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Irregular AttractorsVADIM S. ANISHCHENKO* and GALINA I. STRELKOVA

Laboratory of Nonlinear Dynamics, Department of Physics, Saratov State University,Astrakhanskaya str., 83, 410026 Saratov, Russia

(Received 20 September 1997)

In this paper the definition of attractor of a dissipative dynamical system is introduced. Theclassification of the existing types of attractors and the analysis of their characteristics arepresented. The discussed problems are illustrated by the results of numerical simulationsusing a number of real examples that provides the possibility to understand easily the mainproperties, similarities and differences of the considered types of attractors.

Keywords." Dynamical system, Attractor, Hyperbolic attractors, Lorenz type attractors,Quasiattractors, Strange nonchaotic attractors, Nonstrange chaotic attractors

1 INTRODUCTION

One of the main methods for investigation of self-oscillatory systems is the statement of equationsdescribing their dynamics and analysis of theirsolutions. Therefore, the study of the field ofmathematics named "Dynamical systems" is thebasic part of fundamental training on the theory ofnonlinear oscillations. In the classical theory ofoscillations the study of periodic and quasiperiodicregimes which are important for the description ofthe phenomena of generation and modulation ofoscillations was and remains the central problem.From this point of view, the mathematical imagesof these oscillatory regimes (a limit cycle and ann-dimensional torus) were not the main objects forinvestigation. They only provided an alternative

representation of the above-mentioned regimes inthe phase space of the systems under consideration.Really, one can define the main .properties ofperiodic oscillations using a segment of one of thephase coordinates x(t) on a finite time interval0 _< < T (T- period of oscillations) togetherwith the Fourier-spectrum or auto-correlationfunction of initial regime. In this sense quasiperi-odic oscillations are only slightly different. For theobservation time T of the realization x(t) it is

necessary to choose the largest of the characteristictimes that corresponds to the minimal basicfrequency in the spectrum. In other words, theavailability of a good oscillograph and spectrumanalyzer allows investigators to get a completeinformation about the properties of generatorsincluding modulation effects.

Corresponding author.

53

54 V.S. ANISHCHENKO AND G.I. STRELKOVA

When the dynamical chaos was discovered thesituation dramatically changed (Schuster, 1984;Lichtenberg and Lieberman, 1983; Anishchenko,1990; 1995). Chaotic oscillations are not periodicor quasiperiodic. Therefore, observation x(t) dur-ing any finite time interval does not providecomplete information. Moreover, it is very difficultto predict the specific observation times duringwhich it is possible to determine the features ofoscillatory regime. In this situation it is useful toanalyze in detail the geometric image of a self-oscillatory regime in the system phase space, i.e.,attractor. Note, that analyzing of the geometricstructure of attractors being the images of self-oscillations in dissipative dynamical systems can-not provide full information about oscillationsbeing, however, substantially more effective com-pared to the time series analysis.As known, a so-called strange attractor (Lorenz,

1963; Ruelle and Takens, 1971) is associated withthe image of dynamical chaos. Originally all non-trivial self-oscillatory regimes, whose general prop-erty is the absence of periodicity in time, wererelated with the image of the strange attractor.Later there came the understanding that chaoticself-oscillations may be substantially different intheir properties. And it definitely leads to thedifference in structure and properties of thecorresponding attractors. So, for example, it hasbecome clear that strange attractor is the image ofsome "ideal" chaos satisfying a number of rigorousmathematical requirements. It has been establishedthat in real systems the regime of strange attractorin the strict sense of mathematical definitioncannot be realized. What we observe in experi-ments is more often the regimes of a so-calledquasihyperbolic attractor or quasiattractor, whichare more complicated and cannot be rigorouslydescribed in terms of mathematics (Afraimovichand Shil’nikov, 1983; Afraimovich, 1989; 1990). Adistinctive feature of strange, quasihyperbolic andquasiattractors is exponential instability of phasetrajectories and the fractal dimension. Exponentialinstability is a criterium of chaotic behavior of thesystem in time. The fractal metric dimension shows

that the attractor is a complex geometric objectwhich is not a manifold. Since our knowledge ofdeterministic chaos is related just with theseproperties, one does not pay a significant attentionto the differences between geometric characteristicsof the attractor and temporal characteristics of thesystem’s dynamics. Nevertheless, recently the atten-tion of researchers has been attracted by the factthat non-periodic oscillations can possess asymp-totic stability in the presence of complex geometryof the attractor and, on the contrary, they can beexponentially unstable and correspond to theattractor which is a simple geometric object(a manifold) (see, Farmer et al., 1983; Grebogiet al., 1984).

It is appropriate to introduce a definition of"strangeness" of the attractor in terms of its

geometric structure and without connection withthe system’s dynamics. In the paper by Grebogiet al. (1984) such a definition is formulated:"Strange attractor is an attractor which is not a

finite set of points and not piecewise differentiable.We say that an attractor is piecewise differentiableif it is a piecewise differentiable curve or surface, ora volume bounded by a piecewise differentiableclosed surface".

Taking into account the importance of theanalysis of complex non-periodic regimes of oscil-lations in dynamical systems it is very useful to

classify in detail the types of attractors andformulate their definitions and basic properties.

In this paper we present definitions, propertiesand examples of non-trivial attractors of differenttypes which are realized in differential and discrete

dissipative nonlinear dynamical systems with finitenumber of degrees of freedom.The paper is organized as follows. We start with

a definition of a dynamical system attractor inSection 1. Regular attractors are discussed inSection 2. The main part of the work is Section 3that is devoted to the analysis of strange chaoticattractors. In this section we describe robusthyperbolic strange attractors, quasi-hyperbolicattractors (Lorenz type attractors) and quasiat-tractors. In Section 4 chaotic nonstrange and

IRREGULAR ATTRACTORS 55

strange nonchaotic attractors are analyzed. Andfinally, in Section 5 we formulate conclusions.The results presented in this paper have the aim

to provide the necessary information available forresearchers being specialists in the experimentswith nonlinear dynamical systems.

2 WHAT IS AN ATTRACTOR?

The time evolution of the state of a system with N/2degrees of freedom is described by either adeterministic system of differential equations orN-dimensional maps:

dxi )i-fi(xl,...,XN,#I,... #k), (1)

or

n2 N

Xn+ fi(x ,Xn,...,Xn,#l,...,k),i--l,2,...,N.

Here, xi(t) (or xn) are variables uniquely describ-ing the system’s state (its phase coordinates),/zt aresystem parameters, j(x,#) are smooth and, ingeneral case, nonlinear functions. A solution ofthe system (1) exists, it is unique for the given initialconditions xi(O) (or x) and smoothly depends onthe initial conditions (Cauchy theorem).Time evolution of the system can be uniquely

related with the phase trajectory in N-dimensionalCartesian space 9N, whose coordinates are phasevariables. The trajectory starts from the giveninitial condition xi(O) (or x), 1,2,..., N.We will consider only self-oscillatory regimes of

the system motion. From the physical point ofview, the latter means that in the system there existsome steady-state oscillations whose characteristicsdo not depend, to a certain extent, on the choice ofinitial state. We shall also consider the regime of astable equilibrium state being a limit case of self-oscillatory regime. As we will see, the notion self-oscillatory regime introduced by Andronov(Andronov et al., 1981) is the classical physical

interpretation of the definition of a dynamicalsystem attractor.

Let us examine the phase space 91u of system (1).All the values of system’s parameters mk are fixed.Let G be some finite (or infinite) region belongingto 9tu and including a subregion Go. The regions Gand Go satisfy the following conditions" 1. For anyinitial conditions xi(0) (or x) from the region Gall phase trajectories will reach sooner or later (intheory as oo (or n--+ oc)) the region Go. 2. If aphase trajectory belongs to the region Go at themoment t- tl (n- hi), then it will always belong to

Go, i.e., for any t>t (or n>_n) the phasetrajectory will be in the region Go (Afraimovich,1989; 1990).

If the conditions and 2 are satisfied, then theregion Go is called an attractor of a dynamicalsystem (1). In other words, the attractor Go is theinvariant with respect to the law (1) bounded set ofsystem’s trajectories, which any trajectories from

G approach and remain in. The region G is calledthe region (or basin) of attraction for the attractor

Go. According to the definition, only transientnonstationary types of motions can exist in theregion G1. The region Go corresponds to the steady-state (limit) types of motions. In this sense one cansay that the attractor Go is the isolated limit set ofthe phase trajectories of system (1). Any types ofthe system motion in the vicinity of the attractorhave only a transient character and as a result, thephase trajectories are attracted by the region Goas oe (n- oc). Hence, the name appears"attractor".The given definition of the attractor requires

some comments. Let us examine a stable ergodictwo-dimensional torus as an example of attractor.Any point on the ergodic torus surface belongs tothe attractor. By varying the system’s parameter,turn to the regime of the resonance structure on thetorus (let it be the resonance 1"1). In the Poincaresection the resonance 1"1 corresponds to theclosure of unstable separatrixes of a saddle on astable node. The problem is whether these separa-trixes belong to the attractor or the attractor is astable point being an image of the resonant limit

56 V.S. ANISHCHENKO AND G.I. STRELKOVA

cycle. From the viewpoint of rigorous mathe-matics, the first is correct! But in experiments wewill observe only the stable limit cycle that isconnected with the attractor in our understanding(Andronov et al., 1981; Arnold et al., 1986).

3 REGULAR ATTRACTORS

Before deterministic chaos was discovered onlythree types of stable steady-state solutions of thedynamical system (1) were known: an equilibriumstate when after a transient process the systemreaches a stationary (non-changing in time) state; astable periodic solution and a stable quasiperiodicsolution. In these cases the corresponding attrac-tors are a point in the phase space, a limit cycle anda limit n-dimensional torus. The Lyapunov char-acteristic exponents (LCE) signature of a phasetrajectory will be as follows (Anishchenko, 1990;1995):

for an equilibrium state,

for a limit cycle,

for an n-dimensional torus, n >_ 2.

As we will see further, strange chaotic attractorsof a complex geometrical structure correspond tonon-periodic solutions of the system (1). They haveat least one positive Lyapunov exponent and, as aconsequence, a fractional dimension that can beevaluated by using Kaplan-Yorke’s definition(Kaplan and Yorke, 1979):

-i: /iD --j+

I /ll(2)

where j is the largest integer number for which thesum % + 2 +"" + & >_ 0. The dimension D calcu-lated from formula (2) is one of the fractal

dimensions of the set and is called Lyapunov’sdimension. In general case, it is a lower bound forthe metric dimension of the attractor. If we applyformula (2) to the three types of the attractorsindicated above, we will then have the dimensionthat is equal to zero for the attractor being a point,D for the limit cycle attractor and D n for then-dimensional torus. It is very interesting to notethat in all cases the fractal dimension is strictlyequal to the metric dimension of the attractors. Theindicated types of the solutions are asymptoticallystable. The dimension D of the correspondingattractors is defined by an integer number andstrongly coincides with the metric dimension. Allthese facts allow us to say that the attractorsindicated above are regular. If one of the formu-lated conditions is violated, then the attractor isexcluded from the group of regular attractors. As ithas become clear now non-regular (strange) attrac-tors require special classification (Afraimovich,1984; 1989; 1990; Shil’nikov, 1993).

4 STRANGE CHAOTIC ATTRACTORS

A new type of attractor of the dynamical system (1)was first revealed by Lorenz in 1963 when hewas investigating numerically the Lorenz model(Lorenz, 1963). A rigorous proof of the existence ofnon-periodic solutions of system (1) was given byRuelle and Takens in 1971. They also introducedthe notion of strange attractor as the image ofdeterministic chaos (Ruelle and Takens, 1971).Since that time, very often the phenomenon ofdeterministic chaos and the concept of strangeattractor are definitely connected to each other.However, this is not always correct and needs someexplanations.

4.1 Robust Hyperbolic Attractors

If we read the work (Ruelle and Takens, 1971) verycarefully, we will realize that a proof of the strangeattractor existence was given under the strongsuggestion that the dynamical system (1) was

IRREGULAR ATTRACTORS 57

robust hyperbolic. What does it mean? The systemis hyperbolic if all of its phase trajectories aresaddle. A point as an image of a trajectory in thePoincare section is always a saddle. Robustnessmeans that when the right-hand parts in (1) areslightly perturbed or control parameters are

slightly varied, all the trajectories remain saddle.Speaking more strictly, hyberbolic attractors

should satisfy the following three conditions

(Afraimovich, 1989; 1990):

1. A hyperbolic attractor consists of a continuumof "unstable leaves", or curves, which aredense in the attractor and along which closetrajectories exponentially diverge.

2. A hyperbolic attractor (in the neighborhood ofeach point) has the same geometry defined as a

product of the Cantor set on an interval.3. A hyperbolic attractor has a neighborhood

foliated into "stable leaves" along which theclose trajectories converge to the attractor.

Robustness means that properties 1-3 hold underperturbations.

Figure represents a saddle trajectory I" and thecorresponding points Qi of its intersection withthe secant Poincare surface S and also illustratesthe local behavior of stable and unstable manifoldsof a saddle point Qi. But the condition that thepoint Qi of intersection of I’ with S is locally arobust saddle is not enough for robust hyperboli-city! Certain conditions upon global (non-local)properties of stable and unstable manifolds areneeded. Let us consider Fig. 2. Due to the presenceof attractor, stable and unstable manifolds Ws and

Wu are to be concentrated in the region of theattractor Go. At the same time they can intersectwith the appearance of homoclinic points (sur-faces) by forming so-called homoclinic structures.These structures in robust hyperbolic systems mustbe robust. This means that from the topologicalviewpoint, the intersection structure of Ws and Wumust correspond to Fig. 2(a) and should notchange qualitatively under perturbations! Thecases in Figs. 2(b) and (c) are excluded as theycharacterize two non-robust phenomena, namely,

the phenomenon of closure of the manifolds withthe loop formation (Fig. 2(b)) and the phenomenonof tangency of the stable and unstable manifolds(Fig. 2(c)).

FIGURE A saddle point Qi as the image of a hyperbolictrajectory in the Poincare section.

FIGURE 2 Three possible cases of the intersection of thestable and unstable separatrixes of the saddle point Qi in thePoincare section.

58 V.S. ANISHCHENKO AND G.I. STRELKOVA

If non-local properties of the manifolds lead tothe non-robust situations shown in Figs. 2(b) and(c) when the dynamical system is perturbed,bifurcations of system’s solutions are possible(Gavrilov and Shil’nikov, 1972; 1973). In robusthyperbolic systems no bifurcations should occur. Ifsmall perturbations are introduced, the trajectory 1always remains saddle, the latter correspondingto the case shown in Fig. 2(a). As we will see later,the non-robust cases (Figs. 2(b) and (c)) cause theappearance of more complicated chaotic attractingsets, i.e., quasiattractors (Afraimovich, 1984; 1989;1990).

Therefore, it is necessary to understand thatstrange (according to Ruelle-Takens) attractorsare always robust hyperbolic limit sets. The mainfeature in which strange chaotic attractors differfrom regular ones is exponential instability of thephase trajectory on the attractor. In this case theLCE spectrum includes at least one positiveexponent:

D- 2 + a+/la l > 2.

As seen from Kaplan-Yorke’s formula, fractaldimension of an attractor will always be more than2 and, in general case, will not be defined by an

integer number. A minimal dimension of the phasespace in which a strange attractor can be"embedded" equals 3. Therefore, the regime ofdeterministic chaos can be observed in differentialdynamical systems which have the dimensionalityN>3.

In mathematics at least two examples of robusthyberbolic attractors are known. These areSmale-Williams attractor (Smale, 1967) andPlykin attractor (Plykin, 1980). Unfortunately, upto now in real systems the regime of rigorouslyhyperbolic robust chaos has not been revealed!"Truly" strange attractors are an ideal but stillunattainable model of deterministic chaos. In reallife, as usual, everything is more complicatedcompared with idealization.

4.2 Quasihyperbolic Attractors. LorenzType Attractors

There is a finite number of dynamical systems thathave almost hyperbolic attractors (quasihyperbolicattractors). Such attractors do not contain stableregular trajectories (points, cycles etc.) and areclosest in their structure and properties to robusthyperbolic attractors. As examples, we canindicate here the Lorenz attractor (Lorenz, 1963;Shil’nikov, 1980), Belykh and Lozi attractors

(Lozi, 1978; Belykh, 1982; 1995). For quasihyper-bolic attractors at least one of three conditions ofhyperbolicity is violated. In particular, for theLorenz attractor the second condition is not valid.However, Lorenz type attractors were revealed in a

number of systems and from the experimentalpoint of view, we can treat them as the examplesof "truly" strange attractors (Shil’nikov, 1980;Williams, 1977; Cook and Roberts, 1970).

It is surprising but it is a fact that it is the chaoticattractor in the Lorenz model that is closest in itsproperties and structure to robust hyperbolicattractors. In the Lorenz attractor all trajectoriesare saddle and when one varies parameters, no

stable points or cycles are born (Bykov andShil’nikov, 1989; Afraimovich, 1984; 1989; 1990).The Lorenz equations were first obtained from

the Navier-Stokes’s equations while solving theproblem of thermal convection and have thefollowing form:

:---o-(x-y), j:-- rx- y- xz, - xy- bz,

where or, b and r are control parameters. Some lasermodels as well as the model of disc dynamo can bereduced to the equations of the type (3) (Shil’nikov,1980; Cook and Roberts, 1970).System (3) is invariant with respect to the

transformation (-x, -y, z) + (x, y, z) and is char-acterized by three equilibrium states. Let us fix thevalues of the parameters as a 10, b 8/3. Whilestudying the behavior of system’s trajectories whenthe parameter r is varied, over the critical point

IRREGULAR ATTRACTORS 59

rc--- 24, 74 the only chaotic attractor, i.e., theLorenz attractor, is realized. The basin of itsattraction is the entire phase space. The Lorenzattractor is the attracting set consisting of the phasetrajectories which are characterized by the indi-vidual exponential instability. The properties indi-cated above do not change when one varies theparameters in a finite range of their values and donot depend on initial conditions. Figure 3 shows a

projection of the Lorenz attractor on the plane(x, z) and the basin of its attraction.The Lorenz attractor demonstrates practically

all properties and qualities of robust hyperbolicattractors:

1. The presence of a denumerable set of separa-trix loops of equilibrium states of the Lorenzsystem does not lead to the birth of stableregular attractors when any perturbations areadded.

2. When the parameters of the system (3) arevaried in a finite range of their values, nobifurcations occur in the Lorenz attractor andno other stable attracting subsets appear.

3. When the equations are slightly perturbed orexternal noise of small intensity is added, the

FIGURE 3 Projection of the Lorenz attractor on the plane(x, z) (black color) and the basin of its attraction (grey color)in the Poincare section y=0 for r=28, c7= 10, b=8/3.

changes in the attractor’s structure are alsosmall. The own dynamical nature of the chaoticbehavior is much stronger than that of thenondynamical chaos added from outside.

4. For the Lorenz attractor it is possible toconstruct a smooth function of distribution

density p (x, y, z), i.e., the probability measureof the attractor. Small perturbations of the flow(3) cause small changes in the probabilitymeasure according to the theoretical resultswhich were formulated for hyperbolic systemsby Kifer (1974).

The Lorenz attractor has the classical spectrumof the Lyapunov characteristic exponents (LCE):

=0.9, ,k2=O,(4)

,’3 14.57 (r 26).

It gives Lyapunov’s dimension D 2.06. The factthat the fractal part of dimension is close to 0 is

explained by the strong contraction of the flow indissipative system (3)

divV -(or + b + 1). (5)

This circumstance explains the fact that thePoincare map of the Lorenz attractor is very closeto a one-dimensional map. Due to the fact thatdiv F(see (5)) does not depend on phase variables,the birth of the regime of two-frequency quasipe-riodic oscillations in system (3) is impossible.Therefore, the set of non-singular phase trajec-tories of the system includes only points, cycles andthe Lorenz attractor.Note that from experimental viewpoint, system

(3) displays practically robust hyperbolic chaos in a

finite range of values of its control parameters. Thebifurcation diagram of system (3) is shown in Fig. 4(Bykov and Shil’nikov, 1989). The shaded region inthe parametric space corresponds to the existenceof Lorenz attractor, while outside of this region theproperties of the chaotic attractor will be essen-

tially different. In particular, bifurcation line 13 inFig. 4 corresponds to the transition "Lorenzattractor- quasiattractor".

60 V.S. ANISHCHENKO AND G.I. STRELKOVA

Let us illustrate the typical characteristics andproperties of Lorenz attractor. The LCE spectrumdoes not change under variation of initial condi-tions because the Lorenz attractor is the only oneand the basin of its attraction is the entire phasespace (see Figs. 4 and 5).The LCE spectrum does not practically change

as one varies the system’s control parameters inthe region of Lorenz attractor existence (seeFig. 4). These properties visually illustrate therobustness of Lorenz attractor in experiments:

10-

Z,orenzQuastattractor

010 20 30 40

FIGURE 4 Bifurcation diagram of the Lorenz system onthe plane of parameters z, cr for b=8/3, ll is the line of theexistence of a symmetrical separatrix loop of zero equilibriumstate; 12 is the line of the birth of the Lorenz attractor; 13 isthe line of the bifurcation transition to quasiattractor.

-5

5 10 15 20 25 30

FIGURE 5 Dependence of the LCE spectrum of the Lorenzattractor on the values of x coordinate for r-28, or= 10,b= 8/3.

general properties of the attractor hold undervariation of the parameters and initial conditions;no bifurcations of the attractor occur.

Autocorrelation function and power spectrum ofLorenz attractor presented in Fig. 6 are typical forintermixing systems. The autocorrelation functionalmost exponentially decreases without oscillationswith the increase of time (see Fig. 6(a)). The powerspectrum is a continuous decreasing function offrequency and does not contain pronounced peaksat any characteristic frequencies (see Fig. 6(b)).

All the characteristics and properties of theLorenz attractor do not practically change in thepresence of additive (or multiplicative) noise withsmall intensity. Figure 7 represents the plots for thestationary two-dimensional probability densityp(x,z) of Lorenz attractor in the absence and inthe presence of additive noise introduced into thethree equations of system (3) (Anishchenko, 1995).

4.3 Quasiattractors

The so-called quasiattractors (Afraimovich andShil’nikov 1983; Shil’nikov, 1993) are most typicalin experiments. They illustrate experimentallyobserved chaos in the majority ofdynamical systems(Schuster, 1984; Lichtenberg and Lieberman, 1983;Anishchenko, 1990; 1995; Neimark and Landa,1989; Rabinovich and Trubetskov, 1984). In thesystems with quasiattractors the regimes of deter-ministic chaos, which are characterized by theexponential instability of trajectories and a fractalgeometry of the attractor, are realized. From thispoint of view, the characteristics of the indicatedregimes of self-oscillations are identical to thegeneral ones of robust hyperbolic attractors andLorenz type attractors. However, there are veryessential and principal differences which it isnecessary to take into consideration to avoidincorrect explanations of experimental results. Afeature of quasiattractors is the co-existence of adenumerable set of different chaotic and regularattractors in a bounded element ofthe system phasespace volume when the system’s parameters arefixed. This set of all co-existing limit subsets of

IRREGULAR ATTRACTORS 61

1.0 1.0 [3

0. 0..

"0. "0.

vO., 0.4

0.2 0.2

0.0 0.00 3 6 9 12 15 0 3 6 9 12 15

T T

FIGURE 6 Autocorrelation function and power spectrum for the Lorenz attractor (a, c) and for Anishchenko-Astakhov’soscillator (b, d).

trajectories in the bounded region Go ofphase space,which all or almost all the trajectories from theregion G1 including Go approach, is called thequasiattractor of the dynamical system.

Quasiattractors are characterized by a verycomplex structure of embedded basins of attrac-tion. But the complexity is wider than this fact.Under variation of system’s parameters in a finiterange of their values the cascades of differentbifurcations of both regular and chaotic attractorsare realized. Accordingly, the bifurcational re-

organization of their basins of attraction takesplace. The reason for such a complexity ofquasiattractors is the effects of homoclinic tan-gency of stable and unstable manifolds of saddlepoints in the Poincare section which take place onthe set of parameter values of non-zero measure

(Gavrilov and Shil’nikov, 1972; 1973; Afraimovich,1984; 1989; 1990).

If one takes into account that the basins ofattraction of co-existing limit sets can have fractalboundaries and occupy very narrow regions inphase space, then it becomes clear how importantthe role of accuracy in numerical experiments andthe influence of external noises is. Let us demon-strate the properties of quasiattractors using aseries of examples.

Let us explore a typical system whose chaoticdynamics fully illustrates Shilnikov’s theoremabout the properties of dynamical systems with asaddle-focus separatrix loop of the equilibriumstate (Anishchenko, 1990; 1995). This system iscalled a modified oscillator with intertial nonlin-earity (Anishchenko-Astakhov’s oscillator) and is

62 V.S. ANISHCHENKO AND G.I. STRELKOVA

FIGURE 7 Stationary two-dimensional probability densityp(x, z) of the Lorenz attractor in the absence of noise (a) and inthe presence of additive white noise with intensity d= 0.8 (b).

a three-dimensional two-parametric differentialsystem described by the following equations:

X mx + y xz, j -x,

-gz + gI(x)x, (6)

where

1, x>0,(x)- 0, x<_0.

Let us fix the parameters as m 1.42, g--0.097and calculate the exponents of the LCE spectrumas a function of initial conditions. The results areshown in Fig. 8 and represent the co-existence ofchaotic and periodic oscillatory regimes. The moredetailed analysis of the results in Fig. 8 shows thatfor -2.0 < x < 0 we can observe the regime of oneof the limit cycles and the chaotic regime, while for-4.0 < x <-2.0 a limit cycle of another family isadded (compare the values of the third exponent ofthe LCE spectrum). More visually this situation

0015

0,010

0,005

-40 -35 -30 -25 -2.0 -10 -05 00X

000

-0 04

-0 12

-0,16

’l0 -3 5 -3 -2.5 -2 5 O -0.5 0 0

x

O0

-00.5

-0 10

-D 15

-0 20

-0 25-4 0 -3 -3 0 -g -g 0 0 -0.5 0.0

x

FIGURE 8 The LCE spectrum of system (6) as a functionof x coordinate for the parameters m 1.42, g-0.097.

can be illustrated by Fig. 9. This figure representsprojections of the three co-existing attractors insystem (6) and basins of their attraction. Indeed,period and period 2 limit cycles and the chaoticattractor co-exist in the system.Due to nonrobustness of system (6) the all of its

limit subsets undergo bifurcations as the param-eters are varied. To illustrate this fact we presentthe dependence of the LCE spectrum exponents onthe parameter m shown in Fig. 10.The fact that the exponent 1 is equal to 0

testifies to the birth of one of the sets of limit cycles

IRREGULAR ATTRACTORS 63

which demonstrate the cascades of period doublingbifurcations. With this, , equals to 0, while ,2takes different negative values. Bifurcations of theattractors are accompanied by the changes in thestructure of their basins of attraction whoseboundaries become fractal.The presence of stable and saddle cycles in a

quasiattractor together with chaotic limit subsetsmanifests itself in the structures of autocorrelationfunction (ACF) and power spectrum. The results

FIGURE 9 Projections of the co-existing attractors in sys-tem (6) for m 1.42, g-0.097 and the structure of the basinsof their attraction in the Poincare section z 1.

FIGURE 10 Lyapunov exponents )1 and X as a functionof parameter m (g 0.2) for system (6).

calculated for the chaotic regime of system (6) atm 1.5, g 0.2 are presented in Figs. 6(b) and (d).The ACF decreases exponentially in average withtime and the power spectrum is continuous.However, under a more careful consideration wecan notice a periodic component in the ACF andsudden peaks at certain characteristic frequenciesin the spectrum. Quasiattractor differs fromLorenz attractor by these peculiarities of ACF andpower spectrum of chaotic regime, the latter beingtypical (compare the results presented in Fig. 6).

Fractality and riddling of the basin boundariesof the set of co-existing regular and chaoticattractors of the system cause a high sensitivity tonoise perturbations. Let us consider the regime ofthe chaotic attractor in system (6) where rn 1.5,g 0.2. Figure 11 represents the plots of the two-dimensional probability density p(x,y) in theabsence of noise and in the case when Gaussiannoise is introduced additively to the right-hand

FIGURE 11 Probability distribution density p(x,y) for thechaotic regime in Anishchenko-Astakhov’s oscillator(m= 1.5, g=0.2) in the absence of noise (a) and in the pres-ence of additive noise with intensity d 10 introduced to allthe system (6) equations (b).

64 V.S. ANISHCHENKO AND G.I. STRELKOVA

parts of the three system equations. As seen fromthe figures, the introduction of the noise of smallintensity leads to explicit changes in the structure ofprobability function.We have chosen the regime where the finite

number of attractors co-exist as a visual illustrationof complexity of a quasiattractor. Theoretically a

quasiattractor includes an infinite number of co-

existing limit regimes which undergo an infinitesequence of different bifurcations when parametersare varied slightly. Also there can be the ranges ofparameter values where (from the experimentalpoint of view) the system has only one chaoticattractor that attracts all trajectories in the phasespace. If we are able to observe such a regime inexperiments and determine the region of param-eters where it exists, we can then speak about theregime that is close to robust hyperbolic attractor.As an example, examine a discrete dynamical

system in the form of two coupled logistic maps(Strelkova and Anishchenko, 1997):

x,+, ox2,, + 7(Yn X,,),(7)

Yn+l oy2n + 7(Xn Yn).The only regime of hyperchaos can be realized insystem (7) whose basin of attraction is a boundedrhombus on the parameter plane (x, y). However,if we change the control parameters, the number ofco-existing attractors increases abruptly and thestructure ofbasins of their attraction becomes morecomplicated. The results are shown in Fig. 12.Figure 12(b), in particular, visually illustrates theinfluence of uncertainty in the choice of initialconditions on the system’s behavior. For this pur-pose choose a small box denoted by 2 inFig. 12(b) as a region of uncertainty in initialconditions from the basin of attraction of thequasiattractor and examine its evolution in time.We will obtain a combination of all three attractorsof the system as the stationary regime! In experi-ments, in the presence of noise one of the three co-

existing regimes will dominate randomly in thesystem’s dynamics. This means that by choosinginitial conditions with a finite accuracy the generalproperty of dynamical chaos, i.e. the reproduc-

FIGURE 12 Hyperchaos in system (7) for parametersc =0.9, 7 =0.285 (a), the regime of the co-existence of attrac-tors and the structure of the basins of their attraction in sys-tem (7) as c =0.78, g =0.2876 (b) in phase space (xn, yn). Thenumbers 1,2,3 denote the regions of uncertainty in initialconditions leading to the corresponding limit sets indicated byarrows.

ibility from initial conditions, will be violated insuch systems.

STRANGE NONCHAOTIC AND CHAOTICNONSTRANGE ATTRACTORS

Chaotic attractors of the three types describedabove have two common principal properties.The first one is the complex geometric structureof an attractor (and, as a consequence, fractality ofits metric dimension). The second property is theexponential instability of individual trajectories on

IRREGULAR ATTRACTORS 65

the attractor. It is these properties that are used byresearchers as a criterium for diagnostics of theregimes of deterministic chaos.However, nonregular attractors as the mathe-

matical images of complex dynamics are notrestricted by the chaotic attractors describedabove. It has become clear that chaotic behaviorin the sense of intermixing and the geometric"strangeness" of an attractor cannot be relatedwith each other. Strange attractors in terms of theirgeometry can be nonchaotic due to the absence ofexponential instability of phase trajectories. On theother hand, there are examples of intermixingdissipative systems whose attractors are not strangein a strict sense, that is, they are not characterizedby the fractal structure and the fractal metricdimension.

In other words, there are examples of concretedissipative dynamical systems whose attractors arecharacterized by the following properties:

1. An attractor has a regular geometric structurefrom the viewpoint of integer metric dimen-sion. In addition, individual phase trajectorieson the attractor are exponentially unstable inaverage.

2. An attractor is characterized by a complicatedgeometric structure. Here, trajectories on it areasymptotically stable. There is no intermixing.

The first type is called a chaotic nonstrangeattractor (CNA). The second one is called a strangenonchaotic attractor (SNA).

5.1 Chaotic Nonstrange Attractors

Chaotic attractors which are not strange from theviewpoint of their geometry have been known for along time (Farmer et al., 1983; Grebogi et al., 1984;1985), but by now they nave been studied insuffi-ciently. The modified Arnold’s map (Farmer et al.,1983) is an example of a dynamical system withCNA. This map is a well-known "cat map" with anonlinear periodic term:

xn+l xn + yn + cos 2ry, modl,

y+l x + 2y, mod 1. (8)

If < 1/2r, then map (8) is a diffeomorphism ona torus. In other words, map (8) is one-to-one

(reversible) and transforms a unit square on theplane (xn, yn) into itself. Map (8) is dissipative, thatis an area element contracts with each iteration.This property is proved easily if one calculates theJacobian:

27r6 sin 2ryn2 -0, f<-. (9)

The average (in time)value [Jl< 1. The LCEspectrum is "+ ", "-", i.e., there is intermixing.

It might seem that we are dealing with an

ordinary chaotic strange attractor, but it is notso. A distinctive feature of the considered case isthat, despite the contraction, the motion of a

representative point of the map (8) is ergodic! Asn--+ oc, the point visits any element of the unitsquare! The evidence of this fact is that the metricdimension of the attractor (the capacity accordingto Kolmogorov) equals 2. Although the density ofpoints of the attractor is not uniform in the unitsquare but it is nowhere equal to 0. Therefore,inspite of the contraction, the attractor of thesystem (8) is the whole unit square. In this senseArnold’s attractor is not strange as its geometry isnot fractal.

Let us consider how the attractor is formed inorder to understand in more detail the peculiaritiesof its structure. Let us choose a small element of thearea as a region of initial conditions (0 < x < 0.2,0<yn<0.2) and observe the evolution of thiselement while iterating map (8).

Figure 13 represents the sequential images of theinitial small square that display the following. Dueto the contraction along one direction and theextension along another the initial square evolvesinto a finite set of "bands" which tend to cover theentire surface of the unit square when iterating. Asn oc we have "the black square".

But as seen from the phase diagram of theattractor shown in Fig. 14, although the pointscover the square practically entirely its distributiondensity is explicitly inhomogeneous! As a quanti-tative measure of such an inhomogeneity we use the

66 V.S. ANISHCHENKO AND G.I. STRELKOVA

oo

0.75

0.50

0 25

0.00000 0.25 0.50 075 O0

X

1.00 1.00

Y Y0,75 0.75

0.50 0.50

0 25 r 025

Lo00.00 0.25 O.r’oO O. 75 .00 0.00

"AZ.’.,.’.>:;{/.. : ." ."/.;." / ,t.,’.::. ,...3"o. .." :..:U.’ _.."/J,":" "...a’. I a" ]" .;,i." /.9:.,,/..-,, ,., ,,,’,,: .Z,, .-’.’_’Y"’,:Z" "/ ,: 2 i g’;" /.’:;.’" i:l>S’’," " ,.".’:;" A-".,.".,". : e" ,’.:::,-

i -".>:’." Ag /.--)2:’1I -" .::;;g" ,.;’-,".??"’1

.,.,’.; ,’.&::.’" :: ,:’;;.; ,’z, ," :- .:,,’/;; *.>.:’A7"," 5 .-’/.;-t "Z" ’,*;; /..-’;4v.," .,,,z-_.".)...-. ." _"" -:L-’..

"z" :g--" z...7:; 4;."" g’’." ,;_..,e’, .*" ZI_._____

0.00 0.25 0.50 0.-/5 1.00x x

FIGURE 13 Evolution of an initial square element in Arnold’s map (8) in 1, 3 and 5 iterations, respectively.

FIGURE 14 Phase diagram of the chaotic nonstrangeattractor in the Arnold map (8) for h-0.15.

information dimension <_ D _< 2. For instance,for 5-0.05, D 1.96, for 5--0.10, D 1.84. Inaddition, as we have said, the capacity Dc-2.0(this is a rigorous result of Y. Sinai). As a

consequence of inhomogeneity of the probabilitydistribution density of the points on the attractor,the values of an probability-metric dimensions ofthe Arnold’s attractor will lie in the interval< D< 2. These dimensions take into account

not only geometric but also dynamical propertiesof the attractor.CNAs were revealed in a number of other maps

on a torus. One can assume that ergodic chaoticmotions are typical for diffeomorphisms on a

torus. A proof of the existence of CNA in suchmaps gives the possibility to state that there are

flow (differential) system in 91N, N_> 4 which havethe regimes of CNA. However, by the present timeCNAs have not been discovered in differentialdynamical systems. In this connection, in particu-lar, by now a problem of the possibility of theexistence of the chaotic attractor on a three-dimensional torus surface embedded into phasespace with the dimensionality N_> 4 is still open.

Let us pay attention to the following importantfact. As seen from Fig. 14, the chaotic set of thepoints of map (8) cover densely the unit square ofthe surface with some continuous probabilitymeasure. Therefore, the attractor is the entire unitsquare! Then how is this fact adjusted with thedefinition of the attractor as the isolated limit set?What is the region of attraction in this case? Suchquestions arise because map (8) describes only theattractor and does not contain any informationabout transient processes. Here we should discuss as

follows. Let us choose some differential system inN(N> 4) that has a three-dimensional torus T as

its attractor (the system is dissipative!). Consider thestructure of phase trajectories on T. In order to dothat let us introduce the Poincare section on it. Inthis case we have a map on a two-dimensional torusT2. System (8) is such a map. It models theproperties of the limit trajectories lying in theoriginal system on T, i.e., on the attractor. There-fore, Fig. 14 illustrates the structure of the attractorand the region ofits attraction is outside ofthe limitsof possibilities described by the discrete model.

IRREGULAR ATTRACTORS 67

5.2 Strange Nonchaotic Attractors

As we have said, strange chaotic attractors possessgeometric "strangeness" and intermixing. In otherwords, complex dynamics of an intermixing systemis the reason for the geometric complexity of thecorresponding attractor. Nevertheless, in the caseof CNA we had to divide these properties, sinceintermixing cannot always lead to geometrical"strangeness" of the attractor. In this part we willconsider the possibility of realizing the oppositesituation when the system demonstrates compli-cated non-periodical oscillatory regime that isasymptotically stable (without intermixing) butthe attractor is not regular from the viewpoint ofits geometric structure.One can easily think of examples of nonrobust

SNAs. Any strange chaotic attractor at the criticalpoint of its transition to chaos is an example ofSNA. Indeed, let us explore, for instance, theFeigenbaum attractor in the well-known logisticmap:

xn+ rxn(1 x). (10)

At the critical point r* 3 569945 there appears alimit set of points that has the fractal dimension

Dc 0.548... (a so-called Feigenbaum attractor).In addition, the Lyapunov exponent is equal tozero (there is no chaos!). According to thedefinition, such an attractor is strange nonchaotic.But it is nonrobust. From the physical point ofview, it is interesting to study robust attractorswhich exist on the set of parameter values of non-zero measure and hold their structure underperturbations. As it has become clear, robustSNAs exist both in differential and discretedynamical systems (Grebogi et al., 1984;Kapitaniak and Wojewoda, 1993; Anishchenkoet al., 1996).SNAs are typical for dynamical systems driven

by quasiperiodic force. Here it is useful to elucidatewhat the attractor of a non-autonomous systemmeans in our understanding. Assume that anautonomous dynamical system in :N is driven bya periodic force with period To---27r/020. We will

analyze the Poincare section in a period of theexternal force. In a secant surface nTo each time

(for any n) we will observe some set of points. Inthis case the attractor is a projection of the set ofpoints in the secant planes, which was obtained fora sequence n , on the initial secant surface forn-0. It is also possible to use the followingmethod. The original non-autonomous systemcan be reduced to an autonomous system byintroducing new variables and limiting the regionsof new variable values by the period duration.Thus, the phase space is expanded and one can usethe attractor definition introduced in Section 2.A feature of the systems with quasiperiodic force

is that the introduction of two new independentvariables means that we take into considerationtwo time scales which are not related with the statevariables of the original autonomous system andare independent of each other.

In the simplest case, a map where SNA is realizedcan be written in the form

Xn+l =f(Xn, qSn, r), qSn+l co + qSn, modl,

where x is a dynamical variable, b is a phase ofexternal force, r is a system parameter (or param-eters), f is a nonlinear function that is periodicwith respect to b, with the period 1, 02 is anirrational number. If 02 is irrational, then theforcing will be quasiperiodic because there is no

period k such thatf(bn + k) =f(b). Thus, the mapsin the form (11) model the dynamics of differentialsystems with quasiperiodic (two-frequency) exter-nal force. There are two characteristic time scales,namely, k (a map iteration) and k2 1/02, 02 isa phase shift during one iteration. Therefore, 02 iscalled the rotation number that is characterized bythe ratio of two frequencies of the quasiperiodicforce.SNA was first revealed and studied in the

following map (Grebogi et al., 1984):

Xn+ Ath(x) cos 2rb, b+ 02 + bn, modl.

(12)

68 V.S. ANISHCHENKO AND G.I. STRELKOVA

An irrational value of the parameter a; is moreoften chosen to be equal to the so-called "goldenmean": - 0.5(x/-1). For the values > inthe map (12) the existence of SNA was rigorouslyproved. Besides system (12), SNAs were revealedwhen quasiperiodic excitation had been applied tothe circle map, logistic map, Henon map, etc. Theexamples of SNA in map (12) and in Feigenbaummap (Heagy and Hammel, 1994)

xn+l a(1 scos27rn)Xn(1 xn),b+l bn + v, modl

(13)

are presented in Fig. 15.The main features of SNA which allow us to

extract these objects as a separate class are asfollows:

1. Geometric characteristics of SNA. The attrac-tor (for example, on the phase plane) is formedby a curve of an infinite length that is non-differentiable on the dense set of points. Thiscurve like Peano’s curve covers densely a partof the phase plane so that the metric dimen-sion (the capacity) of SNA is strongly equalto 2. But unlike the map (8), in this case onecannot consider that a part of the plane is theattractor since the total measure of pointsbelonging to the attractor is equal to 0. Thefact that information dimension D equals(the latter corresponding to the line but not tothe plane) indicates this circumstance. Sincethere is no positive exponent in the LCEspectrum, the Lyapunov dimension of SNAequals 1. Despite the integer metric dimension,SNA demonstrates as a rule a self-similarity ofthe structure and, as a consequence, the prop-erties of scaling. All the properties indicatedabove allow us to speak about the "strange"geometry of SNA.

2. The LCE spectrum of the strange nonchaoticattractor. The system dynamics in the SNAregime is not chaotic as there is no intermixing.In average there is no exponential instability oftrajectories on the attractor. The LCE spectrumdoes not contain a positive exponent. The LCE

spectrum signature of phase trajectories onSNA does not differ from the correspondingone of a quasiperiodic motion. However, SNAcannot be considered as a quasiperiodic attrac-tor because, in particular, the local (calculatedon an finite time interval) largest LCE spectrumexponent of a trajectory on SNA will bepositive. Particularly, it has been proved thatthe probability that the largest local Lyapunovexponent will be positive is not equal to zero.

Spectrum and autocorrelationfunction. As thereis no intermixing in the regime of SNA, thepower spectrum does not contain in a rigoroussense the continuous component. At the same

time, the spectrum of a trajectory on SNA isnot discrete! The spectrum of SNA that isintermediate between discrete and continu-ous cases has a specific name: a singular con-tinuous spectrum. A feature of the singularcontinuous spectrum is that it includes a denseset of 6-peaks of the self-similar structure andhas the properties of fractals.

Since the spectrum of SNA is not continuous,the autocorrelation function (-) does not tendto zero as -. For the trajectories on SNA,(-) decreases to some limit nonzero level. More-over, (-) will demonstrate the scale-invariantproperties in the same way as spectrum.

As an example, Fig. 16 represents the spectrumofSNA in system (12) calculated for the coordinatex(n) of the attractor shown in Fig. 15(a) (Pikovskyand Feudel, 1995). As seen from the graph, thespectrum is really an everywhere-dense set of6-peaks and does not contain a pronouncedcontinuous component. Looking at the shape ofthe spectrum function [SN it is difficult to get surethat the distribution of spectrum componentsobeys the scale-invariant properties. For thispurpose let us consider the autocorrelation func-tion (-) for the attractor in Fig. 15(a) which isshown in Fig. 17 (Pikovsky and Feudel, 1994).

If we compare the dependencies (-) the timeintervals -1000 _< - < 1000 and 1584 < - _< 3584,then we can conclude about the complete

IRREGULAR ATTRACTORS 69

4.0

0.0

-2.0

-,4.00.0

a)

0.2 0.4 0.6 0.8 d)

0.0 0.2 0.4 0.6 0.8

FIGURE 15 Phase diagrams of strange nonchaotic attractors in the map (12) for = 1.5 (a) and in the map (13) for s--0.1,oz 3.277, co 0.5(x/ 1) (b).

70 V.S. ANISHCHENKO AND G.I. STRELKOVA

0 4000 8000 N

FIGURE 16 Singular-continuous spectrum of SNA in themap (12) for A-- 1.5.

difficult and nonstandard task and needs precisecalculations to be carried out using a good modernequipment. Otherwise, it is impossible to distin-

guish the SNA regime and a quasiperiodic regimewith a large number of combinative frequencies inthe spectrum.

6 CONCLUSIONS

The analysis of the structure and properties ofattractors of nonlinear dissipative systems as theimages of nonperiodic self-oscillations presented inthis paper allows us to make the followingconclusions:

0.5

0.0

-0.5

-500 0 500

-1.0

1584 2084 2584 3084

FIGURE 17 Self-similarity of the autocorrelation functionof SNA in the system (12).

correspondence of the autocorrelation functionstructure. The plot for (7-) presented in Fig. 17(a)is fully reproduced in Fig. 17(b). This is a con-

sequence of the property of scale invariance. Theenvelop 1(7-)1 in the SNA regime is a decreasingfunction that tends to some nonzero limit as

7---+00.

It is important to note that the diagnostics of theSNA regime in numeric simulations is a very

1. Robust hyperbolic systems and Lorenz typesystems demonstrate classical properties ofdeterministic chaos as nonperiodic exponen-tially unstable solutions of the correspondingdynamical systems. Strange (or practicallystrange) attractors are their mathematicalimages. Their distinctive feature is the fractalityof their geometrical structure, the fractal metricdimension and the presence of at least one

positive exponent in the LCE spectrum that is aconsequence of the intermixing. Robust hyper-bolic attractors and Lorenz type attractors areless sensitive to the influence of noise. Thebasins or attraction of such attractors are

smooth and homogeneous. The attractor’sproperties are not sensitive to the variation ofinitial conditions.

2. Quasiattractors which include a finite or infiniteset of regular and chaotic attracting subsets co-

existing for the fixed values of system param-eters are more complicated objects. Variation ofsystem parameters can lead to bifurcations ofthese subsets, whose number may be infinitewhile there is a finite variation of parameters.The basins of attraction of the co-existing attrac-tors have a fractal geometry. As a result, quasi-attractors demonstrate high sensitivity to thechanges in initial conditions and the influence ofnoise.

IRREGULAR ATTRACTORS 71

3. Exponential instability of individual trajectoriesand "strange" geometry of an attractor cannotbe connected uniquely. There exist the regimesof chaotic (unstable) self-oscillations to whichregular, in geometrical sense, attractors corre-

spond. These are the so-called chaotic non-

strange attractors. On the other hand, it ispossible to observe nonperiodic stable, accord-ing to Lyapunov, oscillations whose corre-

sponding attractor is a strange geometricalobject. Here, we deal with the strange noncha-otic attractors.

The ideas and concepts presented in this papercannot be considered as absolutely noncontradic-tory and generally accepted. A number ofproblemsdescribed here is up to the present moment a

subject of detailed studies and scientific discus-sions, the latter proving a fundamental significanceof the subject under investigation.

Acknowledgements

The authors express their sincere acknowledge-ments to Prof. V.N. Belykh, Prof. V. Afraimovichand Dr. T. Vadivasova for the numerous fruitfuldiscussions on a number of mathematical prob-lems. We are also grateful to our colleaguesDr. I. Khovanov and Dr. N. Janson for their helpin carrying out some experiments and preparingthe manuscript for publication.

This work was partially supported by the grantof the Russian State Committee of High EducationN 95-0-8.3-66 and by the common research projectof DFG and RFBR N 436 RUS 113/334.

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