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International Journal of Bifurcation and Chaos, Vol. 16, No. 3 (2006) 497–522c© World Scientific Publishing Company
COMPETITIVE MODES AND THEIR APPLICATION
WEIGUANG YAO, PEI YU∗, CHRISTOPHER ESSEX andMATT DAVISON
Department of Applied Mathematics, The University of Western Ontario,London, Ontario, Canada N6A 5B7
Received June 23, 2003; Revised March 15, 2005
We investigate nonlinear dynamical systems from the mode competition point of view, and pro-pose the necessary conditions for a system to be chaotic. We conjecture that a chaotic systemhas at least two competitive modes (CM’s). For a general nonlinear dynamical system, we give asimple, dynamically motivated definition of mode suitable for this concept. Since for most chaoticsystems it is difficult to obtain the form of a CM, we focus on the competition between the cor-responding modulated frequency components of the CM’s. Some direct applications result fromthe explicit form of the frequency functions. One application is to estimate parameter regimeswhich may lead to chaos. It is shown that chaos may be found by analyzing the frequency func-tion of the CM’s without applying a numerical integration scheme. Another application is to cre-ate new chaotic systems using custom-designed CM’s. Several new chaotic systems are reported.
Keywords : Mode competition; competitive mode; mode analysis; “smooth” Chua’s system; newchaotic system.
1. Introduction
Chaos has been extensively studied in the pastthree decades and many significant results havebeen obtained. However, there are many open ques-tions. Even a general rigorous definition of chaoshas not been universally accepted, even if its broadproperties are uncontroversial. For example, chaoticmotion is bounded and sensitive to initial con-ditions. The fractal nature of attractors is alsoaccepted even if it is not essential. However, someauthors (e.g. [Wiggins, 1990]) do not regard a pos-itive Lyapunov exponent as necessary, even if suffi-cient for sensitivity to initial conditions.
Nevertheless, the well-accepted characteristicsof chaos are good enough to be used to determinewhether a motion is chaotic or not, but are not help-ful in predicting the location in parameter spaceof a chaotic regime for a given nonlinear system.
Often, to find chaos for a given system, we must firststudy the stability of the fixed points of the system.Then we identify typical behaviors in known routesleading to chaos [Weiss & Vilaseca, 1991] by apply-ing a numerical approach and varying one or moreparameters to move away from the stable regimes ofthe fixed points. In the context of numerical inves-tigation, the largest Lyapunov exponent and thePoincare map are used to confirm the occurrenceof chaos.
However, the numerical approach is often sotime consuming as to border on hopeless, espe-cially when the system has many parameters. Forexample, in the 1980 Texas experiment, chaos wasdefinitely confirmed in the Belousov–Zhabotinskiireaction (B–Z reaction) [Turner et al., 1981], whichwas first reported in 1958. In order to give a the-oretical explanation of the phenomenon, a popular
∗Author for correspondence.
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model, called the Oregonator, was derived by Fieldand Noyes [1974]. This model involves only threevariables, but has four adjustable parameters. Tofind the chaotic parameter regimes of the model,much numerical work was carried out, but failed[Richetti et al., 1987]. This presented a dilemma:Does such a simplified model really fail to repro-duce the chaos of the original system or should wecontinue to search the parameter space because it isyet to be found [Richetti et al., 1987]? If we cannotfind chaos with such methods in a specific knownsystem, it is thus no trivial matter to find parameterregimes where chaos exists in general dynamical sys-tems. It becomes even more problematic to designdynamical systems that have chaotic regimes, letalone find them.
Creation of complicated dynamical systemshas practical value in studies of chaos-based com-munication [Pecora & Carroll, 1990; Cuomo &Oppenheim, 1993; Short, 1994; Pecora et al., 1997;Yao et al., 2002a; Yao et al., 2004]. These stud-ies suggest that the security of such communicationrelies on the randomness (complexity) of the chaoticcarrier generated by a chaotic system. Althoughone may use a number of coupled nonlinear sys-tems, such as spatially extended systems [Sundar& Minai, 2000; Wei & Zhao, 2002], partial differ-ential equations [Kocarev et al., 1997] to increasethe randomness (complexity) of the carrier, sincesuch systems may have more than one positiveLyapunov exponent and the fractal dimension oftheir attractors may be large. Nevertheless, lowdimensional but highly chaotic systems are stilldesired not only for improving the security of thecommunication but for understanding the nature ofcomplexity and chaos.
This paper introduces a new strategy in identi-fying chaotic regimes which we have found useful increating new chaotic systems. Nicolis and Prigogine[1977] speculated that competition in species willresult in the variety and complexity of species.Haken, in his book Synergetics [Haken, 1983], fur-ther suggested that coexistence of multiple compet-ing modes may result in complex behavior such aschaos. According to Haken, the competitors neednot be living things. They can be any generalizedmodes that compete for limited sources, for exam-ple, the energy of the system. A fundamental dif-ference between Haken’s mode concept and the moretraditional idea of mode is that Haken’s mode deter-mines the form of the motion rather than describingit. To be precise, we are accustomed to expressing
the solution of some problem in terms of super-position of normal modes. Haken’s modes appearnot in the solution of a differential equation but inthe differential equation itself. As pointed out byHaken, the concept of mode has numerous advan-tages because it allows people to better understandcomplexity and chaos.
Can we give a mathematical definition for thismode concept so that we may meaningfully con-sider competition between them for our purpose?In answering this question, we need to take intoaccount the variety of definitions of “mode” thatexist in many fields such as physics, biology, ecologyand music. However, these “modes” are not suitableas “competing modes”.
In the next section, we will give a brief reviewof definitions of mode in several fields, and thenin Sec. 3 propose necessary conditions of a sys-tem to be chaotic from the mode competition pointof view. In order to use these conditions to studynonlinear dynamical systems, we propose a sim-ple definition of competitive mode (CM) in theremainder of Sec. 3. Because for most nonlinearsystems it is difficult to obtain the exact formof a CM (e.g. “mode shape”, frequently used inengineering literature), we consider the competi-tion between CM’s via their corresponding modu-lated frequency components. It is found that thereis a relationship between the competition of the fre-quency components and the dynamical behavior ofthe investigated system. The applications of the CMfor estimating chaotic parameter regimes of somedynamical systems, including the “smooth” Chua’ssystem are given in Secs. 4 and 5 respectively. InSec. 6, this concept is used for constructing newchaotic systems. Finally, results are discussed andconclusions are made in Sec. 7.
2. Modes
The word “mode” comes from modus, which is aLatin word for “measure” or “size”. In physics,a mode often denotes an oscillation of single fre-quency. A related usage is the term, normal mode,which refers to an oscillation in which all par-ticles move with the same frequency and phase.Recent studies consider a mode as the solution of amodel system. The model system captures the maindynamical behavior of the original system. Thus,the solution of the original system could be wellapproximated by the solution of the model system.For a complex dynamical system, there can be more
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Competitive Modes and Their Application 499
than one model system, and the solution of the orig-inal system may be obtained by combining the solu-tions of the model systems.
When one uses a model system to find theapproximate solution of the original system, themodel system must be at least approximately solv-able (integrable). Shaw and Pierre [1993] extendedthe concept of mode to some invariant manifold andused their normal mode to analyze nonlinear vibra-tions. Their modes usually stand for the motionin the invariant manifolds, and the solution of thesystem is a linear or nonlinear combination of themodes. Thus, the solution of the system is deter-mined by the property of the modes. When thebehavior of the model system is nonchaotic, thesolution of the original system is also nonchaotic,because the solution is linear or nonlinear com-bination of finite number of modes implying thatthe power spectrum of the solution is not broadband as required for chaos. On the other hand, ifthe model system is chaotic, their method [Shaw &Pierre, 1993] can also be used to study the behav-ior of the original chaotic system. However, in thiscase, the model system itself cannot be analyti-cally solved. Furthermore, their work mainly focuseson simplifying the analysis of higher- and infinite-dimensional systems, not on mode competition. Forthe latest development of their work, see [Shaw &Pierre, 1994] and [Pesheck et al., 2002].
Every field of science and engineering seems tohave its own definition of mode. For example, whena system is described by a partial differential equa-tion (PDE), in order to simplify the computationand analysis, one often reduces the PDE into someordinary differential equations (ODE’s). Modes areoften considered as the key to constructing a Fourierseries decomposition. To reflect the complexity ofthe PDE, more than one mode should be includedin the decomposition. As an example, let us con-sider the following PDE describing a thin plate ina flow [Yu et al., 2001]:
D∇4w + ρh∂2w
∂t2− ∂2w
∂x2
∂2φ
∂y2− ∂2w
∂y2
∂2φ
∂x2
+ 2∂2w
∂x∂y
∂2φ
∂x∂y− µ
∂w
∂t= F (x, y) cos Ω1t, (1)
∇4φ = Eh
[(∂2w
∂x∂y
)2
− ∂2w
∂x2
∂2w
∂y2
], (2)
where w is the displacement of a point in the cen-ter plane of the thin plate in the z direction, φ isthe stress function, ρ, h, µ,E are parameters, and
F (x, y) cos Ω1t is the transverse force of the flow onthe plate.
By considering the boundary conditions andthe first two modes,
m1(x, y) = sinπx
asin
3πy
b,
m2(x, y) = sin3πx
asin
πy
b,
(3)
one may construct an approximate solution of w inthe form of
w(x, y, t) = u1(t)m1(x, y) + u2(t)m2(x, y), (4)
where ui(t) (i = 1, 2) are, respectively, the ampli-tudes of the two modes [Yu et al., 2001].
Correspondingly, the transverse force can alsobe expressed approximately by
F (x, y) = F1m1(x, y) + F2m2(x, y), (5)
where F1 and F2 are parameters. By substitutingEqs. (4) and (5) into Eqs. (1) and (2), and thenusing a technique like the Galerkin method [Peshecket al., 2002] to determine the expression of u1(t) andu2(t), one finally obtains the following dimension-less equations:
x1 + εµx1 + (ω21 + 2εF1 cos Ω2t)x1
+ ε(α1x31 + α2x1x
22) = εF1 cos Ω1t, (6)
x2 + εµx2 + (ω22 + 2εF2 cos Ω2t)x2
+ ε(β1x32 + β2x2x
21) = εF2 cos Ω1t, (7)
where dot denotes differentiation with respect totime t, xi = (
√ab/h2)ui (i = 1, 2), ω1 and ω2 are
the two linear natural frequencies of the thin plate,while Ω1 and Ω2 are the frequencies of the externalforces. µ > 0 is the damping coefficient, and ε is asmall perturbation parameter. For a more detaileddescription of the equations and parameters, see [Yuet al., 2001].
Now, instead of the original system describedby Eqs. (1) and (2), Eqs. (6) and (7) are usedto determine the parameter regimes where chaosappears, which agrees with the result obtained fromthe original system.
We have reviewed some definitions of modes.However, these definitions either cannot be directlyused to study chaos and complexity, or have notbeen used to study the dynamical behavior fromthe mode competition point of view. For example,the modes given in Eq. (3) help separate the spa-tial mode from the temporal mode of the originalPDEs (1) and (2), but the dynamical behavior of theODEs (6) and (7) for the temporal mode still needs
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to be studied using other approaches such as normalforms and numerical simulation [Yu et al., 2001].Here, the problem is: if we want to use mode compe-tition to simplify the study of a system’s dynamicalbehavior, then what kind of modes can be used ascompetitive modes? In the next section, we proposethe necessary conditions for a system to be chaoticbased on mode competition, and an expression ofCM for nonlinear dynamical systems.
3. A Definition of CM
In the previous section, we did not emphasize thecompetition among modes. In the sense of Haken[1983], Nicolis and Prigogine [1977], modes may beconsidered to be competing with each other in someway, for example, to occupy the maximal resourceof a system. A fundamental problem in the study ofmodes is how to mathematically define such a com-petitive mode (CM). First of all, certainly we needto know what form a CM should have. For a sim-ple sinusoidal mode, all components of the mode,including the frequency, phase and amplitude, areconstants. The result of competition between suchmodes is straightforward, and therefore such a modecannot be expected to imply complex dynamicalphenomena such as chaos, since it is expected thatthe results of competition in a chaotic system arecomplicated. In fact, it has been observed that thepower spectrum of a chaotic system is broad band,which strongly suggests that frequencies associatedwith CM might be expected to be functions of evo-lution variables.
Secondly, since we attribute the behavior ofa dynamical system to the competition betweenmodes, a system with complicated behavior suchas chaos must have at least two CM’s in compe-tition. Lack of competition for example, when thesystem has at most one CM, or the CM’s nevercompete with another, does not provide appropriatecircumstances.
Finally, we need to establish a “competitionrule” between CM’s, which can be used to classifythe dynamical behavior of the system. Convenientlyif practical methods are sought, the simplest rule isthe comparison of the values of their correspond-ing components: the bigger dominates the smaller.As a result of competition, the system’s dynamicalbehavior is mainly determined by the characteristicof the bigger components. For an oscillating sys-tem such as a chaotic system, the components of amode include the frequency, phase and amplitude
of the oscillation. If there exist two modes, A andB, in the system, and if at time t the amplitudeof A is larger than that of B, the amplitude of thesystem at t is mainly dominated by the mode A.However, for the other components (frequency andphase) of a mode, it is hard to say whether they arebigger or not. However, from the dynamical pointof view, we are interested only in finding if theyare competitive. Therefore, for the sake of simplic-ity, we use the following as a simple “rule”: If thecorresponding components of CM’s are not equal atany time, they are considered not competitive andwe say one dominates the other; otherwise if theyare equal at some time, then they are consideredcompetitive at that time. For a dynamical system,the competition of modes should be tested after thetransient period ends.
Thus, we assume that the modes determine thedynamical behavior of a system. If one mode domi-nates all others, the system’s behavior is controlledby the dominant mode, which seizes the most effi-cient way of occupying the system’s resources. If,on the other hand, no mode dominates the othersfor all time, the competition amongst the modesresults in complicated behavior. It should be notedthat the classical definition of “mode shape” usedin engineering emphasizes the shape/solution of themotion, while in this paper by “mode competition”we emphasize the characteristics of the motion.
Based on the above discussions, before givingour definition of CM, we propose a conjecture forthe necessary conditions under which a system ischaotic.
Conjecture 1. The necessary conditions for adynamical system to be chaotic are given asfollows:
(i) there exist at least two modes in the system;(ii) at least two such modes are competitive via
their corresponding components; and(iii) at least one component of such a mode is the
function of evolution variables such as t.
Remark 1. The first condition is obvious sinceone mode can only generate one type of simplemotion. The second one requires the modes to befairly matched with each other. The last condi-tion excludes a system from being chaotic if all themodes are constants, such as in the linear case.
As discussed, there are many definitions ofmodes, but some of them cannot be used as CM’s.That is, if the system is chaotic, the competition of
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Competitive Modes and Their Application 501
its modes cannot display the complicated dynamicalbehavior. In the remainder of this section, we givea simple definition of mode which can be used asa CM.
Let us start by considering the linear system:
x + ω2x = A sin Ωt, (8)
where A sin Ωt is the external force (A = 0), and ωis a parameter not equal to Ω. In system (8) thereare two modes m1(t) and m2(t) which satisfy
m1 + ω2m1 = 0,m2 + Ω2m2 = 0.
(9)
The solution of the system can be expressed asx = c1m1 + c2m2, where c1 and c2 are constantsdetermined by the initial condition of system (8).Thus, the mode m1 corresponds to the frequencyω, and m2 to the frequency Ω. We may call m1 theinternal mode, and m2 the external mode. Clearly,the system’s dynamical behavior (periodic or diver-gent) is determined by these two modes.
Now we rewrite system (8), by introducing y =sinΩt, as
x + ω2x = Ay,
y + Ω2y = 0,(10)
which can be further put in the vector form:
x + gx = h, (11)
where in general g and h are functions of t. (For sys-tem (10) g is a constant vector.) As we have men-tioned, ω corresponds to the internal mode, and Ωto the external mode. For system (10), we restatethat ω is associated with the mode in the x directionand Ω with the mode in the y direction.
Now, we define our competitive mode. Considerthe general nonlinear autonomous system, given by
xi = fi(x1, x2, . . . , xn), i = 1, 2, . . . , n, (12)
where fi ∈ C1(R) and for any j ∈ [1, n], |∂fi/∂xj |is bounded.
To study the general system (12) using modes,one first constructs some modes mk(t), k ≤ n basedon some properties of the system such as the invari-ant manifold [Shaw & Pierre, 1993]. In general,these modes satisfy the following model equations.
M + AM + BM = 0, (13)
where M = [m1,m2, . . . ,mk]T , both A and B arek × k matrices which can be functions of M and t.
The solution of system (13) may be expressedas a linear or nonlinear combination of these modes.In the linear case, one has
x = CM, (14)
where x = [x1, x2, . . . , xn]T and C is an n × kcoefficient matrix, which can be determined usinga method such as the Galerkin approach [Peshecket al., 2002].
The main difficulties of the above mode proce-dure are to find the appropriate manifold, namely,to construct the mode equation (13) based on themanifold, and to obtain the modes by solving theequation, especially when the system is chaotic. Toovercome the difficulties, we construct our modeequation below.
Differentiating Eq. (12) with respect to t yields
xi =n∑
j=1
fj∂fi
∂xj
= −xigi(x1, x2, . . . , xi, . . . , xn)+ hi(x1, x2, . . . , xi−1, xi+1, . . . , xn), (15)
where gi and hi are functions of x, and gi isbounded. Comparing Eq. (15) to system (10) showsthat hi should not contain the variable xi.
It can be seen from Eq. (15) that the dynami-cal behavior of xi is determined by the functions gi
and hi, as well as the system’s initial condition. Fur-thermore, the dynamical behaviors of systems (12)and (15) are related. If the motion of system (15)is periodic, i.e. fj(t + T ) = fj(t), j ∈ [1, n] forsome T > 0, then system (12) must be periodictoo; suppose system (15) is chaotic, implying thatfj, j ∈ [1, n] must be chaotic, and thus system (12)is chaotic too. Hence, system (15) can be taken asa model system of system (12).
Definition 1. For system (12), the competitivemodes are defined to be the solution, xi, i ∈ [1, n],of the model system, (15).
Remark 2. The difference between systems (12) and(15) is obvious. First, the dimension of system (12)is n, but that of system (15) is 2n. Therefore, sec-ond, these two systems cannot be related to oneanother by a one-to-one map. Third, system (15) isconservative in the sense of ∇ ·F = 0, where F is thevector field of system (15), no matter whether sys-tem (12) is conservative or not. However, as arguedabove, when the model system is periodic or chaotic,its dynamical behavior is related to the original
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system. The model system (15) does not have anycorresponding stable equilibrium point because itis conservative. But here we are not interested inequilibrium points. We will present more discussionlater on the relationship between the model systemand the original system.
The above mode definition is simple but maynot be perfect. For some specific systems, one canhave a better mode definition. Even with such a sim-ple definition, we still, in general, cannot solve themode equations except when the gi’s and hi’s takevery simple forms. However, if we focus on the modecompetition via their functional frequency compo-nents gi’s, namely, gi’s are not constants but vari-ables (i.e. explicitly or implicitly functions of time),we shall show that the dynamical behavior of thesystem is relative to the competition. In this sensewith a comparison to system (9), we have
Definition 2. For system (15), if gi > 0 at some t,we call gi the frequency component of the competi-tive mode xi. If gi ≤ 0 at any time, we shall simplystate that gi does not exist.
Remark 3. When the model system is nonlinearlycoupled, the components of CM’s affect each other.It then makes sense to focus on gi’s and the abovedefinition is reasonable.
Now, we conjecture necessary conditions forchaos based on the competition between CM’s viatheir frequency components, g’s.
Conjecture 2. The conditions for a dynamical sys-tem to be chaotic are given below:
(i) there exist at least two g’s in the system;(ii) at least two g’s are competitive or nearly
competitive, that is, there are gi gj > 0 atsome t;
(iii) at least one g is the function of evolution vari-ables such as t; and
(iv) at least one h is the function of the systemvariables.
Remark 4
(a) Under the conditions (i) and (ii), if there existgi(t) gj(t) > 0 at some time t = t0, the sub-model systems associated with xi and xj havethe same or approximately the same frequency,√
gi(t0). Further, according to condition (iv),if xi and xj are coupled via hi and/or hj , themodel system, (15), is in or close to resonanceat t0. (Condition (iv) is for coupling system
variables so that resonance could exist. If themodel system is built upon some manifold, con-dition (iv) may not be necessary.)
(b) Since the model system is conservative, theeigenvalues of the Jacobian at any pointon the orbit are in the form of ±λi, withRe(λi) ≥ 0, where i = 1, 2, . . . , n/2 and nthe dimension of the model system. Condition(ii) implies that the orbit repeatedly returns toor approaches the resonant regime. Thus whenresonance occurs at least at one of Re(λi) > 0,resulting in the existence of an unstable man-ifold, while the eigenvalue, −λi leads to a sta-ble manifold. We have shown that at resonance,determined from the equation gi = gj > 0,there exist saddles and the orbits passing thesesaddles are homoclinic-saddle orbits.
(c) A chaotic orbit has at least one saddle becauseotherwise the largest Lyapunov exponent of theorbit cannot be positive. Because of continu-ity, there are infinite many saddles in chaoticorbits, and usually this also holds for a periodicorbit. However, saddles are usually invisiblebecause they are easily destroyed by perturba-tions [Haller, 1999]. By using condition (ii), onecan determine the special saddles existing in theresonant regime. They make a greater contri-bution to the sensitivity of the initial conditionthan the other saddles in the orbits because,besides their positive eigenvalues, the resonancealso greatly helps increase the sensitivity.
(d) Condition (iii) is necessary for chaos becauseotherwise the model system is always in reso-nance under conditions (i) and (ii). The originalsystem (12) is different from the model sys-tem but our numerical results obtained fromthe original system show that condition (iii)is indeed necessary. Particularly, condition (iii)guarantees the existence of perturbations nearthe resonant regime so that the resonant pointsare saddles.
(e) One should note the close relationship betweenthe model system and the original system. Asa conservative system, the model system can,in general, have three types of states: periodic,chaotic and divergent. When the original sys-tem is periodic, namely, f ’s in Eqs. (12) and(15) are periodic, then the model system canbe periodic or divergent depending upon howfj and ∂fi/∂xj (see Eq. (15)) interact, but notchaotic. Similarly, when the original system ischaotic, the model system can be chaotic or
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Competitive Modes and Their Application 503
divergent but not periodic. On the other hand,based on the model system one can constructinfinite first-order dissipative systems. In otherwords, given
xi = −xigi(x1, x2, . . . , xi, . . . , xn)+ hi(x1, x2, . . . , xi−1, xi+1, . . . , xn), (16)
one can have infinite solutions of xi,k, k =1, 2, 3, . . . . But only one of them, say xi,1 = fi,satisfies
−xigi(x1, x2, . . . , xi, . . . , xn)+ hi(x1, x2, . . . , xi−1, xi+1, . . . , xn)
=n∑
j=1
fj∂fi
∂xj, (17)
where fi’s uniquely determine the first-orderdifferential equations, the original system. Fur-ther, all xi,k, k = 1, 2, 3, . . . (including xi,1 = fi)share the same functions gi and hi. Therefore, itis safe to use g’s and h’s to study the dynamicalbehavior of the original system.
We have noticed that the proof for the aboveconditions is not an easy task and requires muchmore work. However, readers may find motivationfrom the resonant point of view. Chaos has beenfound near resonance in many physical, chemicaland biological systems [Haller, 1999]. Mode com-petition might be considered as a generalization ofresonance with at least two frequencies. Certainly,these frequencies should not be all constants. There-fore, naturally one can imagine that the system mayexhibit complex phenomena when these frequenciesare close or interactive, regardless of the effect fromother components (e.g. phase and amplitude).
In the following, we use the well-known Lorenzand Rossler systems to show by example howg’s determine the dynamical behavior of nonlinearsystems.
The Lorenz system [Lorenz, 1963] isdescribed by
x = −10x + 10y, (18)y = −y + αx − xz, (19)
z = −βz + xy, (20)
where α and β are parameters. The system hasthree fixed points, given by (x0, y0, z0) = (0, 0, 0)and (±√
β(α − 1), ±√β(α − 1), α − 1).
Differentiating Eq. (18) with respect to t andthen substituting Eqs. (18) and (19) into the
resulting equation yields
x = −10x + 10y= (100 + 10α − 10z)x − 110y, (21)
which indicates that
gx = −100 − 10α + 10z. (22)
Similarly, one can obtain gy and gz as
gy = −1 − 10α + 10z + x2, (23)gz = −β2 + x2. (24)
The competition may occur in three pairs ofg’s, namely, (gx, gy), (gx, gz) and (gy, gz). Obviously,gx < gy for any t. Thus, we see that the fre-quency component of the CM in the x directionis “dominated” by that in the y direction, and weneed only consider the competition in the othertwo pairs. It should be pointed out that hereafterby “dominated” or “dominating” we mean the twocorresponding modes are neither interactive (i.e.separated) nor very close to each other since, in gen-eral, a mode with larger g need not always “domi-nate” the one with smaller g.
Figures 1(a) and 1(b) show the functions gx, gy
and gz when the Lorenz system exhibits chaoticand period-1 motions, respectively (gx (< 0) is notdepicted in Fig. 1(b)). The values of gi displayedin Fig. 1 are obtained by first numerically inte-grating the Lorenz system under the initial condi-tion taken near the fixed point, for parameter valuesβ = 2.6667, and α = 25 and 24.5, respectively, toget x(t), y(t) and z(t), and then using x(t), y(t) andz(t) to calculate gx, gy and gz given in Eqs. (22)–(24). It is seen that (i) gx < gz in both cases, whichindicates that there is no competition between themodes in the x and z directions; (ii) gz > 0 in bothcases, while gy > 0 for some times in Fig. 1(a) andall time in Fig. 1(b), which means that CM’s existin the y and z directions; (iii) in the chaotic case(see Fig. 1(a)), gy and gz are predominant over eachother alternatively, while in the periodic case (seeFig. 1(b)), gz dominates gy so that only one g is ineffect. Therefore, we may conclude that the compe-tition between gy and gz results in chaotic motion(see Fig. 1(c)), while the lack of competition leadsto the simple dynamical behavior (see Fig. 1(d)).
The Rossler system [Rossler, 1976] is given by
x = −y − z,
y = x + 0.2y, (25)z = β + z(x − α),
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(a) (b)
(c) (d)
Fig. 1. The CM’s components, gy and gz, and phase portraits of the Lorenz system (18)–(20): (a) and (c) chaos when α = 25,β = 2.6667; (b) and (d) periodic motion when α = 24.5, β = 2.6667.
which has two fixed points (x0, y0, z0) =(0.5γ,−2.5γ, 2.5γ), γ = α ±
√α2 − 0.8β, where α
and β are parameters.From system (25), one obtains
gx = 1 + z,
gy = 0.96, (26)gz = −(x − α)2 + y + z.
The functions gx, gy and gz versus t are depicted inFigs. 2(a) and 2(b) when system (25) is in a chaoticstate and a period-1 state, respectively. It is seen
from Fig. 2(a) that gy is dominated by gx but com-petition exists in both pairs, (gx, gz) and (gy , gz),and the motion is chaotic (see the phase portraitgiven in Fig. 2(c)); while in Fig. 2(b) g’s are sepa-rated at any time for the period motion, as shownin Fig. 2(d), indicating no mode competition.
The Lorenz and Rossler systems are goodexamples for demonstrating how g’s interact inchaotic or periodic motions. Applications of theCM defined in this section to more compli-cated chaotic systems are given in the next twosections.
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Competitive Modes and Their Application 505
(a) (b)
(c) (d)
Fig. 2. The CM’s components, gx, gy and gz of the Rossler system (25) when β = 0.2 for (a) and (c) chaos when α = 0.5;(b) and (d) periodic motion when α = 5.7.
4. Application of CM: EstimatingChaotic Parameter Regimes
In [Yao et al., 2002b], we used our CM con-cept to find chaos of the system described byEqs. (6) and (7), where the form of g’s can beanalyzed without any numerical simulation. Byintroducing
y1 = x1, y2 = x1,
y3 = x2, y4 = x2,
y5 = cos(Ω1t), y6 = cos(Ω2t),
we can rewrite Eqs. (6) and (7) as
y1 = y2,
y2 = εF1y5 − εµy2 − (ω21 + 2εf1y6)y1
− ε(α1y31 + α2y1y
23),
y3 = y4,
y4 = εF2y5 − εµy4 − (ω22 + 2εf2y6)y3
− ε(β1y33 + β2y3y
21),
(27)
andy5 = −Ω2
1y5,
y6 = −Ω22y6.
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The g-functions are easily found to be:
gy1 = ω21 + ε(2f1y6 + α1y
21 + α2y
23),
gy2 = ω21 + ε(2f1y6 + 3α1y
21 + α2y
23) − ε2µ2,
gy3 = ω22 + ε(2f2y6 + β1y
23 + β2y
21),
gy4 = ω22 + ε(2f2y6 + 3β1y
23 + β2y
21) − ε2µ2,
gy5 = Ω21,
gy6 = Ω22.
(28)
When, as it should be, the term ε2µ2 is very small, agy1 or gy2 will dominate the other, depending uponthe sign of α1. Similarly gy3 or gy4 dominates theother depending upon the sign of β1. For gy5 andgy6 , only the larger of Ω1 and Ω2 need be con-sidered because the smaller will be dominated bythe larger. Without loss of generality, assume thatα1 > 0, β1 > 0, and Ω2 ≥ Ω1, then only thecompetition among gy2 , gy4 and gy6 need be stud-ied because if gyi < 0, then gyi−1 < 0 too, wherei = 2, 4, 6.
For instance, when ε ∼ O(1) and Ω2 = 0, butω1 or ω2 Ω2, then gy6 is out of competition.Only gy2 and gy4 need to be considered. Further,if ω1 = ω2, then the difference between f1 and f2
should be large so that gy2 and gy4 can predom-inate over each other alternatively. For example,when ε = 0.1, µ = 0.01, ω1 = ω2 = 1.0, f1 =0.9, f2 = 6.0, Ω1 = Ω2 = 0.1, α1 = α2 = 1.0, β1 =0.1, β2 = 1.0, F1 = 0.5, F2 = 0.8, the systemexhibits chaotic motions, as shown in Figs. 3(a)and 3(b). Figure 3(c) depicts the functions gy2 andgy4 .
For most nonlinear systems the relation-ship between g’s cannot be directly analyzed.But since g’s are given in analytic form, wemay roughly analyze their behavior from theaveraged form over the evolution variable twhere t is taken from t0 to ∞. We pro-pose the following heuristic: When there exist atleast two positive averaged g’s, the system mayexhibit chaotic motions. Otherwise, chaos is notexpected.
For example, consider the Lorenz system again.We may define 〈gy〉 = −1 − 10α + 10〈z〉 + 〈x2〉,where 〈· · ·〉 denotes an average of · · · over t from t0to ∞. For instance, 〈z〉 = limt→∞(1/t)
∫ t0 z(t′)dt′
is an average value — the mean. If we restrictattention to the local behavior of the system, thenx ≈ x0 + A cos(ω1t + φ1), z ≈ z0 + B cos(ω2t + φ2),where A,B, ω1, ω2, φ1 and φ2 are constants, and x0
and y0 represent the fixed points. Around the fixed
points, we have 〈z〉 ≈ z0, 〈x2〉 x20. Therefore,
〈gy〉 αβ − β − 11, (29)
when the nontrivial fixed points are considered. Asimilar discussion gives
〈gz〉 β(α − 1 − β). (30)
If both 〈gy〉 and 〈gz〉 are positive, chaotic motionmay exist. For example, when α = 25, β = 2.6667,both 〈gy〉 and 〈gz〉 are greater than zero, andthe system is truly chaotic. However, when α =24.5, β = 2.6667, both 〈gy〉 and 〈gz〉 are greaterthan zero, but the system is periodic. This indicatesthat the averaged g’s condition for predicting chaosis at most necessary. Even though the condition isnot sufficient, it is still very useful in estimatingchaotic parameter regimes easily and efficiently, inparticular for high dimensional systems involving alarge number of parameters. We demonstrate thisusing the following two examples.
4.1. Example 1
Our first example is a six-dimensional psychologi-cal model for stress and coping [Neufeld, 1999]. Thesix-dimensional equation is given by
y1 = a − by3y4 − cy1,
y2 = y5(a − by3y4 − cy1)(1 + y6)− ey2 − fy3y4y1 + g,
y3 = h − iy2,
y4 = y6(a − by3y4 − cy1) − jy4
+ k(fy3y4y1 − g) + d,
y5 = 1 − y5(y1 + y4) − y5,
y6 = 1 − y6y1 − y6,
(31)
where a, b, c, d, e, f, g, h, i, j, k are non-negativeparameters, see [Neufeld, 1999] and [Levy et al.,2003] for the meaning of yi, i = 1, . . . , 6. The vari-ables y2 and y4, which describe the stress arousallevel and coping activity, are most important. Thefixed points of system (31) are(
1 − r
r,
h
i,
ij(−c + ar + cr)b(id − hek)r
,
id − hek
ij,
ijr
idr − hekr + ij, r
),
where
r =afi + 2cfi ±
√a2f2i2 + 4bcfieh − 4bcfgi2
2(afi + cfi − beh + bgi).
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Competitive Modes and Their Application 507
(a) (b)
(c)
Fig. 3. The phase portraits and function g’s of system (6)–(7). (a) x1 versus x1; (b) x2 versus x2; (c) gy2 and gy4 versus t.
Numerical work has shown that it is difficult tofind chaotic and even periodic motions of the sys-tem (31). It is also very complicated to analyticallydetermine the parameter boundary where the fixedpoints lose stability.
Let us apply our CM approach to system (31)and find the following six g’s:
gy1 = bkfy23y4 − bcy3y6 − c2,
gy2 = −biy4y5(1 + y6) − e2 − fiy1y4,
gy3 = −biy4y5(1 + y6) − fiy1y4,
gy4 = by6y3 + by3y6 − kfy3y1 − kfy1y3]− bcy3y6 − (j + by3y6 − kfy1y3)2,
gy5 = y1 + y4 − (1 + y1 + y4)2,gy6 = y1 − (1 + y1)2.
(32)
Note that gy2 < gy3 . To estimate the chaotic param-eter regimes, we further consider the averaged g’s. Itis always true that for a bounded variable x(t), theaverage of x over t → ∞ is 0 (otherwise, x(t) willdiverge). Using 〈y1〉 = 〈y4〉 = 0, we have 〈gy5〉 < 0and 〈gy6〉 < 0. Therefore, we may only need toconsider 〈gy1〉, 〈gy2〉, 〈gy3〉 and 〈gy4〉 expressedby the parameters. We use a simple program tosearch the values of the parameters for which at
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least two of 〈gy1〉, 〈gy2〉, 〈gy3〉 and 〈gy4〉 are greaterthan zero. Executing the program gives the resultsalmost instantaneously because the program doesnot need to numerically integrate system (31). Theoutput contains many groups values of parame-ters, for instance, the group (a, b, c, d, e, f, g, h, i,j, k) = (0.2, 0.1, 0.1, 0.1, 0.1, 0.2, 0.1, 2.1, 1.1, 2.4,4.93). Figure 4 shows some phase portraits of thesystem under the group of parameters. Figure 5shows the functions gy3 , gy4 and gy5 versus t (gy2
is close to gy3). While gy3 is predominant most ofthe time, gy4 and either gy5 dominate the others in
some intervals of t. gy1 and gy6 are not shown inFig. 5 because they are always less than zero.
We have also applied the CM approach tostudy a number of chaotic systems such as theRossler hyper-chaotic system [Rossler, 1991] andthe disk dynamo system [Hardy & Steeb, 1999].The results are very encouraging: Only the param-eter regimes where the nonlinear system has atleast two positive averaged g’s may exhibit chaoticbehavior. This suggests that one may, instead of g’s,use the averaged g’s to estimate chaotic parameterregimes.
(a) (b)
Fig. 4. The phase portraits of system (31): (a) y1 versus y3; (b) y2 versus y6.
(a) (b)
Fig. 5. (a) The function gy3 , gy4 and gy5 of system (31) in chaos; and (b) a piece of data from Part (a).
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Competitive Modes and Their Application 509
4.2. Example 2
The next example is the Oregonator model [Field& Noyes, 1974]:
α = s(η − ηα + α − qα2), (33)η = s−1(−η − ηα + fρ), (34)
ρ = w(α − ρ), (35)
where α ∝ [HBrO2], η ∝ [B−r ], ρ ∝ [Ce(IV )],
and s,w, q and f are parameters. All variables andparameters are non-negative. A group of frequentlyused parameters is s = 77.27, w = 0.1610, q =8.375×10−6, f = 1.0. Under these values of param-eters, the system’s fixed point is α0 = ρ0 =488.68, η0 = 0.99796. Because the scales of theseparameters are very different, it is extremely diffi-cult to find chaos via numerical simulation. Here weconsider the general case.
From Eqs. (33)–(35), we can find
gα = fρ − αη − s2(2q2α2 + 3qαη − 3qα+ η2 − 2η − 2qη + 1), (36)
gη =η − ηα + α − qα2 − (1 + α)2
s2, (37)
gρ = −w2. (38)
The mode associated with the direction ρ does notexist because gρ is strictly negative. Thus, we onlyneed to consider those associated with the direc-tions α and η. Because all the variables and param-eters are non-negative, in order to have gη greaterthan 0, Eq. (37) suggests that q be small and s belarge. This conclusion agrees with the selection ofparameter values often used in the literature. How-ever, such a choice may cause gα < 0.
We further consider the averaged gη,
〈gη〉 =⟨
η − ηα + α − qα2 − (1 + α)2
s2
⟩. (39)
It follows from Eq. (33) that
〈α〉 = s〈η − ηα + α − qα2〉 = 0. (40)
Substituting Eq. (40) into Eq. (39) yields
〈gη〉 = −⟨
(1 + α)2
s2
⟩< 0. (41)
Because both 〈gη〉 and 〈gρ〉 are less than zero, thesystem has at most one positive averaged g function,〈gα〉. Based on this analysis, we conclude that chaosin the Oregonator model is rare though we cannotabsolutely exclude the existence of chaos from this
model. This agrees with the results from numericalexperiment [Field & Noyes, 1974].
5. Application of CM: The “Smooth”Chua’s Systems
Chua’s famous circuit [Matsumoto, 1984; Chuaet al., 1986] is the first real electronic system usedto exhibit chaos. Unlike the Lorenz and Rossler sys-tems which have nonlinear coupled terms, Chua’ssystem is piecewise-linear. This means that our CMapproach cannot be applied because the piecewise-linear term is not differentiable at the turningpoints. Our CM approach can however be applied tothe recently proposed “smooth” Chua’s system. See[Tsuneda, 2005] for the details of how the originalChua’s system is transformed to the smooth Chua’ssystem, which is described by
x = k1y + k2x + k3x3,
y = k4(x − y + z), (42)z = k5y + k6z,
where ki, i = 1, . . . , 6 are parameters. We notethat the system has only one nonlinear term, k3x
3,and the system equations are not nonlinearly butlinearly coupled, even though simple Chua’s sys-tem still displays very interesting and complicateddynamical behavior, as well as many chaotic attrac-tors, even when the nonlinear term is very weak,namely, |k3/k1| 1, |k3/k2| 1 [Tsuneda, 2005].
Performing the CM on the system, we obtain
gx = −[k1k4 + k22 + k2k3x
2
+ 3k3x(k1y + k2x + k3x3)],
gy = −k4(k1 + k4 + k5),gz = −(k4k5 + k2
6).
(43)
Since gy and gz are constant, they are not compet-itive. For the system to be chaotic, gx must existand compete with at least one of gy and gz. Sincethe last term of gx can be rewritten as 3k3xx, thisterm oscillates around zero no matter what val-ues of k3 are chosen. Thus, we may interpret thebehavior of gx as an oscillation around the value of−(k1k4 +k2
2 +k2k3x2). If −(k1k4 +k2
2 +k2k3x2) > 0,
gx may exist, and if −(k1k4 + k22 + k2k3x
2) is closeto gy or gz, chaos may appear. Based on the formsof gx, gy and gz, we have the following results.
Case 1. If gx and gy are competitive, then gx gy > 0.
(1a) If k4 < 0 and k1 < 0, then k5 > −k1 − k4 > 0and k2k3 < 0.
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(1b) If k4 < 0 and k1 > 0, then k5 > −k1 − k4, andfurther, if k1k4 + k2
2 > 0, we have k2k3 < 0.(1c) If k4 > 0 and k1 < 0, then k5 < −k1 − k4, and
further, if k1k4 + k22 > 0, we have k2k3 < 0.
(1d) If k4 > 0 and k1 > 0, then k5 < −k1 − k4 < 0,and k2k3 < 0.
Case 2. If gx and gz are competitive, then gx gz > 0, and k4k5 < −k2
6 < 0.
(2a) If k4 < 0 and k1 < 0, then k5 > 0 andk2k3 < 0.
(2b) If k4 > 0 and k1 > 0, then k5 < 0 andk2k3 < 0.
The smooth Chua’s system has been exten-sively studied by Tsuneda [2005] using numeri-cal simulations. Twenty three parameter regions,denoted by C-1 ∼ C-20, and C-7’, C-13’, C-17’, forwhich the system is chaotic or periodic, are iden-tified in [Tsuneda, 2005]. For convenience, Table 2in [Tsuneda, 2005] is copied here as Table 1. Com-paring these 23 regions with the above CM analy-sis, we have found that Case (1a) contains C-4, 8,12, 13, 15, 16, 13’, Case (1b) contains C-18, Case(1c) contains C-3, 5, 9, 10, 14, 20, and Case (1d)contains C-1, 2, 6, 7, 11, 17, 19, 7’, 17’. We have
also noticed that Case (2a) contains C-4, 8, 12, 13,15, 16, 13’, and Case (2b) contains C-1, 2, 6, 7, 11,17, 19, 7’, 17’. Thus, from the CM point of view, inthese 23 parameter regions investigated by Tsuneda[2005] the smooth Chua’s system could exhibit com-plicated dynamical behavior such as chaos. In otherwords, the regions classified by CM agree with thoseidentified by Tsuneda [2005].
Based on the work of Tsuneda [2005], in some ofthese 23 parameter regions the smooth Chua’s sys-tem is not chaotic. In order to filter out some non-chaotic parameter regions and to find new chaoticattractors in new parameter regions, we furthertransform the linear coupled smooth Chua’s sys-tem into a nonlinear coupled smooth Chua’s sys-tem, under which our CM approach may be moreuseful from the resonance viewpoint. From the firstequation of system (42), we have
y =1k2
(x − k2x − k3x3). (44)
(k1 = 0, otherwise there is no chaos.) Substitut-ing Eq. (44) into the other two equations of sys-tem (42), and denoting by x1 = x, x2 = x, we obtainthe following nonlinear coupled smooth Chua’s
Table 1. Parameter values for the smooth Chua’s system (42).
No. k1 k2 k3 k4 k5 k6
C-1 9.3515908493 1.6682747877 −0.6973867753 1.0 −14.7031980540 −0.0160739649C-2 3.7091002664 7.1642927876 −1.6802567087 1.0 −24.0799705758 0.8592556780C-3 −6.6919100000 −0.4602892985 0.1835914401 1.0 1.52061 0.0C-4 −143.1037 14.6570625639 −2.2369405036 −1.0 207.34198 −3.8767721000C-5 −1.301814 1.4311608333 −0.0695653464 1.0 0.0136073 0.0296996800
C-6 8.4562218418 −1.3714231681 0.1384203393 1.0 −12.0732335925 −0.0051631393C-7 6.5792294673 1.5321955041 −0.6247272525 1.0 −10.8976626192 0.0447440294C-8 −4.006 0.6347865975 −0.1504582008 −1.0 54.459671 −0.93435708C-9 −4.08685 −0.3829118599 0.1411663497 1.0 2.0 0.0C-10 −75.0 −50.9876254261 1.3092138284 1.0 −31.25 3.125
C-11 15.6 2.6072521221 −1.0283200040 1.0 −28.58 0.0C-12 −37.195804 3.8096934345 −0.5814301135 −1.0 73.049688 −1.161224C-13 −13.070921 2.0712045602 −0.4909204336 −1.0 53.612186 −0.7508709600C-14 −45012.877058 21.9725901325 −5.0576444486 1.0 14125.7874586260 0.2326833338C-15 −3.505 0.5553986581 −0.1316415362 −1.0 66.672752 −0.9477989200
C-16 −12.141414 1.9239158468 −0.4560098118 −1.0 95.721132 −0.8982235C-17 1800.0 51.2859375 −12.1816406250 1.0 −10000.0 0.0C-18 1.7327033212 −1.5599365845 0.0337768366 −1.0 0.0421159445 0.2973436607C-19 62.3168864230 0.5524169777 −0.0207591313 1.0 −94.7189263 0.6013155140C-20 −1.424557325 1.3377628122 −0.0281432267 1.0 −0.02944201 −0.3226735790
C-7’ 6.62 1.5416902979 −0.6285985969 1.0 −10.8976626192 0.0447440294C-13’ −19.5 3.2779542798 −0.8536807567 −1.0 53.612186 −0.7508709600C-17’ 2300.0 69.5277041101 −18.1433331473 1.0 −10000.0 0.0
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system:
x1 = x2,
x2 = (k2 − k4 + 3k3x21)x2
+ k4(k1 + k2 + k3x21)x1 + k1k4z,
z =k5
k2(x2 − k2x1 − k3x
31) + k6z.
(45)
Then the g’s based on system (45) are found to be
gx1 = −k4(k1 + k2 + k3x21) − 3k3x1x2,
gx2 = −k4(k1 + k2 + k5 + 3k3x21)
− 6k3x1x2 − (k2 − k4 + 3k3x21)
2,
gz = −(k4k5 + k26),
(46)
in which only gz is constant. Again, we can interpretgx1 to oscillate around −k4(k1 + k2 + k3x
21) and gx2
oscillates around −k4(k1 + k2 + k5 + 3k3x21)− (k2 −
k4 +3k3x21)
2 < −k4(k1 +k2 +k5 +3k3x21). Then, for
system (45), we have the following results.
Case 1. If gx1 and gx2 are competitive, thenk4(k1 +k2 +k5 +3k3x
21) < k4(k1 +k2 +k3x
21) < 0, so
k4k5 < 0. Comparing with the 23 parameter regionsidentified by Tsuneda [2005], we have found that allthe regions, except C-3, 5, 9, 14, are contained inthis case.
Case 2. If gx1 and gz are competitive, then k4(k1+k2 + k3x
21) k4k5 + k2
6 < 0. Hence, we havek4(k1 + k2 − k5 + k3x
21) k2
6 > 0, and
(2a) if k4 > 0 and k1 + k2 − k5 < 0, then k3 > 0,k5 < 0, k1 + k2 < 0;
(2b) if k4 < 0 and k1 + k2 − k5 > 0, then k3 < 0,k5 > 0, k1 + k2 > 0.
Thus, it is found that C-9 and C-10 are contained inCase (2a), and none of these 23 regions is containedin Case (2b).
Case 3. If gx2 and gz are competitive, then k4(k1+k2 + k5 + 3k3x
21) k4k5 + k2
6 < 0. Hence, we havek4(k1 + k2 + 3k3x
21) k2
6 > 0, and
(3a) if k4 > 0 and k1 + k2 < 0, then k3 > 0 andk5 < 0;
(3b) if k4 < 0 and k1 + k2 > 0, then k3 < 0 andk5 > 0.
Thus, C-10 is contained in Case (3a) and none inCase (3b).
As stated in Sec. 4, one of the advantages of our CMis to find chaos. From the above analysis, we haveseen that if the parameter values satisfy Case (2b)or Case (3b), chaos may appear in the smoothChua’s system, however these cases were not inves-tigated in [Tsuneda, 2005]. For Case (2b), we founda chaotic attractor when (k1, k2, k3, k4, k5, k6) =(−2, 6.5,−0.15,−1, 4,−1) and the initial conditionis taken near the origin. (In this section, all theinitial conditions are chosen near the origin asdone in [Tsuneda, 2005].) The attractor and cor-responding g’s are displayed in Figs. 6(a) and 6(c),respectively. The attractor obtained from the lin-ear coupled smooth Chua’s system (42), with thesame parameter values and initial condition, isdepicted in Fig. 6(b) for the sake of comparisonwith the attractors in [Tsuneda, 2005]. For Case(3b), another chaotic attractor was found when(k1, k2, k3, k4, k5, k6) = (−2, 3.7,−0.15,−1, 12,−1),as shown in Figs. 7(a) and 7(b), and the
(a) (b)
Fig. 6. For the smooth Chua’s system (42), case (2b) when k(k1, k2, k3, k4, k5, k6) = (−2, 3.7,−0.15,−1, 12,−1): (a) thestrange attractor of system (45); (b) the strange attractor of system (42); and (c) the corresponding g’s versus t to theattractor given in Part (a).
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(c)
Fig. 6. (Continued )
(a) (b)
(c)
Fig. 7. For the smooth Chua’s system (42), case (3b) when k(k1, k2, k3, k4, k5, k6) = (−2, 3.7,−0.15,−1, 12,−1): (a) thestrange attractor of system (45); (b) the strange attractor of system (42); and (c) the corresponding g’s versus t to theattractor given in Part (a).
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Competitive Modes and Their Application 513
(a) (b)
Fig. 8. Results for the smooth Chua’s system (42), case C-1: (a) the strange attractor; and (b) the corresponding g’s.
(a) (b)
Fig. 9. Results for the smooth Chua’s system (42), case C-6: (a) the strange attractor; and (b) the corresponding g’s.
corresponding g’s for Fig. 7(a) is given in Fig. 7(c).It is observed that all g’s are competitive in thesetwo cases and the attractors are somewhat compli-cated. Since these two attractors exist in differentcategories of parameter regions from those investi-gated by Tsuneda [2005] in the viewpoint of CM,the two sets may have some aspects different fromthe others.
We repeated numerical simulations for allthe 23 cases listed in [Tsuneda, 2005] and checkedthe competition of g’s in Eq. (46) obtained from the
nonlinear coupled system (45). We have found thatfor C-6 gx2 and gz are nearly competitive, and forall other chaotic cases, there are at least two g’s incompetition. Figures 8 and 9 display the attractorsand corresponding g’s for C-1 and C-6, respectively.Since for C-6 g’s do not cross although they are closeto each other (near resonance), chaos is not strong.We have also seen that for some periodic casesfound in [Tsuneda, 2005], the g’s are not in compe-tition. One of the examples is C-14, as depicted inFig. 10.
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Fig. 10. Results for the smooth Chua’s system (42), case C-14: (a) the strange attractor; and (b) the corresponding g’s.
6. Application of CM: ConstructingNew Chaotic Systems
In the previous sections, we have shown thatfunction g plays an important role in the studyof chaotic systems. We have estimated chaoticparameter regimes based on the necessary con-dition that a chaotic system should have atleast two CM’s. In this section, we will applythe same rule to actively create new chaoticsystems.
For constructing a chaotic system, we firstdesign n (≥ 2) g’s so that at least two of the g’s arecompetitive in some parameter regime. The n g’sare expressed in n second-order differential equa-tions similar to Eq. (15). These equations form asystem of at least 2n dimensions. If we are interestedin lower dimensional systems, or particularly, lowerdimensional first-order differential systems, we mayadd some constraints to the second-order differen-tial equations. In other words, we seek a subspaceof the 2n-dimensional space so that a lower dimen-sional system exists in the subspace. Note that theconstraints only help derive a lower dimensionalsystem from the higher one, but do not affect thedesigned g’s.
For example, to obtain a three-dimensionalfirst-order differential system, a general approachcan be described as follows.
First, we design the functions gx and gy inthe following equations to be competitive in someparameter regime so that the competition may
result in chaos.
x = −gx(x, y, z)x + h1(y, z), (47)y = −gy(x, y, z)y + h2(x, z). (48)
In general, Eqs. (47) and (48) construct a dynamicalsystem in R5 spanned by x, x, y, y, z because weare going to use z as a variable. In other words, weneed five independent initial values for, respectively,x, x, y, y and z to solve the system. In this space, weseek a three-dimensional subspace in which x, x, y, yand z are not independent but constrained, and thesystem may be chaotic.
Second, we design the constraint, given by
x = f(x, y, z). (49)
At this stage, the system, which consists ofEqs. (47)–(49), is not five- but, in general, four-dimensional because the state of x is determined byx, y and z. Also, the constraint has to be designed insuch a form that the competition between gx and gy
holds in some part of the parameter space inducedby these equations. Actually, by these two steps, wecan obtain y and z.
Next, differentiating Eq. (49) with respect to tresults in
x =∂f
∂xx +
∂f
∂yy +
∂f
∂zz
= f(x, y, z)∂f
∂x+
∂f
∂yy +
∂f
∂zz. (50)
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Then combining Eqs. (47) and (50) yields
∂f
∂yy +
∂f
∂zz = −gx(x, y, z)x + h1(y, z)
− f(x, y, z)∂f
∂x. (51)
Now, if ∂f/∂z = 0 and ∂f/∂y = 0, then we have
y =
[−gx(x, y, z)x + h1(y, z) − f(x, y, z)
∂f
∂x
]∂f
∂y
.
(52)
y is uniquely determined by x, y, z. In this case, thesystem which consists of Eqs. (48), (49) and (52) isthree-dimensional because we can solve the systemby using the initial conditions for x, y and z. Thatis, our new system no longer includes Eq. (47).
Similarly, differentiating Eq. (52) with respectto t and combining with Eq. (48), we obtain
F
(z, x, y, z, h1, h2, f,
∂2f
∂x∂y, . . .
)= 0. (53)
Finally, we may solve z from Eq. (53).Equations (49), (52) and (53) form the expected
system. From this system, we can obtain gx, gy andgz, in which gx and gy are the same as in Eqs. (47)and (48), respectively, while gz is new. Since we needonly two competing g’s, the form of gz will not resistthe appearance of chaos.
For chaos, the forms of gx and gy should bedesigned to be greater than 0 in some parameterregimes. The requirement on the forms of h1(y, z)and h2(x, z) by condition (iv) given in Conjecture 2is easily satisfied, and we choose the forms as simpleas possible. We present three examples in the fol-lowing, some of which have been used in improvingthe security of communications via chaos synchro-nization [Yao et al., 2002a].
6.1. Example 1
We design gx, gy and the corresponding second-order differential equations as follows.
x = −(α − y)x − z, (54)y = −(1 − β2)y + βx − z, (55)
where α and β are parameters. gx = α − y andgy = 1 − β2. When |β| < 1, gy exists. Then onecan adjust α so that gx will oscillate around gy
because gx contains y. The competition of these twog’s may result in chaos. The forms of h1 = −z andh2 = βx − z are very simple and relative to z fromwhich we can deduce z. Except for these require-ments, all of these forms are chosen arbitrarily.
To obtain a three-dimensional first-order sys-tem, we add the following constraint,
y = −βy − x, (56)
to Eqs. (54) and (55). Differentiating Eq. (56) withrespect to t and then using Eq. (55), we obtain
x = y + z. (57)
Then differentiating Eq. (57) with respect to t andthen using Eq. (54) gives
z = −z + (1 − α)x + βy + xy. (58)
Equations (56)–(58) form the required system.From this system, we have
gz = −2 + α − y, (59)
which is dominated by gx.To estimate the chaotic parameter regimes, we
further use the averaged g’s. The nontrivial fixedpoint of the system is (−1 − αβ, (α + 1)/β, (−α −1)/β) near which
〈gx〉 ≈ − 1β
,
〈gy〉 = 1 − β2, (60)
〈gz〉 ≈ − 1β− 2.
We note that α does not appear in 〈gx〉, 〈gy〉 and〈gz〉. This is because we use first-order approxima-tion of y to get 〈y〉. When β ∈ (−1, 0), there exist atleast two averaged g’s. Especially, for β ∈ (−0.5, 0),three averaged g’s exist. Because α is free, we canfirst choose β in the regime, then adjust α to findthe chaos, as shown in Fig. 11.
This example shows that it is easy to applyCM’s to construct a chaotic system which allowsmuch freedom to make parameter changes. Whenthe form of the function gi is complicated, the resultof competition among CM’s may become intricatetoo. Next, we show an example with more compli-cated forms of gi.
6.2. Example 2
In this example, we construct a four-dimensionalsystem. In this case, one may use four differen-tial equations, for example, two second-order dif-ferential equations which define two g’s and two
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Fig. 11. (a) The strange attractors of the system (56)–(58) when α = 8.3 and β = −0.48; and (b) the corresponding g’sversus t.
first-order differential equations. The two second-order differential equations contains two other freevariables. The dimension of the subsystem is thensix. However, the two first-order differential equa-tions constrain the second-order equations. There-fore, the whole system is four-dimensional. Thus, let
x = −(w + α − 1 − y2)x + β1y + γw − (1 + β2)z,
(61)y = −(w + x − γ − β2
1)y − (β1 − β2)(x − γ)w− zw − xz + γz, (62)
x = y + z, (63)y = β1y − xw + γw, (64)
where α, β1, β2 and γ are parameters, and x, y, zand w are variables. From Eqs. (61), (63) and (64),we find
z = (1 − α)x − (1 + β2)z + xy2. (65)
From Eqs. (62)–(64), we can similarly obtain
w = −β2w + y + z. (66)
Equations (63)–(66) form a possibly chaotic system.The function g’s of the system are
gx = w + α − 1 − y2,
gy = w + x − γ − β21 ,
gz = α − 1 − (1 + β2)2 − y2,
gw = x − β22 − γ.
(67)
One may adjust the parameters so that the g’s arecompetitive. Because the only fixed point of the sys-tem is (0, 0, 0, 0), one can easily get the averaged g’s,given by
〈gx〉 α − 1,〈gy〉 ≈ −β2
1 − γ,
〈gz〉 α − 1 − (1 + β2)2,〈gw〉 ≈ −β2
2 − γ.
(68)
Therefore, when α > 1 and γ < −max(β21 , β2
2),there exist at least two averaged g’s. The localstructure of the chaotic attractors of the systemis expected to be complicated. For example, whenα = 20.2, β1 = 0.322, β2 = 0.2 and γ = −1.4,the system is chaotic as shown in Fig. 12(a). Theattractor is more entangled than that of the sys-tem consisting of Eqs. (56)–(58) (see Fig. 13(a)).The corresponding gx, gy, gz and gw versus t aredisplayed in Fig. 12(b). It is seen that all g’s arein competition with each other. The competitionresults in the complicated structure of the chaoticattractor.
When β1 = β2, 〈gy〉 〈gw〉, then the com-petition between gy and gw may become stronger.The structure of the chaotic attractor is expectedto be more complicated than that when β1 = β2.Figures 13(a) and 13(b) show, respectively, thechaotic attractor and the functions gx, gy, gz andgw versus t when α = 38.2, β1 = β2 = 0.2 andγ = −0.6. It is observed that the local structure ofthe attractor is even more entangled than that inFig. 12(a), and that all g’s are in competition.
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(a) (b)
Fig. 12. (a) The chaotic attractor of the system (63)–(66) when α = 20.2, β1 = 0.322, β2 = 0.2 and γ = −1.4; and(b) gx, gy, gz and gw versus t in this case.
(a) (b)
Fig. 13. (a) The chaotic attractor of the system (63)–(66) when α = 38.2, β1 = β2 = 0.2 and γ = −0.6; and (b) gx, gy, gz
and gw versus t in this case.
6.3. Example 3
In the last example, we create a class of systemsexhibiting complex chaotic attractors using compli-cated g’s. In addition, we wish to use this exampleto further show that CM’s reveal the nature of chaosto some extent. Let
xi = −αi +
n∑j=1,j =i
x2j
xi +
n∑j=1,j =i
xj (69)
where αi are parameters, i = 1, 2, . . . , n (n ≥ 2).When these parameters are greater than 0, we may
have at least n positive g’s. As it may not beeasy to find a corresponding n-dimensional first-order differential equations, we use this second-order differential equations directly, which turn outto be 2n-dimensional in phase space. If all theparameters and initial conditions take the samecorresponding values, the 2n-dimensional systemsreduce to two-dimensional no matter the value ofn (>1), and the dynamical behavior of the systemsis trivial. However, complex motions can be trig-gered by any small perturbations in the initial con-ditions or the parameters.
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The functions gi’s are
gxi = gxi = αi +n∑
j=1,j =i
x2j . (70)
Therefore, we have 2n positive g’s when the param-eters are greater than 0, but n pairs of them areequal.
These systems can have very complicatedattractors. Here, we consider the simplest case ofthe system when n = 2. It has been found thatwith larger n, the dynamical behavior of the sys-tems is more complicated. For instance, when n = 3we have observed an interesting attractor similar
to a three-dimensional poodle [Essex & McKitrick,2002].
When n = 2, the system is rewritten as:
x = −(α + y2)x + y,
y = −(β + x2)y + x,(71)
where α and β are parameters.Obviously, when α is close to β, and the mag-
nitudes of x and y are approximately equal (whichis possible because the system is symmetric), theg’s of the system are in full competition. Highlychaotic motions may result from the competition.On the other hand, when the difference between αand β is large, the magnitudes of x and y may be
(a) (b)
(c) (d)
Fig. 14. (a) A chaotic attractor of system (71) when α = β = 0.1 with the initial condition (x(0), x(0), y(0), y(0)) =(−0.17, 0.86,−0.97,−0.2); (b) the g’s corresponding to (a); (c) a quasi-periodic attractor when α = 6.0, β = 0.1 and the sameinitial condition used in (a); and (d) the g’s corresponding to (c).
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quite different. In this case, there may be no compe-tition among g’s except in the equal pairs (gx, gx)and (gy, gy), and thus the behavior of the systembecomes simpler. Figures 14 and 15 show the attrac-tors and the g’s of the system when α = β andα β, respectively. It is seen that the numericalresults agree with our analytical predictions basedon g’s.
To show the complexity of the chaotic attrac-tors obtained above, we further consider the relationbetween the competition of the g’s and the largestLyapunov exponent (λ) of the system. In general,for a given system, a larger positive λ implies a
more complex chaotic attractor. Figure 16 depictsλ when α increases from 0.1 to 10 while β ≡ 0.1.It clearly indicates that the maximum λ appearswhen α = β for which the g’s are in good competi-tion (see Fig. 14(b)). As α increases (leaving awayfrom β), λ becomes smaller, which agrees with thatobserved from Fig. 14(d): the competition amongthe g’s is getting weaker, when α > 5.8, λ → 0+,and the state of the system is indeed quasi-periodic.The complete Lyapunov exponent spectrum of thesystem is (λ, 0, 0,−λ) because ∇ ·F= 0.
Further, to see how g’s determine the dynami-cal behavior of a nonlinear system, we construct a
(a) (b)
(c) (d)
Fig. 15. (a) A chaotic attractor of system (71) when α = β = 0.5 with the initial condition (x(0), x(0), y(0), y(0)) =(−0.5, 0.7,−2.1,−1.6); (b) the g’s corresponding to (a); (c) a quasi-periodic attractor when α = 0.5, β = 8.0 and the sameinitial condition used in (a); and (d) the g’s corresponding to (c).
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Fig. 16. The largest Lyapunov exponent of system (71)with respect to α when β = 0.1 with the initial condition:(x(0), x(0), y(0), y(0)) = (−0.17, 0.86,−0.97, 0.20).
(a)
(b)
Fig. 17. The attractors of system (72) when c1 = 1.0, c2 =1.1 with the initial condition (x1(0), x2(0), y1(0), y2(0)) =(0.13,−0.51,−8.78, 11.73) for (a) α = β = 1.5; and(b) α = 1.7, β = 1.5.
dissipative system which has similar g’s as those ofsystem (71). To do this, we modify system (71) as
x1 = c1x1 + x2,
x2 = −c2x2 − (α + y21)x1 − y1,
y1 = c1y1 + y2,
y2 = −c2y2 − (β + x21)y1 + x1,
(72)
where α, β, c1 and c2 are parameters.The functions g’s now become
gx1 = α + y21 − c2
1,
gx2 = α + y21 − c2
2,
gy1 = β + x21 − c2
1,
gy2 = β + x21 − c2
2.
(73)
When c1 < c2, system (72) is dissipative. The com-petition of the g’s is similar to that of system (71)except that gx1 > gx2 and gy1 > gy2 . When α = β,the competition among the g’s can be strong, andotherwise may be weak. Figure 17(a) displays ahighly chaotic attractor when α = β = 1.5, whileFig. 17(b) shows a quasi-periodic attractor whenα = 1.7, β = 1.5. In both cases, we have takenc1 = 1.0, c2 = 1.1. Therefore, we may conclude thatthe complexity of an attractor depends more on thecompetition of the CM’s than on the dissipation ofthe system.
7. Conclusion and Discussion
In this paper, we have investigated nonlineardynamical systems from the mode competitionpoint of view, and proposed a simple approachto construct competitive modes. We have focusedon the competition between the corresponding fre-quency components of the modes. The frequencycomponents, g’s, are in general a function of t. Fora given ODE, it is easy to find the g’s and theaveraged g’s which can be used to estimate chaoticparameter regimes. From this mode point of view,chaotic motion is the result of competition betweenmodes. At least two competitive modes must existin order to have chaotic motions. Although this con-dition is at most only a necessary condition, it isvery useful to identify the parameter regimes wherechaos may exhibit. The mode competition may leadto complicated motion, but not necessarily chaos.It is hoped that a sufficient and necessary conditionbased on mode competition can be found in future.This is a challenging task.
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Competitive Modes and Their Application 521
In particular, two applications of g’s have beenpresented in this paper. One is to estimate chaoticparameter regimes for a given system such as the“smooth” Chua’s system, and the other is to con-struct more complex chaotic systems. It has beenfound from several examples that the complexityof a chaotic attractor is related to the form andthe number of g’s. Because it is not difficult to useg’s to design any complex form and any number ofg’s, more complex chaotic systems could be easilydesigned.
There are still many fundamental problemsunsolved in mode competition. Some include: howto prove that the competition condition is necessaryfor a system to be chaotic, how to quantitativelymeasure the competition, and how to find more rig-orous definition of competitive modes. We believethat these studies of mode competition will helpunderstand the nature of complexity and chaos.
Acknowledgments
We thank Drs. Gail Wolkowicz, Lindi Wahl andSteve Feng for their valuable suggestions. This workwas supported by the Natural Sciences and Engi-neering Research Council of Canada (NSERC).
References
Chua, L. O., Komuro, M. & Matsumoto, T. [1986] “Thedouble scroll family,” IEEE Trans. Circuits Syst. 33,1073–1118.
Cuomo, K. M. & Oppenheim, A. V. [1993] “Circuitimplementation of synchronized chaos with applica-tions to communications,” Phys. Rev. Lett. 71, 65–68.
Essex, C. & McKitrick, R. [2002] Taken by Storm: TheTroubled Science, Policy and Politics of Global Warm-ing (Key Porter Books, Toronto).
Field, R. J. & Noyes, R. M. [1974] “Oscillations in chem-ical systems. IV. Limit cycle behavior in a model ofa real chemical reaction,” J. Chem. Phys. 60, 1877–1884.
Haken, H. [1983] Synergetics, An Introduction, Nonequi-librium Phase Transitions and Self-Organizationin Physics, Chemistry and Biology, 3rd edition(Springer-Verlag, Berlin).
Haller, G. [1999] Chaos Near Resonance (Springer-Verlag, NY).
Hardy, Y. & Steeb, W.-H. [1999] “The Rikitake two-diskdynamo system and domains with periodic orbits,”Int. J. Theoret. Phys. 38, 2413–2417.
Kocarev, L., Tasev, Z. & Parlitz, U. [1997] “Synchroniz-ing spatiotemporal chaos of partial differential equa-tions,” Phys. Rev. Lett. 79, 51–54.
Levy, L. R., Neufeld, R. W. J. & Yao, W. [2003]“Methodological challenges to capturing dynamicalaspects of health-care acquisition,” in Enriching theArt of Care with the Science of Care: Emotionaland Interpersonal Dimensions of Health Services, eds.Dube, L. & Moscowitz, D. (McGill-Queen’s UniversityPress, Montreal), pp. 273–293.
Lorenz, E. [1963] “Deterministic nonperiodic flow,” J.Atmosph. Sci. 20, 130–141.
Matsumoto, T. [1984] “A chaotic attractor from Chua’scircuit,” IEEE Trans. Circuits Syst. 31, 1055–1058.
Neufeld, R. W. J. [1999] “Dynamic differentials of stressand coping,” Psychol. Rev. 106, 385–397.
Nicolis, G. & Prigogine, I. [1977] Self-Organization inNonequilibrium Systems (Wiley, NY).
Pecora, L. M. & Carroll, T. L. [1990] “Synchronizationin chaotic systems,” Phys. Rev. Lett. 64, 821–824.
Pecora, L. M., Corroll, T. L., Johnson, G. & Mar, D.[1997] “Volume-preserving and volume-expandingsynchronized chaotic systems,” Phys. Rev. E 56,5090–5100.
Pesheck, E., Pierre, C. & Shaw, S. W. [2002] “A newGalerkin-based approach for accurate nonlinear nor-mal modes through invariant manifolds,” J. SoundVibr. 249, 971–993.
Richetti, P., Roux, J. C., Argoul, F. & Arneodo, A. [1987]“From quasiperiodicity to chaos in the Belousov–Zhabotinskii reaction II. Modeling and theory,” J.Chem. Phys. 86, 3339–3356.
Rossler, O. E. [1976] “An equation for continuous chaos,”Phys. Lett. A 57, 397–398.
Rossler, O. E. [1991] A Chaotic Hierarchy, eds. Baier, G.& Klein, M. (World Scientific, Singapore).
Shaw, S. W. & Pierre, C. [1993] “Normal modes for non-linear vibratory systems,” J. Sound Vibr. 164, 85–124.
Shaw, S. W. & Pierre, C. [1994] “Normal modes of vibra-tion for non-linear continuous systems,” J. SoundVibr. 169, 319–347.
Short, K. M. [1994] “Steps toward unmasking securecommunications,” Int. J. Bifurcation and Chaos 4,959–977.
Sundar, S. & Minai, A. A. [2000] “Synchronization ofrandomly multiplexed chaotic systems with applica-tion to communication,” Phys. Rev. Lett. 85, 5456–5459.
Tsuneda, A. [2005] “A gallery of attractors from smoothChua’s equation,” Int. J. Bifurcation and Chaos 15,1–49.
Turner, J. S., Roux, J. C., McCormick, W. D. & Swinney,H. L. [1981] “Alternating periodic and chaotic regimesin a chemical reaction-experiment and theory,” Phys.Lett. A 85, 9–12.
Wei, G. W. & Zhao, S. [2002] “Synchronization andinformation processing by an on-off coupling,” Phys.Rev. E 65, 056210.
Int.
J. B
ifur
catio
n C
haos
200
6.16
:497
-522
. Dow
nloa
ded
from
ww
w.w
orld
scie
ntif
ic.c
omby
UN
IVE
RSI
TY
OF
WE
STE
RN
ON
TA
RIO
WE
STE
RN
LIB
RA
RIE
S on
07/
24/1
2. F
or p
erso
nal u
se o
nly.
522 W. Yao et al.
Weiss, C. O. & Vilaseca, R. [1991] Dynamics of Lasers(VCH, NY), pp. 59–66.
Wiggins, S. [1990] Introduction to Applied Nonlin-ear Dynamical Systems and Chaos (Springer-Verlag,NY).
Yao, W., Essex, C. & Yu, P. [2002a] A New ChaoticSystem for Better Secure Communication, Dynam-ics of Continuous, Discrete and Impulsive Systems,Series B: Applications and Algorithms, Vol. 10,pp. 221–234.
Yao, W., Yu, P. & Essex, C. [2002b] “Estimating chaoticparameter regimes by generalized competition modesapproach,” Commun. Nonlin. Sci. Numer. Simul. 7,197–205.
Yao, W., Yu, P. & Essex, C. [2004] “Communicationbetween synchronized random number generators,”Int. J. Bifurcation and Chaos 14, 3995–4008.
Yu, P., Zhang, W. & Bi, Q. [2001] “Vibration analysison a thin plate with the aid of computation of normalforms,” Int. J. Non-Lin. Mech. 36, 597–627.
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J. B
ifur
catio
n C
haos
200
6.16
:497
-522
. Dow
nloa
ded
from
ww
w.w
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UN
IVE
RSI
TY
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WE
STE
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ON
TA
RIO
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or p
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se o
nly.
