) z(t).

are presented. Forα = 3.2, the oscillation is highly coherent:is characterized by a dominant frequency atωC ≈ 0.9 and itsharmonics over a noisy background. This coherence is raplost asα increases, and forα = 6.0 no sign of coherent rotatiois observed. Similar conclusions can be drawn fromPy(ω), andPz(ω), although for these signals the loss of coherence is sowhat slower, as shown inFig. 3(b), (c). So, we conclude thatmajor effect of the increase ofα is a significant lost of coherence of the chaotic dynamics.

For an oscillator like this, whose dynamics occurs alonproper rotation[7], it is possible to define the phase of the mtion by means of[12]

(5)φ(t) = arctan

[z(t) − zF

x(t) − xF

],

with xF andzF the coordinates of the center of rotation. Foclockwise rotation,φ(t), decreases towards minus infinity. Tbehavior ofφ(t) for different α, as illustrated byFig. 4(a), issuch that the decrease of the phase can be described aswith an aperiodic oscillation superimposed. For a linear oslator we will have the case of a completely coherent rotatIn this case the phase would evolve asφ(t) = φ0 − Ωt , withΩ a constant equal to the frequency of the rotation. Theplitude of the oscillation superimposed to the linear decaymeasure of the degree of incoherence of the rotation. To qtify this dispersion it is common[12] to define an instantaneou

86 J.M. González-Miranda / Physics Letters A 352 (2006) 83–88

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Fig. 4. (a) Time dependence of the phase, and (b) distribution function otime derivative of the phase for the SprottE flow at three values ofα: 3.2 (thickline), 5.0 (thin line), 6.0 (dashed line). (c) The increase of the dispersion ofvalues ofΩ with the bifurcation parameter.

frequency as

(6)Ω(t) =∣∣∣∣dφ

dt

∣∣∣∣.For a linear oscillation this frequency is a constant, fochaotic oscillator there is a certain distribution of frequencfΩ(|dφ/dt |), which gives us the probability of the observatiof a certain value ofΩ . In Fig. 4(b) they are displayed resulfor this distribution function, obtained numerically for the savalues ofα as the curves inFig. 4(a). They show how the distribution extends further asα increases. To quantify this increawe have measured the dispersion of the distribution by mof

(7)Ω = |ΩM − Ωm|,with ΩM and Ωm the maximum and minimum values oserved for|dφ/dt | in a significantly large run of Eqs.(1)–(3).In this case, taking 2π/ωC (ωC = 0.9) as the time unit, theruns spanned a time interval equal to 105. The results shownin Fig. 4(c) present an steady increase of the dispersion ofΩ

with α. This backs up our conclusion that a major effect ofincrease ofα is a loss of phase coherence.

The mechanism behind this loss of coherence can be trout by means of the study of the first return maps of the sysvariables. Plots of series of values of the relative maxima ofz(t)

as functions of the value of its predecessor,Zn+1 = Zn+1(Zn),presented inFig. 5(a)–(d) show that, forα = 3.2, we have a one

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Fig. 5. (a)–(c) First return maps for representative values of the bifurcaparameter around the first and second crisis: (a) 3.2, (b) 3.6, (c) 4.0, (d) 4.1.(e) The first return map forα = 6.0, and (f) a detail of the top tip of this map.

valued function with a single maximum, a shape that is aciated with a standard map like behavior when the increasthe bifurcation parameter moves the maxima up, as it hapin this case. Forα = 3.6, above the first crisis, the functiohas become bivalued because a new branch above theing one develops. This continues its development until a tbranch, close to the second, develops at the second cristweenα = 4.0, andα = 4.1. This branch-adding cascadebifurcations develops apparently without limit asα increasesThis is illustrated inFig. 5(e), (f), where it is shown how thmap for α = 6.0 presents a large number of branches whtend to accumulate at the top of the map as seen in the dpresented inFig. 5(f). The loss of coherence and increasechaoticity then results from the superposition of dynamics oa series of different standard like maps, embedded in the chattractor.

The relationships between the height,ZM , and position,ZM−1, of the peak in the first return map and the value of thefurcation parameter,α, presented inFig. 6(a), show an overalexponential increase withα, despite some discontinuities asociated to crises. This suggests that the dynamics on sestandard like maps shown inFig. 5, can be seen as an extensof the standard map dynamics beyond its more chaotic sA plot of such maps (forα = 4.5) is presented inFig. 6(b). Thisis composed of four branches, being each one an exampa standard-like map with a different degree of developmentname these branches, starting from the inner one, as thesecond, third, and fourth branches. All of them merge in agle line o–d , placed to the right of the map. The pointo is anunstable fixed point, that we call the main fixed point. The pl

J.M. González-Miranda / Physics Letters A 352 (2006) 83–88 87

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Fig. 6. (a) The height (thick line) and position (thin line) of the peak of the fireturn map as functions ofα, and (b) plot of the standard like maps embeded in the chaotic SprottE flow. A dashed line shows the points that meetconditionZn+1 = Zn, and two doted lines divide theZn–Zn+1 plane in fourquadrants or regions:R0, R1, R2 andR3. Relevant points of the map are indcated by full circles, and labeled by lower case letters. The insert shows aof the left tip of the map.

Zn–Zn+1 can then be divided in four relevant regions, labeR0, R1, R2 andR3, by means of two lines parallel to the axthat cross ato.

The dynamics in this map is as follows. Points of the plaZn–Zn+1 in regionR0 are never visited by a phase space pothat follows the dynamics of the map. Points in the sevbranches of the map that are contained in regionR3, are mappedonto the single branch in regionR1. This contributes to chaoby a folding process that mixes the dynamics of the differbranches. Points in the single branch inR1 contribute to chaosby stretching as follows: (i) the archo–a is mapped into thearcha′–o, in R3, (ii) the archa–c is mapped ontoo1–a′, in R2,and (iii) the right tipc–d is mapped ontoo1–d1, also inR2. Itis to be noted thato1 is a highly unstable fixed point of the mawhose abscissa is slightly above the abscissa of the minimof the single right branch inR1 (pointb).

After a new iteration, points ina′–d1 are mapped onto thsecond branch in a way very much alike as the one justscribed to reach points of the first branch inR2 and R3 (seealso the insert ofFig. 6(b)). Points ina′–a1, map onto points othe second branch withinR3, while points ina1–d1 go to the lefttip of the second branch inR2. Again here we have an unstble fixed point,o2, which is a bit above the minimum,b1, of theo1–d1 curve, and points inc1–d1 are mapped ontoo2–d2, belowthis fixed point. One more iteration sends the points in thetip of the second branch to the third branch, where these p

ail

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t

m

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tts

Fig. 7. Examples of trajectories in a map with four branches: (a) motiothe first branch around the main fixed point, with final injection to the secbranch, and (b) a trip trough all the four branches. In (c) and (d) we show deof the dynamics in the left tip of the map. Points visited are indicated bycircles and distinguished by its ordinal number in the series. Dotted linesthem to guide the eye and arrows show the direction of the motion. A daline shows the points which meet the conditionZn+1 = Zn.

spread between regionsR2 andR3, and again the left tip of thethird branch withinR2 will be mapped onto the fourth and labranch, which lies only inR3 in this case. This dynamics can bstraightforwardly extended to any number of branches. Thephase space point can stay wandering inR2 for a number of in-teractions that increases with the number of branches, untilift up to R3 and then injected toR1 to restart again.

Two examples of trajectories in the planeZn–Zn+1 are givenin Fig. 7. One of them starts close to the main fixed po(Fig. 7(a)), after three iterations in the first branch aroundpoint, it is sent to the neighborhood ofo1 in R2 (Fig. 7(c)), andthen near the top of the second branch inR3, from were it willreturn toR1. The second trajectory (Fig. 7(b)) goes through thefour branches: it starts in the right tip of the map, inR1, andthen moves consecutively to points near the left tip of the fisecond, and third branches inR2 (Fig. 7(d)), to end in the fourthbranch inR3. These examples show how the different branccan be visited by a phase point, accordingly to the map dynics above described.

Further illustration of how phase coherence is destroyedbe obtained from a look to the trajectories of the systemphase space. This is given inFig. 8, where a short part ofrepresentative trajectory is presented in detail. The trajecstarts atx(0) ≈ 0.0, y(0) ≈ 0.0, andz(0) ≈ 8.0, and ends ax(44.6) ≈ 0.0, y(44.6) ≈ 0.0, andz(44.6) ≈ 7.0. During thefirst cycle in the planex–z the phase space point moves falong the positive direction of they axis up to a maximum distance of the fixed pointy ≈ 4. Then, during the next five cyclein the planex–z, the phase point relaxes along they axis to ap-proach to the equilibriaxF at aroundt ≈ 24.2, where a newsimilar episode with only three cycles starts. This sendsphase point only up toy ≈ 1.0, and develops untilt ≈ 44.6. The

88 J.M. González-Miranda / Physics Letters A 352 (2006) 83–88

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Fig. 8. A short fragment of a trajectory of the SprottE flow at α = 5.0. Pro-jections onto (a) thex–z and (b) they–z planes, and time series for the systestate variables (c)x(t), (d) y(t), and (e)z(t). Crosses in (a), (b), and dottelines in (c)–(e) indicate the position of the fixed point,xF .

whole system dynamics is a series of episodes like these, wdiffer in the number of cycles each episode contain and howthe system moves along the stable manifold. For small vaof α, episodes with a low number of cycles and little separaalong they-axis are allowed. Asα increases so increases theversity of the episodes that the system can suffer, and thewhole dynamics results more chaotic and incoherent.

In summary, we have modified the SprottE flow by intro-ducing a bifurcation parameter and observed that, despitsimple mathematical structure, a nonstandard chaotic dyna

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develops for sufficiently large values of the bifurcation pameter. A relevant manifestation of this dynamics is the phbehavior which can become highly incoherent, as show bypower spectrum and the direct analysis of the phase dynaThis behavior is interpreted in terms of the developmentcascade of bifurcations in which the dynamics of severalferent embedded standard maps are superimposed. This rin a dynamics composed of a variety of oscillating episowhich present large quantitative differences between themspeculate that a chaotic flow like this, that oscillates alonproper rotation, and whose coherence can be steadily modby changing a control parameter, can be useful as a test bin chaos theory, for example in the subfield of synchronizaand control of chaos.

Acknowledgement

Research supported by DGI (BFM2003-05106).

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