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Intermittent chaos driven by nonlinear Alfven waves

E. L. Rempel, A. C.-L. Chian, A. J. Preto, S. Stephany

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E. L. Rempel, A. C.-L. Chian, A. J. Preto, S. Stephany. Intermittent chaos driven by nonlinearAlfven waves. Nonlinear Processes in Geophysics, European Geosciences Union (EGU), 2004,11 (5/6), pp.691-700. <hal-00302411>

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Nonlinear Processes in Geophysics (2004) 11: 691–700SRef-ID: 1607-7946/npg/2004-11-691 Nonlinear Processes

in Geophysics© European Geosciences Union 2004

Intermittent chaos driven by nonlinear Alfv en waves

E. L. Rempel, A. C.-L. Chian, A. J. Preto, and S. Stephany

National Institute for Space Research (INPE) and World Institute for Space Environment Research (WISER), P.O. Box 515,Sao Jose dos Campos–SP, 12227–010, Brazil

Received: 14 September 2004 – Revised: 9 December 2004 – Accepted: 14 December 2004 – Published: 16 December 2004

Part of Special Issue “Advances in space environment turbulence”

Abstract. We investigate the relevance of chaotic saddlesand unstable periodic orbits at the onset of intermittent chaosin the phase dynamics of nonlinear Alfven waves by us-ing the Kuramoto-Sivashinsky (KS) equation as a modelfor phase dynamics. We focus on the role of nonattractingchaotic solutions of the KS equation, known as chaotic sad-dles, in the transition from weak chaos to strong chaos viaan interior crisis and show how two of these unstable chaoticsaddles can interact to produce the plasma intermittency ob-served in the strongly chaotic regimes. The dynamical sys-tems approach discussed in this work can lead to a better un-derstanding of the mechanisms responsible for the phenom-ena of intermittency in space plasmas.

1 Introduction

Intermittent fluctuations are constantly encountered in spaceplasmas, as reported in several works based on the analysisof solar wind data both at the ecliptic (Burlaga, 1991; Marschand Tu, 1993; Bruno et al., 2003) and high heliographic lat-itude (Ruzmaikin et al., 1995; Pagel and Balogh, 2003). In-termittent events are characterized by time series that dis-play time intervals with low variabilities interrupted by burstsof very high variabilities. As a consequence, the associatedprobability density functions (PDF’s) are non-Gaussian. Ithas been shown by Voros et al. (2002) and Dorotovic andVoros (2004) that intermittent fluctuations in the solar windaffect the geomagnetic response. By comparing solar winddata obtained from the ACE spacecraft with plasma sheetdata from the Geotail mission, Dorotovic and Voros (2004)suggested that the non-Gaussian characteristics of the PDF’sof solar wind data can be interconnected to the occurrenceof intermittency in the magnetic fluctuations in the plasmasheet. Thus, the study of intermittent phenomena in solar

Correspondence to:E. L. Rempel([email protected])

wind plasmas is essential for a better understanding of thesolar wind-magnetosphere coupling processes.

The fluctuations of the plasma velocity and magnetic fieldas revealed by interplanetary data records are frequently wellcorrelated, which is a signature of the presence of Alfvenwaves in the solar wind (Belcher and Davis, 1971; Goldsteinand Roberts, 1995). Alfven waves are low-frequency elec-tromagnetic waves in a plasma with a background magneticfield. From a linear analysis of MHD equations, the disper-sion relation of the Alfven wave is found asω=k||vA, wherek|| is the component of the wave vectork parallel toB0, andvA=B0/(µ0ρ0)

1/2 is the Alfven velocity, whereB0 is thestrength of the ambient magnetic field,µ0 is the permeabilityof vacuum andρ0 is the average mass density of the plasma.The perturbation of the fluid velocityu relates to the mag-netic field’s perturbation vectorb=δB0 byu=∓b/(µ0ρ0)

1/2,where the upper (lower) sign refers to the casek · B0>0(k · B0<0). Thus,u andb are parallel/antiparallel and pro-portional to each other, and the plasma oscillates with themagnetic field lines.

As shown by Lefebvre and Hada (2000), by assumingweak instability the dynamics of quasiparallel Alfven wavescan be studied by a complex Ginzburg-Landau (CGL) equa-tion. The complex Ginzburg-Landau equation has been oneof the most widely studied nonlinear equations in the lastdecades (see Cross and Hohenberg, 1993; Bohr et al., 1998;Aranson and Kramer, 2002, and references therein). It de-scribes the slow modulation of a periodic pattern in spaceand time near the threshold of an instability, where a band ofmodes become unstable.

In this work we study the phase dynamics of nonlinearAlfv en waves modeled by the CGL equation. Phase dynam-ics is of particular interest in systems modeled by the CGLequation, since for a range of values of the control parametersthe dynamics of perturbations of traveling wave solutions isessentially determined by variations of the phase alone, andcan exhibit “phase turbulence”. In such regimes the ampli-tudes are essentially constant and the phase dynamics sat-isfies the Kuramoto-Sivashinsky (KS) equation (Kuramoto

692 E. L. Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves

and Tsuzuki, 1976; Cross and Hohenberg, 1993; Bohr et al.,1998; Aranson and Kramer, 2002; van Baalen, 2004). Thestudy of phase dynamics can elucidate important nonlinearphenomena observed in space plasmas. Finite correlation ofphases in MHD waves upstream of Earth’s bow shock wasfound in Geotail magnetic field data, indicating that non-linear interactions between the waves are in progress (Hadaet al., 2003). This bears important implications in discus-sions of various transport processes of charged particles inspace. In He and Chian (2003) imperfect phase synchro-nization in a nonlinear drift wave system was shown to beresponsible for the origin of bursts in wave energy in a tur-bulent state. Here we focus on the characterization of phaseintermittency in the KS equation. We first describe how thefluctuations of the phase can evolve from periodic to chaoticbehavior through a sequence of bifurcations as the viscosityis varied. We show that in the KS equation, chaotic attractorscoexist with nonattracting chaotic sets responsible for tran-sient chaotic behavior. The collision of a weak chaotic at-tractor with an unstable periodic orbit leads to the generationof a strong chaotic attractor, in an event known as interiorcrisis. The post-crisis strong chaotic attractor can be decom-posed into two nonattracting chaotic sets, responsible for thegeneration of intermittent time series.

Section 2 of this paper contains a brief introduction to ba-sic concepts on nonlinear dynamical systems. In Sect. 3 wedescribe the numerical solutions of the KS equation by theGalerkin spectral method. In Sect. 4 we present our analysisof attracting and nonattracting chaotic sets in the KS equa-tion. The conclusions are given in Sect. 5.

2 Basic concepts of nonlinear dynamics

In this section we review some basic concepts of nonlineardynamical systems (Parker and Chua, 1989; Ott, 1993; Al-ligood et al., 1996) that are essential for understanding theremaining of this paper.

We consider dissipative dynamical systems described byautonomous systems of ODE’s,

x = f (x), (1)

wherex is ann-dimensional vector,f is a vector functionand the dot denotes derivative with respect to time. A “flow”ft (x0) is the solution of Eq. (1) for an initial conditionx0after certain timet . The components of the vector “statevariable”x define a “phase space”, where the flow ofx0 isplotted for increasing values oft , generating the “orbit” ofx0.

A “fixed point” of Eq. (1) is a constant solution, i.e. a pointx for which x=0, or equivalently,ft (x)=x for all t . A “pe-riodic orbit” is a solution of Eq. (1) that always repeats itsbehavior after a fixed time interval, i.e.ft (x)=ft+T (x) forall t and some minimum periodT>0.

Dissipative systems are characterized by the presence of“attractors”. An attractor is a subset of the phase space that

attracts almost all the initial conditions in a certain neighbor-hood, that is, the limit set of the orbits of initial conditions inthe neighborhood as time tends to+∞ is the attractor (Gre-bogi et al., 1984). Attractors may be simple sets like a fixedpoint or periodic orbit, but can also have nonelementary ge-ometrical properties such as noninteger fractal dimension, inwhich case the attractor is called “strange” (Grebogi et al.,1984, 1987b). Fractal sets display scale invariance, whichimplies that continuous blow-up of a tiny portion of the setreveal self-similar structures on arbitrarily small scales (Ott,1993).

Strange attractors are typically (but not always) “chaotic”(Grebogi et al., 1987b). The orbits of random initial con-ditions on a chaotic attractor will display aperiodic behav-ior and “sensitive dependence on initial conditions”, whichmeans that nearby orbits will diverge exponentially withtime. The average rate of divergence can be measured by the“Lyapunov exponent”. Let10 be a small distance separatingtwo initial conditions on the chaotic attractor att=0. Thenfor increasingt the orbits of the two points will diverge onaverage as1t∼10 exp(λt), whereλ is the Lyapunov expo-nent (Grebogi et al., 1987b). For systems withn-dimensionalphase space there aren Lyapunov exponents which measurethe rate of divergence/convergence onn orthogonal direc-tions.

Chaotic sets are not necessarily attracting sets. A strangeset3 might be chaotic and nonattracting. That means thatthe orbits of typical initial conditions in the vicinity of3are eventually repelled from it. Nevertheless,3 contains achaotic orbit (an aperiodic orbit with at least one positiveLyapunov exponent) (Nusse and Yorke, 1989). If the chaoticorbit has also one negative Lyapunov exponent the nonat-tracting chaotic set is known as “chaotic saddle”. Chaoticsaddles, as well as chaotic attractors, contain an infinite num-ber of unstable periodic orbits (UPO’s).

In nonlinear dynamical systems, as one varies some con-trol parameter present in the model equations some dynam-ical changes can occur, such as creation/destruction of fixedpoints and periodic orbits or loss of stability of attracting sets.Thus, periodic attractors can lose their asymptotic stabilityand become unstable periodic orbits. Similarly, chaotic at-tractors can lose their attracting nature and become nonat-tracting chaotic sets, or chaotic saddles. The qualitativechanges in the behavior of solutions of dynamical systemsas a control parameter is varied are called “bifurcations”.When the changes in the phase portrait involve merely thelocal vicinity of fixed points or periodic orbits, one has a “lo-cal bifurcation”. Large changes in the topology of the systemare called “global bifurcations”. An example is the “interiorcrisis” discussed in this paper, whereby a chaotic attractor issuddenly enlarged. The bifurcations of a dynamical systemcan be represented in a “bifurcation diagram”, in which thevalues of one of the state variables are plotted as a functionof one control parameter.

A classical technique to analyze nonlinear dynamicalsystems is the “Poincare map”. It replaces the flow ofannth-order continuous-time system with an(n−1)th-order

E. L. Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 693

discrete-time system, simplifying the analysis and visualiza-tion of the dynamics. There are different forms of defining aPoincare map. At the end of the next section we exemplifyone of them.

3 The Kuramoto-Sivashinsky equation

The derivative nonlinear Schrodinger equation (DNLS)(Rogister, 1971; Mjolhus and Wyller, 1988) describesthe dynamics of quasiparallel Alfven waves of moder-ate amplitudes in a finite-β plasma, taking into accountweak linear dispersion. To describe patterns formedby Alfv en waves subject to damping and growth rate,Lefebvre and Hada (2000) used a model based on amodified version of the DNLS equation, including lin-ear growth rate, linear dissipation and nonlinear Lan-dau damping. Assuming weak instability, Lefebvre andHada (2000) proposed a further simplification to an en-velope equation by writing the complex transverse mag-netic fieldB=By+iBz asB(x, t)=ψ(ξ, t) exp[i(kcx−ωct)],whereωc=ω(kc), ξ=x−v(kc)t andkc is the most unstablemode. Assumingkc>0 (left-hand polarization) the resultingequation forψ is given by

∂tψ = ψ + (1 − ib1)∂ξξψ − iψ(b2 −H)|ψ |2, (2)

whereb1 measures the strength of the dispersion andb2 theeffects of nonlinearity. The termH denotes a function thataccounts for the nonlinear Landau damping (Mjolhus andWyller, 1988). Equation (2) falls into the general class ofthe complex Ginzburg-Landau equation. The envelopeψ isa complex variable. By writingψ=R exp(i8) and assum-ing small perturbationsR=R0+r, 8=80+φ, it is possibleto derive an equation for the phaseφ of the CGL equation.As shown by Kuramoto and Tsuzuki (1976) and Aranson andKramer (2002), the phase equation is given by the Kuramoto-Sivashinsky equation. Here we study the following form ofthe KS equation (LaQuey et al., 1975; Christiansen et al.,1997; Chian et al., 2002; Rempel and Chian, 2003)

∂tφ = −∂2xφ − ν∂4

xφ − ∂xφ2, (3)

whereν is a damping parameter representing viscosity andwe assume thatφ(x, t) is subject to periodic boundary con-ditions

φ(x, t) = φ(x + 2π, t). (4)

To obtain the numerical solution of Eq. (3) we use theGalerkin method (Gottlieb and Orszag, 1977), by applyinga Fourier decomposition for the functionφ(x, t)

φ(x, t) =

∞∑k=−∞

bk(t)eikx . (5)

After substituting Eq. (5) into Eq. (3) and solving thederivatives one obtains

∞∑k=−∞

bkeikx

=

∞∑k=−∞

k2bkeikx

− ν

∞∑k=−∞

k4bkeikx

−

∞∑k=−∞

∞∑m=−∞

i(k +m)bkbmeix(k+m). (6)

Next, multiply Eq. (6) bye−ilx , with l=−∞, ...,∞, andintegrate in the spatial domain,∫ 2π

0

∞∑k=−∞

bkeix(k−l)dx =

∫ 2π

0

∞∑k=−∞

k2bkeix(k−l)dx

−

∫ 2π

0ν

∞∑k=−∞

k4bkeix(k−l)dx

−

∫ 2π

0

∞∑k=−∞

∞∑m=−∞

i (k +m)

× bkbmeix(k+m−l)dx. (7)

Each integral of the exponential functions in Eq. (7) van-ishes except when the exponent is zero. The three first expo-nents are zero whenk=l. For the nonlinear term the expo-nent is null whenk=l−m. After solving the integrals, Eq. (7)becomes

bk = (k2− νk4)bk − ik

N∑m=−N

bmbk−m, (8)

k = −N, ..., N,

whereN is the truncation order. Our choice ofN is explainedlater in this section.

The coefficientsbk in Eq. (8) are complex. We can sim-plify our analysis by restricting to the subspace of odd solu-tions,φ(x, t)=−φ(−x, t). It is possible to prove that if aninitial condition φ(x,0) is an odd function, the solution ofEq. (3) is odd for all time (Temam, 1988)

φ(x, t) = −φ(−x, t), ∀x, t. (9)

The Fourier transform of an odd function has purely imag-inary coefficients, so we can represent odd functions by as-suming thatbk(t) are purely imaginary, setting

bk = −1

2iak, (10)

whereak are real numbers. After substituting Eq. (10) intoEq. (8), we obtain

ak = (k2− νk4)ak −

k

2

N∑m=−N

amak−m, (11)

k = −N, ..., N.

Equation (11) contains unnecessary operations. Sinceφ(x, t) is real,−iak=ia−k, and it is not necessary to com-pute the modes with negativek. Moreover,ak=0 for |k|>N ,and some operations in the nonlinear term can be dropped.Thus Eq. (11) can be written in the form

ak = (k2− νk4)ak +

k

2

(−1∑

m=k−N

a−mak−m (12)

−

k−1∑m=1

amak−m +

N∑m=k+1

amam−k

),

694 E. L. Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves4 Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves

Fig. 1. Period-3 limit cycle solution of Eq. (12) atν = 0.029924and the corresponding Poincare points.

the simulations, due to the sums in the nonlinear term in Eq.(12). On the other hand, low values ofN may result in a dy-namical system whose behavior has no resemblance with theoriginal PDE. A determination of the range of linearly unsta-ble modes is helpful in this case. Considering the linear partof Eq. (12), one finds that the stability eigenvalues are nega-tive for |k| > 1/

√ν and are positive for|k| < 1/

√ν. Thus,

the modes with wave numbersk in the range[−1/√ν, 1/

√ν]

are linearly unstable. These modes excite the short wave-length (high-k) modes through the nonlinear term in Eq.(12), and the excitations are dissipated by the high-k modes(Christiansen et al., 1997). Modes with|k| � 1/

√ν are

strongly damped and their amplitudes remain very close tozero for all times, but intermediate modes must be kept inthe simulations. Figure 2(a) depicts the temporal variationof the Fourier modesak(t) in Eq. (12) for a chaotic regimeat ν = 0.029919 andN = 16 modes. Clearly, the energyis concentrated in the low-k long wavelength modes. Thecorresponding spatiotemporal patternφ(x, t) in real space,obtained with Eq. (5), is shown in Fig. 2(b). The systemdynamics is chaotic in time but coherent in space.

4 Nonlinear dynamics analysis

4.1 The bifurcation diagram

Figure 3(a) depicts the bifurcation diagrama6(ν) for Eq.(12) withN = 16. The diagram is similar for any choice ofak. For each value ofν we drop the initial 100 iterations ofthe Poincare map before we start plotting. These initial iter-ations contain the transient dynamics, before the fluctuationsof the wave phase converge to a regime where the Poincarepoints stay in an attracting subset of the phase space. Forthis range ofν, the attracting set can be either chaotic or pe-riodic. The gray area in Fig. 3(a) depicts another importantsubset of the phase space representing a chaotic saddle. The

Fig. 2. (a) Temporal variation of the Fourier modesak(t) in Eq.(12) for ν = 0.029919 andN = 16 modes; (b) the correspondingspatiotemporal pattern ofφ(x, t). The system dynamics is chaoticin time but coherent in space.

trajectories of random initial conditions are first attracted tothe vicinity of the chaotic saddle, where they display chaoticbehavior for a finite time (chaotic transient), before they con-verge to the attractor (periodic or chaotic). The chaotic sad-dles are obtained by the PIM triple algorithm (Nusse andYorke, 1989).

A saddle-node bifurcation atν = νSNB ≈ 0.02992498,indicated as SNB in Fig. 1, marks the beginning of ape-riodic window in the bifurcation diagram. Forν > νSNB ,random initial conditions converge to a chaotic attractor, andfor ν < νSNB the chaotic attractor no longer exists. At thesaddle-node bifurcation the simultaneous creation of a p-3attractor and a p-3 unstable periodic orbit occurs. The p-3UPO, found with the Newton method, is represented in Fig.3(a) by dashed lines. As the value ofν is decreased, thep-3 attractor undergoes a cascade of period-doubling bifur-cations, whereby the period of the attractor is successivelydoubled. As the period tends towards infinity, a chaotic at-tractor is formed, localized in three separate bands in thebifurcation diagram. We call the region occupied by this“banded” attractor theband region(B), and the region occu-pied by the surrounding chaotic saddle (SCS) thesurround-

Fig. 1. Period-3 limit cycle solution of Eq. (12) atν=0.029924 andthe corresponding Poincare points.

with k=1, ..., N .The dynamics of the system described by Eq. (12) can be

analyzed on a Poincare section defined bya1=0. A trajectoryrepresenting the flow of Eq. (12) in a phase space defined bythe Fourier modesak, can intersect this Poincare section intwo ways: whena1>0 (from “left” to “right”) or when a1<0(from “right” to “left”). We adopt a Poincare mapP de-fined as the(N−1) dimensional hyperplane given bya1=0,with a1>0, so that a Poincare point is plotted every time theflow of Eq. (12) crosses the Hyperplanea1=0 from “left” to“right”, as illustrated in Fig. 1 for a period-3 (p-3) limit cyclesolution atν=0.029924.

The choice of the truncationN for the number of modeshas obvious implications in the numerical solutions ofEq. (3). High values ofN imply high computational costfor the simulations, due to the sums in the nonlinear termin Eq. (12). On the other hand, low values ofN may re-sult in a dynamical system whose behavior has no resem-blance with the original PDE. A determination of the rangeof linearly unstable modes is helpful in this case. Consid-ering the linear part of Eq. (12), one finds that the stabilityeigenvalues are negative for|k|>1/

√ν and are positive for

|k|<1/√ν. Thus, the modes with wave numbersk in the

range[−1/√ν,1/

√ν] are linearly unstable. These modes

excite the short wavelength (high-k) modes through the non-linear term in Eq. (12), and the excitations are dissipated bythe high-k modes (Christiansen et al., 1997). Modes with|k|�1/

√ν are strongly damped and their amplitudes remain

very close to zero for all times, but intermediate modes mustbe kept in the simulations. Figure 2a depicts the tempo-ral variation of the Fourier modesak(t) in Eq. (12) for achaotic regime atν=0.029919 andN=16 modes. Clearly,the energy is concentrated in the low-k long wavelengthmodes. The corresponding spatiotemporal patternφ(x, t) inreal space, obtained with Eq. (5), is shown in Fig. 2b. Thesystem dynamics is chaotic in time but coherent in space.

4 Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves

Fig. 1. Period-3 limit cycle solution of Eq. (12) atν = 0.029924and the corresponding Poincare points.

the simulations, due to the sums in the nonlinear term in Eq.(12). On the other hand, low values ofN may result in a dy-namical system whose behavior has no resemblance with theoriginal PDE. A determination of the range of linearly unsta-ble modes is helpful in this case. Considering the linear partof Eq. (12), one finds that the stability eigenvalues are nega-tive for |k| > 1/

√ν and are positive for|k| < 1/

√ν. Thus,

the modes with wave numbersk in the range[−1/√ν, 1/

√ν]

are linearly unstable. These modes excite the short wave-length (high-k) modes through the nonlinear term in Eq.(12), and the excitations are dissipated by the high-k modes(Christiansen et al., 1997). Modes with|k| � 1/

√ν are

strongly damped and their amplitudes remain very close tozero for all times, but intermediate modes must be kept inthe simulations. Figure 2(a) depicts the temporal variationof the Fourier modesak(t) in Eq. (12) for a chaotic regimeat ν = 0.029919 andN = 16 modes. Clearly, the energyis concentrated in the low-k long wavelength modes. Thecorresponding spatiotemporal patternφ(x, t) in real space,obtained with Eq. (5), is shown in Fig. 2(b). The systemdynamics is chaotic in time but coherent in space.



Figure 3(a) depicts the bifurcation diagrama6(ν) for Eq.(12) withN = 16. The diagram is similar for any choice ofak. For each value ofν we drop the initial 100 iterations ofthe Poincare map before we start plotting. These initial iter-ations contain the transient dynamics, before the fluctuationsof the wave phase converge to a regime where the Poincarepoints stay in an attracting subset of the phase space. Forthis range ofν, the attracting set can be either chaotic or pe-riodic. The gray area in Fig. 3(a) depicts another importantsubset of the phase space representing a chaotic saddle. The

Fig. 2. (a) Temporal variation of the Fourier modesak(t) in Eq.(12) for ν = 0.029919 andN = 16 modes; (b) the correspondingspatiotemporal pattern ofφ(x, t). The system dynamics is chaoticin time but coherent in space.

trajectories of random initial conditions are first attracted tothe vicinity of the chaotic saddle, where they display chaoticbehavior for a finite time (chaotic transient), before they con-verge to the attractor (periodic or chaotic). The chaotic sad-dles are obtained by the PIM triple algorithm (Nusse andYorke, 1989).

A saddle-node bifurcation atν = νSNB ≈ 0.02992498,indicated as SNB in Fig. 1, marks the beginning of ape-riodic window in the bifurcation diagram. Forν > νSNB ,random initial conditions converge to a chaotic attractor, andfor ν < νSNB the chaotic attractor no longer exists. At thesaddle-node bifurcation the simultaneous creation of a p-3attractor and a p-3 unstable periodic orbit occurs. The p-3UPO, found with the Newton method, is represented in Fig.3(a) by dashed lines. As the value ofν is decreased, thep-3 attractor undergoes a cascade of period-doubling bifur-cations, whereby the period of the attractor is successivelydoubled. As the period tends towards infinity, a chaotic at-tractor is formed, localized in three separate bands in thebifurcation diagram. We call the region occupied by this“banded” attractor theband region(B), and the region occu-pied by the surrounding chaotic saddle (SCS) thesurround-

Fig. 2. (a)Temporal variation of the Fourier modesak(t) in Eq. (12)for ν=0.029919 andN=16 modes;(b) the corresponding spa-tiotemporal pattern ofφ(x, t). The system dynamics is chaotic intime but coherent in space.



Figure 3a depicts the bifurcation diagrama6(ν) for Eq. (12)with N=16. The diagram is similar for any choice ofak.For each value ofν we drop the initial 100 iterations of thePoincare map before we start plotting. These initial iterationscontain the transient dynamics, before the fluctuations of thewave phase converge to a regime where the Poincare pointsstay in an attracting subset of the phase space. For this rangeof ν, the attracting set can be either chaotic or periodic. Thegray area in Fig. 3a depicts another important subset of thephase space representing a chaotic saddle. The trajectoriesof random initial conditions are first attracted to the vicinityof the chaotic saddle, where they display chaotic behaviorfor a finite time (“chaotic transient”), before they convergeto the attractor (periodic or chaotic). The chaotic saddlesare obtained by the PIM triple algorithm (Nusse and Yorke,1989).

A saddle-node bifurcation atν=νSNB≈0.02992498, indi-cated as SNB in Fig. 1, marks the beginning of a “peri-odic window” in the bifurcation diagram. Forν>νSNB, ran-

E. L. Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 695Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 5

Fig. 3. (a) Variation ofa6 for the chaotic saddle (gray) as a func-tion of ν, superimposed by the bifurcation diagram of the attractor(black) in a p-3 periodic window. IC denotes interior crisis andSNB denotes saddle-node bifurcation. The dashed lines denote thep-3 mediating unstable periodic orbit. (b) Same as (a), but depictingthe conversion of the three-band weak chaotic attractor into a bandchaotic saddle after (to the left of) IC.

ing region(S), following reference (Szabo et al., 2000). Atν = νIC ≈ 0.02992021 the chaotic attractor collides withthe p-3 UPO created at SNB, called themediating unsta-ble periodic orbit (MPO). This collision is responsible foran interior crisis, which is a sudden enlargement in the sizeof a chaotic attractor (Grebogi et al., 1983). After crisis(ν < νIC), we find a new chaotic saddle embedded in theenlarged chaotic attractor, in the region previously occupiedby the “pre-IC” banded chaotic attractor. In contrast withthe surrounding chaotic saddle, we call this new chaotic sad-dle theband chaotic saddle(BCS). Figure 3(b) illustrates thestructure of BCS after IC, where the bifurcation diagram forthe attractor (black) of Fig. 3(a) is plotted to the right of theIC point and the band chaotic saddle (gray) is plotted to theleft of IC. It is important to stress that although in Fig. 3(a)the surrounding chaotic saddle is plotted only between points

Fig. 4. (b) Variation of the maximum Lyapunov exponentλmax

with ν; (c) variation of the correlation lengthξ with ν.

SNB and IC, it is actually present in the entire bifurcation di-agram. Forν < νIC andν > νSNB SCS is a subset of thechaotic attractor.

In Fig. 4(a) we plot the variation of the maximum Lya-punov exponentλmax of the attracting set as a function ofν. Positive values ofλmax indicate the presence of a chaoticattractor, and negative values indicate that the attractor is pe-riodic. Note thatλmax jumps abruptly atνIC , indicating asudden increase in the attractor’s chaoticity. For that reason,the pre-IC chaotic attractor is calledweak chaotic attractorand the post-IC attractor, thestrong chaotic attractor. Asmentioned before, for the chosen value of the damping pa-rameterν and the spatial system sizeL = 2π, the dynamicsof the Kuramoto-Sivashinsky equation is chaotic in time, butcoherent in space. In fact, the spatial coherence remains basi-cally unaltered throughout the whole range ofν used in Figs.3 and 4, as indicated by the correlation lengthξ (Morris et al.,1993) in Fig. 4(b).

4.2 Chaotic Saddles

Associated to a chaotic saddle, there are stable and unsta-ble manifolds. The stable manifold is the set of points that

Fig. 3. (a) Variation ofa6 for the chaotic saddle (gray) as a func-tion of ν, superimposed by the bifurcation diagram of the attractor(black) in a p-3 periodic window. IC denotes interior crisis andSNB denotes saddle-node bifurcation. The dashed lines denote thep-3 mediating unstable periodic orbit.(b) Same as (a), but depictingthe conversion of the three-band weak chaotic attractor into a bandchaotic saddle after (to the left of) IC.

dom initial conditions converge to a chaotic attractor, and forν<νSNB the chaotic attractor no longer exists. At the saddle-node bifurcation the simultaneous creation of a p-3 attractorand a p-3 unstable periodic orbit occurs. The p-3 UPO, foundwith the Newton method, is represented in Fig. 3a by dashedlines. As the value ofν is decreased, the p-3 attractor under-goes a cascade of period-doubling bifurcations, whereby theperiod of the attractor is successively doubled. As the periodtends towards infinity, a chaotic attractor is formed, local-ized in three separate bands in the bifurcation diagram. Wecall the region occupied by this “banded” attractor the “bandregion” (B), and the region occupied by the surroundingchaotic saddle (SCS) the “surrounding region” (S), follow-ing reference (Szabo et al., 2000). Atν=νIC≈0.02992021the chaotic attractor collides with the p-3 UPO created atSNB, called the “mediating unstable periodic orbit” (MPO).

Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 5

Fig. 3. (a) Variation ofa6 for the chaotic saddle (gray) as a func-tion of ν, superimposed by the bifurcation diagram of the attractor(black) in a p-3 periodic window. IC denotes interior crisis andSNB denotes saddle-node bifurcation. The dashed lines denote thep-3 mediating unstable periodic orbit. (b) Same as (a), but depictingthe conversion of the three-band weak chaotic attractor into a bandchaotic saddle after (to the left of) IC.

ing region(S), following reference (Szabo et al., 2000). Atν = νIC ≈ 0.02992021 the chaotic attractor collides withthe p-3 UPO created at SNB, called themediating unsta-ble periodic orbit (MPO). This collision is responsible foran interior crisis, which is a sudden enlargement in the sizeof a chaotic attractor (Grebogi et al., 1983). After crisis(ν < νIC), we find a new chaotic saddle embedded in theenlarged chaotic attractor, in the region previously occupiedby the “pre-IC” banded chaotic attractor. In contrast withthe surrounding chaotic saddle, we call this new chaotic sad-dle theband chaotic saddle(BCS). Figure 3(b) illustrates thestructure of BCS after IC, where the bifurcation diagram forthe attractor (black) of Fig. 3(a) is plotted to the right of theIC point and the band chaotic saddle (gray) is plotted to theleft of IC. It is important to stress that although in Fig. 3(a)the surrounding chaotic saddle is plotted only between points

Fig. 4. (b) Variation of the maximum Lyapunov exponentλmax

with ν; (c) variation of the correlation lengthξ with ν.

SNB and IC, it is actually present in the entire bifurcation di-agram. Forν < νIC andν > νSNB SCS is a subset of thechaotic attractor.

In Fig. 4(a) we plot the variation of the maximum Lya-punov exponentλmax of the attracting set as a function ofν. Positive values ofλmax indicate the presence of a chaoticattractor, and negative values indicate that the attractor is pe-riodic. Note thatλmax jumps abruptly atνIC , indicating asudden increase in the attractor’s chaoticity. For that reason,the pre-IC chaotic attractor is calledweak chaotic attractorand the post-IC attractor, thestrong chaotic attractor. Asmentioned before, for the chosen value of the damping pa-rameterν and the spatial system sizeL = 2π, the dynamicsof the Kuramoto-Sivashinsky equation is chaotic in time, butcoherent in space. In fact, the spatial coherence remains basi-cally unaltered throughout the whole range ofν used in Figs.3 and 4, as indicated by the correlation lengthξ (Morris et al.,1993) in Fig. 4(b).

4.2 Chaotic Saddles

Associated to a chaotic saddle, there are stable and unsta-ble manifolds. The stable manifold is the set of points that

Fig. 4. (a)Variation of the maximum Lyapunov exponentλmaxwithν; (b) variation of the correlation lengthξ with ν.

This collision is responsible for an “interior crisis”, which isa sudden enlargement in the size of a chaotic attractor (Gre-bogi et al., 1983). After crisis(ν<νIC), we find a new chaoticsaddle embedded in the enlarged chaotic attractor, in the re-gion previously occupied by the “pre-IC” banded chaotic at-tractor. In contrast with the surrounding chaotic saddle, wecall this new chaotic saddle the “band chaotic saddle” (BCS).Figure 3b illustrates the structure of BCS after IC, where thebifurcation diagram for the attractor (black) of Fig. 3a is plot-ted to the right of the IC point and the band chaotic saddle(gray) is plotted to the left of IC. It is important to stress thatalthough in Fig. 3a the surrounding chaotic saddle is plottedonly between points SNB and IC, it is actually present in theentire bifurcation diagram. Forν<νIC andν>νSNB SCS is asubset of the chaotic attractor.

In Fig. 4a we plot the variation of the maximum Lyapunovexponentλmax of the attracting set as a function ofν. Pos-itive values ofλmax indicate the presence of a chaotic at-tractor, and negative values indicate that the attractor is pe-riodic. Note thatλmax jumps abruptly atνIC, indicating asudden increase in the attractor’s chaoticity. For that reason,the pre-IC chaotic attractor is called “weak chaotic attrac-tor” and the post-IC attractor, the “strong chaotic attractor”.As mentioned before, for the chosen value of the damping


Fig. 5. Transient chaos atν = 0.0299248.

converge to the chaotic saddle in forward time dynamics; theunstable manifold is the set of points that converge to thechaotic saddle in the time reversed dynamics. The chaoticsaddle lies on the intersection of its stable and unstable man-ifolds (Nusse and Yorke, 1989; Ziemniak et al., 1994). In-side the periodic window (between SNB and IC in Fig. 3)the trajectories of all initial conditions will eventually con-verge to an attractor, except for initial conditions lying onthe stable manifold of SCS, which is a set of measure zero.Initial conditions close to the stable manifold are first at-tracted to SCS and stay close to its neighborhood for sometime, before they are repelled, following its unstable mani-fold. The closer an initial condition is to the stable manifoldof SCS, the longer its transient time before converging to theattractor (Hsu et al., 1988). In Fig. 5 an example of tran-sient chaos is shown for a time series of Poincare points atν = 0.0299248, where the phase dynamics converges to ap-3 attractor. The average transient timeτ can be estimatedby takingN0 random initial conditions and computingNt,the number of trajectories that are still in the transient stageafter t iterates. Figure 6 shows a graph oflogNt versustat ν = νIC , whereN0 = 10000 different initial conditionswere used. The graph can be fitted with a straight line ofslopeγ = −3.59 × 10−2 ± 1.45 × 10−4, which gives anaverage exit timeτ = −1/γ ≈ 27.9 (Rempel et al., 2004).

Figure 7 shows a three-dimensional projection(a1, a10,a16) of a chaotic saddle defined in the 15-dimensionalPoincare hyperplane atν = 0.029925 > νSNB , to the rightof the saddle-node bifurcation in Fig. 3(a). This chaotic sad-dle is a continuation of the surrounding chaotic saddle shownin Fig. 3(a). The chaotic saddle is not a continuous line. Ithas many gaps, most of which are not visible in Fig. 7 due totheir small size.

4.3 Interior crisis

In this section the dynamics near the interior crisis pointνIC

is investigated in terms of the role of chaotic saddles andUPOs. For details on the numerical algorithms employedsee Rempel and Chian (2003) and Rempel et al. (2004).

Fig. 6. Exponential decay ofNt, the number of trajectories insidethe restraining region at timet, as a function oft. The inverse of theslope of the fitted line gives the average exit timeτ ≈ 27.9.

Fig. 7. Three-dimensional projection(a1, a10, a16) of the chaoticsaddle defined in the 15-dimensional Poincare hyperplane just be-fore the saddle-node bifurcation, atν = 0.029925.

At crisis the collision of the mediating UPO with thetree-band weak chaotic attractor atνIC results in the for-mation of a single-band strong chaotic attractor. Figure 8shows a three-dimensional projection(a1, a10, a16) of thePoincare points of the strong chaotic attractor (SCA, lightline) after crisis(ν = 0.02992006), superimposed by the3-band weak chaotic attractor (WCA, dark lines) at crisis(ν = 0.02992021). An estimation of the fractal dimension ofthe post-crisis chaotic attractor atν = 0.02992006 using theKaplan-Yorke formula (Ott, 1993) results in the dimensionD ≈ 2.08. Note the similarity of the strong chaotic attractorwith the surrounding chaotic saddle shown in Fig. 7.

In Fig. 9 we plot a two-dimensional projection(a5, a6) ofthe upper branch of the chaotic attractor CA (black line) withthe surrounding chaotic saddle SCS (red lines) and its stablemanifold (blue dots) for (a)ν = 0.0299211 (before crisis)

Fig. 5. Transient chaos atν=0.0299248.

parameterν and the spatial system sizeL=2π , the dynam-ics of the Kuramoto-Sivashinsky equation is chaotic in time,but coherent in space. In fact, the spatial coherence remainsbasically unaltered throughout the whole range ofν used inFigs. 3 and 4, as indicated by the correlation lengthξ (Morriset al., 1993) in Fig. 4b.

4.2 Chaotic saddles

Associated to a chaotic saddle, there are stable and unsta-ble manifolds. The stable manifold is the set of points thatconverge to the chaotic saddle in forward time dynamics; theunstable manifold is the set of points that converge to thechaotic saddle in the time reversed dynamics. The chaoticsaddle lies on the intersection of its stable and unstable man-ifolds (Nusse and Yorke, 1989; Ziemniak et al., 1994). In-side the periodic window (between SNB and IC in Fig. 3)the trajectories of all initial conditions will eventually con-verge to an attractor, except for initial conditions lying onthe stable manifold of SCS, which is a set of measure zero.Initial conditions close to the stable manifold are first at-tracted to SCS and stay close to its neighborhood for sometime, before they are repelled, following its unstable mani-fold. The closer an initial condition is to the stable manifoldof SCS, the longer its transient time before converging to theattractor (Hsu et al., 1988). In Fig. 5 an example of tran-sient chaos is shown for a time series of Poincare points atν=0.0299248, where the phase dynamics converges to a p-3 attractor. The average transient timeτ can be estimatedby takingN0 random initial conditions and computingNt ,the number of trajectories that are still in the transient stageafter t iterates. Figure 6 shows a graph of logNt versustatν=νIC, whereN0=10 000 different initial conditions wereused. The graph can be fitted with a straight line of slopeγ=−3.59×10−2

±1.45×10−4, which gives an average exittime τ=−1/γ≈27.9 (Rempel et al., 2004).

Figure 7 shows a three-dimensional projection(a1, a10,a16) of a chaotic saddle defined in the 15-dimensionalPoincare hyperplane atν=0.029925>νSNB, to the right ofthe saddle-node bifurcation in Fig. 3a. This chaotic saddle





4.3 Interior crisis







Fig. 6. Exponential decay ofNt , the number of trajectories insidethe restraining region at timet , as a function oft . The inverse of theslope of the fitted line gives the average exit timeτ≈27.9.





4.3 Interior crisis







Fig. 7. Three-dimensional projection(a1, a10, a16) of the chaoticsaddle defined in the 15-dimensional Poincare hyperplane just be-fore the saddle-node bifurcation, atν=0.029925.

is a continuation of the surrounding chaotic saddle shown inFig. 3a. The chaotic saddle is not a continuous line. It hasmany gaps, most of which are not visible in Fig. 7 due totheir small size.

4.3 Interior crisis

In this section the dynamics near the interior crisis pointνICis investigated in terms of the role of chaotic saddles andUPOs. For details on the numerical algorithms employedsee Rempel and Chian (2003) and Rempel et al. (2004).

At crisis the collision of the mediating UPO with thetree-band weak chaotic attractor atνIC results in the for-mation of a single-band strong chaotic attractor. Figure 8shows a three-dimensional projection(a1, a10, a16) of the

E. L. Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 697Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 7

Fig. 8. Three-dimensional projection(a1, a10, a16) of the strongchaotic attractor (SCA, light line) defined in the 15-dimensionalPoincare hyperplane after crisis atν = 0.02992006, superimposedby the three-band weak chaotic attractor (WCA, dark lines) at crisis(ν = 0.02992021).

and (b)ν = 0.02992021 (at crisis). The upper Poincare pointof the p-3 mediating orbit is represented by the cross. In thePoincare map, an unstable periodic orbit turns into a set ofsaddle points with their associated stable and unstable man-ifolds. The dashed lines in Fig. 9 denote the local branchesof the stable manifold (SM) of the mediating orbit (MPO).Note that SCS has a large gap between the two dashed lines,and many other smaller gaps, or discontinuities. Its gaps aredue to the horizontal white spaces in the background, whichreflect the fractal structure of the stable manifold of SCS.Figure 9(b) reveals that at crisis the S chaotic saddle and theweak chaotic attractor collide. The collision takes place atthe mediating saddle, which belongs to SCS. Likewise, theweak chaotic attractor collides with the stable manifolds ofboth SCS and MPO.

4.4 Crisis-induced intermittency

After the crisis, it is still possible to determine the B and Sregions, which are separated by the stable manifold of MPO.Figure 10 shows the chaotic sets forν = 0.02992006, af-ter the crisis, around the upper branch of the strong chaoticattractor shown in Fig. 8. Figure 10(a) shows the chaoticattractor (CA) and Fig. 10(b) shows the corresponding B(green lines) and S (red lines) chaotic saddles. BCS is local-ized in a region of the phase space previously occupied bythe pre-crisis weak chaotic attractor. SCS is the continuationof the pre-crisis surrounding chaotic saddle. It can be seenfrom Fig. 10 that the post-crisis B and S chaotic saddles aresubsets of the strong chaotic attractor. The gaps in SCS andBCS are filled by a set of coupling UPO’s created at the crisis(Robert et al., 2000).

The two post-IC chaotic saddles are not attracting, but theyexert influence on the dynamics of nearby orbits, since they

Fig. 9. Plots of the upper branch of the chaotic attractor (CA,black line), the surrounding chaotic saddle (SCS, red lines) andits stable manifolds (blue dots): (a) before the interior crisis, atν = 0.0299211; and (b) at the interior crisis,ν = 0.02992021.The cross denotes one of the Poincare points of the p-3 mediat-ing unstable periodic orbit. The dashed lines represent segmentsof the boundary between the band region and the surrounding re-gion, given by the stable manifolds (SM) of the mediating unstableperiodic orbit.

are responsible for chaotic transients. Forν > νIC , trajecto-ries on the chaotic attractor never abandon the band region.Forν slightly less thanνIC a trajectory started in the B regioncan stay in B for a long time, after which it crosses SM andescapes to region S. Once it is in the surrounding region, thetrajectory is in the neighborhood of SCS. Since SCS is nonat-tracting, after some time the trajectory is “re-injected” intoregion B. The “jumps” between regions B and S repeat in-termittently. This crisis-induced intermittency can be viewedas an alternation between two transient behaviors, in whichthe trajectory spends a finite time in the vicinity of eitherBCS or SCS. These transitions between regions B and S aredue to the coupling UPO’s that are located within the gapsof BCS and SCS, and establish the dynamical connectionbetween the two chaotic saddles (Szabo et al., 2000; Rem-pel and Chian, 2004). The crisis-induced intermittency is

Fig. 8. Three-dimensional projection(a1, a10, a16) of the strongchaotic attractor (SCA, light line) defined in the 15-dimensionalPoincare hyperplane after crisis atν=0.02992006, superimposedby the three-band weak chaotic attractor (WCA, dark lines) at crisis(ν=0.02992021).

Poincare points of the strong chaotic attractor (SCA, lightline) after crisis(ν=0.02992006), superimposed by the 3-band weak chaotic attractor (WCA, dark lines) at crisis(ν=0.02992021). An estimation of the fractal dimension ofthe post-crisis chaotic attractor atν=0.02992006 using theKaplan-Yorke formula (Ott, 1993) results in the dimensionD≈2.08. Note the similarity of the strong chaotic attractorwith the surrounding chaotic saddle shown in Fig. 7.

In Fig. 9 we plot a two-dimensional projection (a5, a6) ofthe upper branch of the chaotic attractor CA (black line) withthe surrounding chaotic saddle SCS (red lines) and its stablemanifold (blue dots) for (a)ν=0.0299211 (before crisis) and(b) ν=0.02992021 (at crisis). The upper Poincare point ofthe p-3 mediating orbit is represented by the cross. In thePoincare map, an unstable periodic orbit turns into a set ofsaddle points with their associated stable and unstable man-ifolds. The dashed lines in Fig. 9 denote the local branchesof the stable manifold (SM) of the mediating orbit (MPO).Note that SCS has a large gap between the two dashed lines,and many other smaller gaps, or discontinuities. Its gaps aredue to the horizontal white spaces in the background, whichreflect the fractal structure of the stable manifold of SCS.Figure 1b reveals that at crisis the S chaotic saddle and theweak chaotic attractor collide. The collision takes place atthe mediating saddle, which belongs to SCS. Likewise, theweak chaotic attractor collides with the stable manifolds ofboth SCS and MPO.


After the crisis, it is still possible to determine the B and Sregions, which are separated by the stable manifold of MPO.Figure 10 shows the chaotic sets forν=0.02992006, after thecrisis, around the upper branch of the strong chaotic attrac-

Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 7

Fig. 8. Three-dimensional projection(a1, a10, a16) of the strongchaotic attractor (SCA, light line) defined in the 15-dimensionalPoincare hyperplane after crisis atν = 0.02992006, superimposedby the three-band weak chaotic attractor (WCA, dark lines) at crisis(ν = 0.02992021).

and (b)ν = 0.02992021 (at crisis). The upper Poincare pointof the p-3 mediating orbit is represented by the cross. In thePoincare map, an unstable periodic orbit turns into a set ofsaddle points with their associated stable and unstable man-ifolds. The dashed lines in Fig. 9 denote the local branchesof the stable manifold (SM) of the mediating orbit (MPO).Note that SCS has a large gap between the two dashed lines,and many other smaller gaps, or discontinuities. Its gaps aredue to the horizontal white spaces in the background, whichreflect the fractal structure of the stable manifold of SCS.Figure 9(b) reveals that at crisis the S chaotic saddle and theweak chaotic attractor collide. The collision takes place atthe mediating saddle, which belongs to SCS. Likewise, theweak chaotic attractor collides with the stable manifolds ofboth SCS and MPO.


After the crisis, it is still possible to determine the B and Sregions, which are separated by the stable manifold of MPO.Figure 10 shows the chaotic sets forν = 0.02992006, af-ter the crisis, around the upper branch of the strong chaoticattractor shown in Fig. 8. Figure 10(a) shows the chaoticattractor (CA) and Fig. 10(b) shows the corresponding B(green lines) and S (red lines) chaotic saddles. BCS is local-ized in a region of the phase space previously occupied bythe pre-crisis weak chaotic attractor. SCS is the continuationof the pre-crisis surrounding chaotic saddle. It can be seenfrom Fig. 10 that the post-crisis B and S chaotic saddles aresubsets of the strong chaotic attractor. The gaps in SCS andBCS are filled by a set of coupling UPO’s created at the crisis(Robert et al., 2000).

The two post-IC chaotic saddles are not attracting, but theyexert influence on the dynamics of nearby orbits, since they

Fig. 9. Plots of the upper branch of the chaotic attractor (CA,black line), the surrounding chaotic saddle (SCS, red lines) andits stable manifolds (blue dots): (a) before the interior crisis, atν = 0.0299211; and (b) at the interior crisis,ν = 0.02992021.The cross denotes one of the Poincare points of the p-3 mediat-ing unstable periodic orbit. The dashed lines represent segmentsof the boundary between the band region and the surrounding re-gion, given by the stable manifolds (SM) of the mediating unstableperiodic orbit.

are responsible for chaotic transients. Forν > νIC , trajecto-ries on the chaotic attractor never abandon the band region.Forν slightly less thanνIC a trajectory started in the B regioncan stay in B for a long time, after which it crosses SM andescapes to region S. Once it is in the surrounding region, thetrajectory is in the neighborhood of SCS. Since SCS is nonat-tracting, after some time the trajectory is “re-injected” intoregion B. The “jumps” between regions B and S repeat in-termittently. This crisis-induced intermittency can be viewedas an alternation between two transient behaviors, in whichthe trajectory spends a finite time in the vicinity of eitherBCS or SCS. These transitions between regions B and S aredue to the coupling UPO’s that are located within the gapsof BCS and SCS, and establish the dynamical connectionbetween the two chaotic saddles (Szabo et al., 2000; Rem-pel and Chian, 2004). The crisis-induced intermittency is

Fig. 9. Plots of the upper branch of the chaotic attractor (CA,black line), the surrounding chaotic saddle (SCS, red lines) andits stable manifolds (blue dots):(a) before the interior crisis, atν=0.0299211; and(b) at the interior crisis,ν=0.02992021. Thecross denotes one of the Poincare points of the p-3 mediating un-stable periodic orbit. The dashed lines represent segments of theboundary between the band region and the surrounding region,given by the stable manifolds (SM) of the mediating unstable pe-riodic orbit.

tor shown in Fig. 8. Figure 10a shows the chaotic attractor(CA) and Fig. 10b shows the corresponding B (green lines)and S (red lines) chaotic saddles. BCS is localized in a re-gion of the phase space previously occupied by the pre-crisisweak chaotic attractor. SCS is the continuation of the pre-crisis surrounding chaotic saddle. It can be seen from Fig. 10that the post-crisis B and S chaotic saddles are subsets of thestrong chaotic attractor. The gaps in SCS and BCS are filledby a set of coupling UPO’s created at the crisis (Robert et al.,2000).

The two post-IC chaotic saddles are not attracting, but theyexert influence on the dynamics of nearby orbits, since theyare responsible for chaotic transients. Forν>νIC, trajectorieson the chaotic attractor never abandon the band region. Forν slightly less thanνIC a trajectory started in the B regioncan stay in B for a long time, after which it crosses SM and


Fig. 10. (a) Upper branch of the chaotic attractor (CA) after theinterior crisis, atν = 0.02992006. The dashed lines indicate seg-ments of the stable manifolds (SM) of the mediating unstable pe-riodic orbit (cross); (b) the band chaotic saddle (BCS, green lines)and the surrounding chaotic saddle (SCS, red lines) that composethe chaotic attractor shown in (a).

characterized by time series containing weakly chaotic lami-nar phases that are randomly interrupted by strongly chaoticbursts, as shown in Fig. 11 forν = 0.02992. The weaklychaotic laminar phases correspond to the time spent in re-gion B, and the strongly chaotic bursts correspond to the timespent in region S.

The average duration of the “laminar” phases in the inter-mittent time series (thecharacteristic intermittency time) de-pends on the value ofν. Close toνIC the average time spentin the vicinity of BCS is very long, and decreases asν isdecreased away fromνIC . The characteristic intermittencytime (denoted byτ ) can be obtained as the average over along time series of the time between switches among regionsB and S. Figure 12 is a plot oflog10 τ vs log10(νIC − ν),where the solid line with slopeγ ≈ −0.46 is a linear fitof the values of the characteristic intermittency time com-puted from time series (circles). Figure 12 reveals that thecharacteristic timeτ decreases with the distance from thecritical parameter valueνIC following a power-law decay,

Fig. 11. Crisis-induced intermittency atν = 0.02992.

Fig. 12. log10 τ vs log10(νIC − ν). The solid line with slopeγ ≈−0.46 is a linear fit of the values of the characteristic intermittencytime τ computed from time series (circles).

τ ∼ (νIC − ν)γ , as expected (Grebogi et al., 1987a).

5 Conclusions

We have investigated the relevance of chaotic saddles andunstable periodic orbits at the onset of intermittent chaos inthe phase dynamics of nonlinear Alfven waves by using theKuramoto-Sivashinsky equation as a model equation. We de-scribed how a strong chaotic attractor formed after an in-terior crisis can be naturally decomposed into two nonat-tracting chaotic sets, known as chaotic saddles, dynamicallylinked by a set of coupling unstable periodic orbits. The per-turbed wave phase oscillates irregularly, switching intermit-tently from the neighborhood of one nonattracting chaotic setto the other.

The dynamical scenario described in this paper reflects asituation of intermittency in chaotic spatiotemporal systemsthat is still distant from the well-developed turbulent regimes

Fig. 10. (a)Upper branch of the chaotic attractor (CA) after the inte-rior crisis, atν=0.02992006. The dashed lines indicate segments ofthe stable manifolds (SM) of the mediating unstable periodic orbit(cross);(b) the band chaotic saddle (BCS, green lines) and the sur-rounding chaotic saddle (SCS, red lines) that compose the chaoticattractor shown in (a).

escapes to region S. Once it is in the surrounding region, thetrajectory is in the neighborhood of SCS. Since SCS is nonat-tracting, after some time the trajectory is “re-injected” intoregion B. The “jumps” between regions B and S repeat in-termittently. This crisis-induced intermittency can be viewedas an alternation between two transient behaviors, in whichthe trajectory spends a finite time in the vicinity of eitherBCS or SCS. These transitions between regions B and S aredue to the coupling UPO’s that are located within the gapsof BCS and SCS, and establish the dynamical connectionbetween the two chaotic saddles (Szabo et al., 2000; Rem-pel and Chian, 2004). The crisis-induced intermittency ischaracterized by time series containing weakly chaotic lami-nar phases that are randomly interrupted by strongly chaoticbursts, as shown in Fig. 11 forν = 0.02992. The weaklychaotic laminar phases correspond to the time spent in re-gion B, and the strongly chaotic bursts correspond to the timespent in region S.

The average duration of the “laminar” phases in the inter-








5 Conclusions



Fig. 11. Crisis-induced intermittency atν=0.02992.








5 Conclusions



Fig. 12. log10τ vs. log10(νIC−ν). The solid line with slopeγ≈−0.46 is a linear fit of the values of the characteristic intermit-tency timeτ computed from time series (circles).

mittent time series (the “characteristic intermittency time”)depends on the value ofν. Close toνIC the average time spentin the vicinity of BCS is very long, and decreases asν is de-creased away fromνIC. The characteristic intermittency time(denoted byτ ) can be obtained as the average over a longtime series of the time between switches among regions Band S. Figure 12 is a plot of log10 τ vs. log10(νIC−ν), wherethe solid line with slopeγ≈−0.46 is a linear fit of the val-ues of the characteristic intermittency time computed fromtime series (circles). Figure 12 reveals that the characteristictime τ decreases with the distance from the critical parame-ter valueνIC following a power-law decay,τ∼(νIC−ν)γ , asexpected (Grebogi et al., 1987a).

E. L. Rempel et al.: Intermittent chaos driven by nonlinear Alfven waves 699

5 Conclusions


The dynamical scenario described in this paper reflects asituation of intermittency in chaotic spatiotemporal systemsthat is still distant from the well-developed turbulent regimesfound in space plasmas, where one expects a huge numberof modes to become actively involved in the dynamics. Nev-ertheless, we believe the dynamical systems approach dis-cussed in this work may be helpful to understand more com-plex cases. Several authors have searched for evidence ofdeterministic chaos in space plasmas. Pavlos et al. (1992)found evidence of a low-dimensional chaotic attractor in thesolar wind through the determination of the correlation di-mension from time series of the magnetic field obtained bythe IMP-8 spacecraft at 1 AU. A low value for the correlationdimension obtained from solar wind data was also reported inother works (Marsch and Tu, 1993; Burlaga, 1995). Macekand Obojska (1997), Macek and Redaelli (2000) and Redaelliand Macek (2001) estimated the Lyapunov exponents, frac-tal dimension and the Kolmogorov entropy from solar winddata obtained by the Helios 1 spacecraft in the inner helio-sphere, concluding that the solar wind is likely a determinis-tic chaotic system. Sorriso-Valvo et al. (2001) studied plasmaintermittency by using a simplified shell model which mim-ics 3-D MHD equations. They showed that, just like in solarwind data, the intermittency displayed by the shell model ischaracterized by a strong departure from Gaussian behaviorin the probability density functions for the velocity and mag-netic field fluctuations at small scales. Although it is not pos-sible to state that the erratic fluctuations observed in the realworld are simple manifestations of chaos, in the works men-tioned above the behavior observed in nature seems to have astrong deterministic component. Thus, the study of dynam-ical systems can lead to a better understanding of the mech-anisms responsible for the onset of intermittent turbulencein space plasmas. In fact, space data is a mixture of bothstationary and nonstationary components. Therefore we canperform space data analysis using either deterministic or sta-tistical techniques. Most works on solar wind intermittencyare based on statistical analysis (Burlaga, 1991; Marsch andTu, 1993; Voros et al., 2002; Bruno et al., 2003; Dorotovicand Voros, 2004). Our theoretical study of solar wind in-termittency based on chaos theory provides new insights toextend the deterministic analysis of solar wind data (Macekand Obojska, 1997; Macek and Redaelli, 2000; Redaelli andMacek, 2001) to deepen our understanding of solar wind in-

termittency. In particular, our theoretical results can be usedto reconstruct chaotic transients in phase space from spacedata and to calculate the average duration of laminar phasesin solar wind intermittency.

Acknowledgements.This work is supported by FAPESP and CNPq.

Edited by: T. PassotReviewed by: two referees

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