Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 46, 1, pp. 185-204, Warsaw 2008 SYNCHRONIZATION OF MECHANICAL OSCILLATORS EXCITED KINEMATICALLY Przemyslaw Perlikowski Technical University of Lodz, Division of Dynamics, Lodz, Poland e-mail: przemyslaw.perlikowski@p.lodz.pl In this paper, the behaviour of a system of coupled mechanical oscilla- tors excited kinematically was studied numerically. Several methods of detection of synchronization were shown and advantages of each were mentioned. The relation between Lyapunov exponents and the synchro- nization state under different kinds of the excitation signal (harmonic, periodic and chaotic ones) was presented. It was demonstrated that the mode lockingwith excitationwas dependent only on inner damping and completely uncorrelated with the connection between oscillators. Key words: synchronization,mechanical oscillators, kinematic excitation 1. Introduction Thephenomenonof synchronization in dynamics has beenknown for a long ti- me. Recently, the idea of synchronization has been also adopted for non-linear systems. It has been demonstrated that two ormore non-linear systems canbe synchronized by linking them with mutual coupling or with a common signal or signals (Blekhman et al., 1995;Boccaletti et al., 2002;Kapitaniak, 1994; Pe- cora andCarroll, 1990; Pyragas, 1993). In the case of linking, a set of identical chaotic systems (the same set of ODEs and values of the system parameters) Complete Synchronization (CS) can be obtained. The complete synchroniza- tion takes place when all trajectories converge to the same value and remain in step with each other during further evolution (i.e. limt→∞ |x(t)−y(t)|=0 for two arbitrarily chosen trajectories x(t) and y(t)). In such a situation, all subsystems of the augmented system evolve on the same invariant mani- fold on which one of these subsystems evolves (the phase space is reduced to the synchronization manifold). The linking of homochaotic systems (i.e. 186 P. Perlikowski systems given by the same set of ODEs but with different values of the sys- tem parameters) can lead to Imperfect Complete Synchronization (ICS) (i.e. limt→∞ |x(t)−y(t)| ¬ ε, where ε is a vector of small parameters) (Kapitaniak et al., 1996). A significant change of the chaotic behaviour of one ormore sys- tems can be also observed in such linked systems. This so-called ”controlling chaos by chaos” procedure has some potential importance formechanical and electrical systems. An attractor of such two systems coupled by a negative feedback mechanism can be even reduced to the fixed point (Stefański and Kapitaniak, 1996). Recently, another type of synchronization has been detected in non-linear systems called Phase Synchronization (PS) (Rosenblum et al., 1996; Rosen- blum et al., 2001). It can be introduced as synchronization of periodic oscil- lators, where only the phase locking is necessary, while no requirements on the amplitudes is imposed. The PS in non-linear systems is defined as the ap- pearance of reaction between phases of subsystems (or between the phase of subsystemsand thedriving signal),while theamplitudes can still benon-linear and non-correlated. Fromtheviewpoint of practical considerations, a periodicandnon-periodic excitation can be met in mechanical systems. For that reason, this paper is concentrated on analysis of the CS of a set of identical mechanical oscillators linked by a common excitation. Interactions between phases of the excitation and oscillators were taken under consideration. The presented analysis is ba- sed on connections between the appearance of synchronization and Lyapunov Exponents (LEs), and it is motivated by Pecora and Carroll’s theoretical and experimental studies (Pecora and Carroll, 1990, 1991). It has been discovered that uncoupled non-linear systems linked by common signals can synchronize if LEs for the subsystems reaches negatives values. In the next paper byPeco- ra and Carroll (1998) a concept called Master Stability Function (MSF) was introduced. It has been proved that the system of coupled non-linear oscilla- tors can synchronize when all Transversal Lyapunov Exponents (TLEs) are negative. Considerations presented in this paper, which are supported by nu- merical analysis, shown that even a non-periodic nature of common external excitation can lead to synchronization of driven mechanical oscillators. The necessary, but not sufficient, condition for the occurrence of synchronization is the negative sign of LEs associated with the response of the system. In the numerical experiment, a pair of non-linear mechanical oscillators of the Duf- fing type, coupled or uncoupled, forced by different types of excitation has been used. All numerical simulations have been carried out with programs written in C++, and for calculating LEs and TLEs algorithms by Parker and Synchronization of mechanical oscillators excited kinematically 187 Chua (1989) have been used.The paper is organized as follows. In Sec. 2 some theoretical fundamentals are recalled. Section 3 describes types of oscillators which are used in numerical investigations. Section 4 is devoted to numerical examples. A brief conclusion is presented in Sec. 5. 2. Complete synchronization, phase synchronization and Lyapunov exponents 2.1. Complete synchronization of uncoupled systems Consider a set of l separate identical j-degree-of-freedommechanical oscil- lators forced by a common signal as shown in Figure 1. The character of excitation (function e(t)) and differential equations describing the motion of oscillators can be chosen arbitrarily. The dynamical state each of these oscillators is determined by the j-dimensional vector xi = [xi1, xi2, . . . ,xij], (i=1,2, . . . , l). This vector describes a response generated by the system.The r-dimensional vector e= [e1,e2, . . . ,er] describes evolution of the excitation. Thus, the state of a separate subsystem is described in the phase space of h= j+r dimension, and the equations of motion of such a subsystem can be written in a general first order differential equation autonomous form ẋi = f(xi,e) ė= f(e) (2.1) where xi ∈ R j (i=1,2, . . . , l) and e∈ Rr. Fig. 1. The scheme of the system under consideration Equation (2.1) describes the dynamical evolution of excitation in the s- dimensional subspace of the system phase space (excitation subspace). From the formofEqs. (2.1) it results that the time evolution of excitation is indepen- dent of the remaining state variables and is characterized by Lyapunov expo- nents, where at least one of them is equal to zero. Equation (2.1)1 describes 188 P. Perlikowski the evolution of the system response (response subspace) in the j-dimensional subspace of the phase space which is transversal to the abovementioned exci- tation subspace and is characterized by a series of j LEs. A two-dimensional visualization of the system phase space is presented in Fig.2. This idea was proposed by Stefański and Kapitaniak (2003b). Fig. 2. Two-dimensional visualization of synchronization (a) and desynchronization (b); attractor of the chaotic system; α(A) basin of attraction of the attractor A; E excitation subspace; R response subspace The common excitation causes that the distance between trajectories in the direction associated with zero LE amounts to zero and the trajectories can be found in the same j-dimensional response subspace at each moment. This fact leads to the conclusion that for a set of negative RLEs there exists a point in the response subspace which is a stable sub-attractor. This point is a trace of the system attractor in the response subspace. If the trajectories start from different points of the basin of attraction, then they evolve to the same state and the oscillators will synchronize (Fig.2a). In otherwords, an invariant subspace representing the synchronized state (x1 =x2 = . . .=xl) is a stable attractor. Such synchronization is caused by the common excitation only, and it occurs without any additional coupling between the oscillators. If at least oneRLEs is positive, then the synchronization between the oscil- lators under consideration is impossible. It means that instability associated with positive RLEs causes divergence of nearby trajectories (Fig.2b) in the response subspace, and the sub-attractor becomes a sub-repeller representing an unstable orbit in this subspace. Synchronization of mechanical oscillators excited kinematically 189 2.2. Complete synchronization of coupled systems Systemswith couplings between oscillators can not be treated in the same way as uncoupled ones. It is impossible to describe their behaviour as a system composed of independent oscillators. The idea of RLEs is not sufficient to determine synchronization. Different criteria have to be applied. Pecora and Caroll (1998) introduced a concept called theMaster StabilityFunction,which allows one to determine the stability of synchronization by means of TLEs. The stability problem in this idea can be formulated in a very general way by addressing the question of stability of the CS manifold x1 ≡ x2 ≡ . . . ≡ xi. The CS exists when the synchronization manifold is asymptotically stable for all possible trajectories. Considering a dynamical uncoupled system given by equations ẋi = f(xi) (2.2) H : Rm → mIRm is the output function of each oscillator variables that is used in the coupling, x=(x1,x2, . . . ,xN)∈ R m, F(x)= (f(x1), . . . ,f(xN)), G is the matrix of coupling coefficients and σ is the coupling strength. An arbitrary set of N coupled (linear coupling) identical systems can be created ẋ=F(xi)+(σG⊗H) (2.3) where ⊗ is the direct (Kronecker) product. In this place, it should be recalled that the direct product of two matrices A and B is given by A⊗B= [aijB] (2.4) where aij are elements of thematrix A. Note also that themanifold invariant requires ∑ jGij =0. The variational equation of Eq. (2.3) is ϕ̇= [IN ⊗DF+σG⊗DH]ϕ (2.5) where ϕ = (ϕ1,ϕ2, . . . ,ϕN) are perturbations and IN is the (N × N)- dimensional identity matrix. After diagonalization, each block has the form ϕ̇k = [DF+σγkDH]ϕk (2.6) where γk is the eigenvalue of the connectivity matrix G, (k = 0,1,2 . . . , N−1). For k=0→ γ0, one gets variational equation for the synchronization manifold. All k correspond to transverse eigenvectors, so one has succeed in separating the synchronizationmanifold from the other, transverse directions. 190 P. Perlikowski TheJacobian functions DFand DH are the same for eachblock, since they are evaluated on the synchronized state. Thus, for each k, the form of each block (Eq. (2.6)) is the same onlywith only the scalarmultiplier σγk different for each of them.This leads to the following formulation of themaster stability equation and the associated MSF. One can calculate the maximum TLE λ1 for the generic variational equation ϕ̇k = [DF+(δ+ iβ)DH]ϕk (2.7) where σγk = δ+iβ. Inmechanical systems,we havemutual interaction, hence σγk = δ (real coupling β=0). 2.3. Phase synchronization In chaotic oscillators, there is no unique phase of oscillations. The idea introduced byRosenblum et al. (Rosenblum et al., 1996, 2001) was applied to detect the phase. It enabled one to detect the PS between the oscillators and the driven signal. The amplitude and the phase of an arbitrary signal s(t) has to be determined. A general approach was introduced by Gabor (1946) and it was based on the analytic signal concept by Rosenblum et al. (1996). The analytic signal x(t) is a complex function of time defined as x(t)= s(t)+ js̃=A(t)ejφ(t) (2.8) where the function s̃(t) is the Hilbert transform of s(t) s̃= 1 π P.V. ∞∫ −∞ s(t) t− τ dτ (2.9) (where P.V. means that the integral is taken in the sense of the Cauchy prin- cipal value). The instantaneous amplitude A(t) and the instantaneous pha- se φ(t) of the signal s(t) are thus uniquely defined by equation (2.8). From (2.9), the Hilbert transform s̃(t) of s(t)may be considered as the convolution of the functions s(t) and t/π. Hence, the Fourier transform S̃(jω) of s̃(t) is the product of the Fourier transforms of s(t) and t/π. For physically relevant frequencies ω > 0, S̃(jω) =−jS(jω); i.e. ideally, s̃(t) may be obtained from s(t) by a filter whose amplitude response is unity, and whose phase response is a constant π/2 lag at all frequencies. Synchronization of mechanical oscillators excited kinematically 191 3. Duffing oscillator The Duffing system has been chosen to present an example of a mechanical oscillator. The considered system is excited kinematically, what was shown in Fig.3. Its evolution is described by mÿ−dy[ė(t)− ẏ]+ky[e(t)−y]−kd[e(t)−y] 3 =0 (3.1) where m,dy,ky,kd areviscousdamping, linear andnon-linearpartsof stiffness of the spring, respectively, and e(t) is the signal of excitation. Fig. 3. Duffing oscillator After introducing dimensionless variables, the following form of the equ- ation of motion is obtained ξ̈− c1[ċ4(τ)− ξ̇]+ [c4(τ)− ξ]− c2[c4(τ)− ξ] 3 =0 (3.2) where ω= ky m c1 = dy mω c2 = kdmy 2 st ω2k2y c4(τ)= e(t) yst ċ4(τ)= ė(t) ystω yst = mg ky In numerical analysis it is assumed that c2 = 2.0. The inner damping c1 is chosen as a controlling parameter. The excitation signal c4(τ) depends on the type of excitation transmitted to the oscillator. It is described in Sec. 4 in more detail. 3.1. Two Duffing oscillators excited by common signal A system of two Duffing oscillators excited by a common signal e(t) is shown in Fig.4. 192 P. Perlikowski Fig. 4. TwoDuffing oscillators with common excitation This is the simplest example of the system given by Eqs. (???). According to Eqs. (3.2), the system shown in Fig.4 can be described by the following dimensionless equations ξ̈1− c1[ċ4(τ)− ξ̇1]+ [c4(τ)− ξ1]− c2[c4(τ)− ξ1] 3 =0 (3.3) ξ̈2− c1[ċ4(τ)− ξ̇2]+ [c4(τ)− ξ2]− c2[c4(τ)− ξ2] 3 =0 Common dynamics is realized by the kinematic excitation applied to the ri- gid suspension of oscillators, which ensures the same amplitude and phase of forcing. 3.2. Two dissipatively coupled Duffing oscillators excited by common signal Thesystemof two coupledDuffingoscillators excitedby the commonsignal is taken under consideration. Due to mutual coupling, there are interactions between subsystems. Fig. 5. Two coupled Duffing with common excitation Synchronization of mechanical oscillators excited kinematically 193 This system is a generalisation of the previous problem.When d1 is equal to zero the system given by Eq. (3.3) is obtained. Motion of the system is given by dimensionless equations ξ̈1− c1[ċ4(τ)− ξ̇1]+ [c4(τ)− ξ1]− c2[c4(τ)− ξ1] 3+d1[ξ̇1− ξ̇2] = 0 (3.4) ξ̈1− c1[ċ4(τ)− ξ̇1]+ [c4(τ)− ξ1]− c2[c4(τ)− ξ1] 3−d1[ξ̇1− ξ̇2] = 0 where: d1 = d/(mω) and ω, c1, c2, c4(τ), ċ4(τ) are the same as in Eq. (3.2). 4. Numerical examples 4.1. Harmonic excitation To show the idea ofRLEon the simplest example, the harmonic excitation is applied to the Duffing oscillators. The excitation signal e(t) is given by the equation e(t) = asin(Ωt). Hence, for the whole system, the dimensionless equations of motion are as follows ξ̈1− c1[ċ4(τ)− ξ̇1]+ [c4(τ)− ξ1]− c2[c4(τ)− ξ1] 3 =0 (4.1) ξ̈2− c1[ċ4(τ)− ξ̇2]+ [c4(τ)− ξ2]− c2[c4(τ)− ξ2] 3 =0 where c4(τ)= a yst sin(ητ) ċ4(τ)= a yst ηcos(ητ) τ =ωt η= Ω ω =1 other parameters are the same as in theEq. (3.2). As it wasmentioned before, the controlling parameter is the dimensionless viscous damping c1. As can be seen in Figs. 6 and 7, there is a wide range of chaotic behaviour. It is well known that due to sensitivity to the initial condition, it is impossible to synchronize systemsworking in the chaotic regimewithout additional coupling between them. In Figs. 6 and 7, bifurcation diagrams of ξ1 and the difference ξ1 − ξ2 as a function of c1 are shown. As can be, the seen subsystems reach synchronization in ranges where a single oscillator has periodic behaviour. For each subsystem, there is a boundary value of c1 after which the state of synchronization is stable and unchangeable despite an increase of c1. For the system with the amplitude of excitation c4(τ) = 1.9sinτ (Fig.6) the boundary value is c1 =0.2, and for c4(τ) = 2.2sinτ (Fig.7) is c1 =0.27. In 194 P. Perlikowski the second example, it is worthmentioning that ”synchronization windows” in which synchronization appears exist between two ranges of desynchronization. Such a type of synchronization has been also detected in networks of coupled oscillators and has been called Ragged Synchronizability (RSA) (Stefański et al., 2007). Fig. 6. Bifurcation diagram of a single oscillator excited by the periodic signal Eq. (4.1). Parameters: c1 ∈ (0.0,0.5), c4(τ)= 1.9cosτ 4.2. Non-linear excitation In this Section, the systemwith the excitation signal given by a non-linear signal is taken under consideration .As a systemgenerating non-linear forcing, the Duffing oscillator has been chosen. It is described by the dimensionless equation β̈+0.1β̇−β+β3 = q sinτ (4.2) Synchronization of mechanical oscillators excited kinematically 195 Fig. 7. Bifurcation diagram of a single oscillator excited by the periodic signal Eq. (4.1). Parameters:c1∈ (0.0,0.5), c4(τ)= 2.2cosτ In this case, the investigated system is excited using a signal given by the variables β and β̇, i.e. c4(τ) = β and ċ4(τ) = β̇. For numerical simulations, three different values of the forcing amplitude are chosen: 1. q=4.0 – periodic motion with the period one (Fig.8) 2. q=5.6 – chaotic motion (Fig.9) 3. q=5.8 – periodic motion with the period three (Fig.10). In Figures 8-10, three bifurcation diagrams depicting the relation between the controlling parameter c1 versus ξ1 − ξ2, ξ1 and Lyapunov exponents of the single subsystem (two RLEs and two ELEs) are presented respectively. In Sec. 2, the rule that the system reaches synchronization when RLEs reaches negative values is introduced. Thus, it is sufficient to calculate LEs for one subsystem only (in the case when subsystems are identical). It is worth to mention that the subsystems tend to behave like the excitation. For the perio- dic excitation with the period one, the oscillators reach period onemotion and 196 P. Perlikowski Fig. 8. Non-linear excitation – period one (q=4.0), bifurcation diagrams of ξ1−ξ2, ξ1 and Lyapunov exponents Synchronization of mechanical oscillators excited kinematically 197 Fig. 9. Non-linear excitation – chaotic behaviour (q=5.6), bifurcation diagrams of ξ1− ξ2, ξ1 and Lyapunov exponents 198 P. Perlikowski Fig. 10. Non-linear excitation – period three (q=5.8), bifurcation diagrams of ξ1− ξ2, ξ1 and Lyapunov exponents Synchronization of mechanical oscillators excited kinematically 199 so on. The relation between the largest RLE and synchronization phenomena is confirmed. In Fig.9, the synchronization is reached despite chaotic motion – in this way the oscillators lose their sensitivity to the initial conditions. It is possible because the phenomena ofmode locking with the signal of excitation occur (it is proved in the next Subsection and shown in Fig.16c). Pioncaré maps of the systemwith chaotic excitation (q=5.6) are presented in Fig.11. In Fig.11a a map before synchronization range (c1 = 0.3) where hyperchaos exists (two positive LEs – λr1 and λe1) is shown. Figure 11b concerns the sys- tem working in the synchronous regime (c1 = 0.5) and the chaotic attractor can be seen (mode locking regime with excitation occurrs and only one LE is positive: λe1). In Fig.11 a comparison of Pioncaré maps of excitation (grey color) and the response (black color) is shown. As can be seen, the transition from hyperchaos to chaos is exhibited by noticeable reduction of the fractal dimension from DKY =2.76 (Fig.11a – hyperchaos) to DKY =1.89 (Fig.11b – chaos). Fig. 11. Pioncaré maps of one subsystem excited by the chaotic signal (black dots) and a comparisonmap of the excitation (grey dots): (a) c1 =0.3 and DKY =2.76 (before synchronization), (b) c1 =0.5, DKY =1.89 (after synchronization) 200 P. Perlikowski In the last Figure in this subsection, the influence of perturbation on the stability of the synchronous state is shown. Every 50 periods of the driving system, the response systems have been perturbed by a small displacement (δ = 0.1). After a short time, the response systems reach synchronization, which means that this state is stable. Fig. 12. Synchronization error ξ1− ξ2 for c1 =0.4 and q=5.6, after every 50 periods of the excitation signal, a small perturbation is added to the system 4.3. Coupled oscillators Synchronization between forced coupled Duffing oscillators has been inve- stigated in Perlikowski and Stefańsk (2006), Stefański andKapitaniak (2003), Stefański et al. (2007). In this Section, two coupled Duffing oscillators exci- ted kinematically, described byEqs. (3.4), are taken under consideration. The excitation signal is given by the forcedDuffingoscillator introduced in the pre- vious Section. In these simulations, only one value of the parameter q: q=5.6 (chaotic motion) has been used. The results of different kinds of analysis of the synchronization process in the examined system, are presented below. Fig. 13. Plot of the synchronization error ξ1− ξ2 as a function of c1 and d1 Synchronization of mechanical oscillators excited kinematically 201 In Fig.13, a plot of the evolution of the synchronization error (ǫ= ξ1−ξ2) is shown. Then, in Fig.14, values of RLE for the whole system presented in Fig.5 are plotted. At the end (Fig.15), the maximal TLE is presented. All figures exhibit the dependence on the dissipative parameters c1 versus d1. Fig. 14. Lyapunov exponents for the whole system for all cases: λr3 < 0, λr4 < 0, λe1 > 0 and λe2 < 0 The comparison of those threemethods arises their interchangeability, but on the other hand, much more information about the system is possible to obtain by calculating LE. In Fig.14 areas with three possible configurations of RLE are presented. The first range (dark grey) corresponds to desynchro- nization and hyperchaotic behaviour of the response systems. When the pa- rameters are located in the light grey area, then the oscillators reach syn- chronization, but still there is no correlation with the signal of the excita- tion. In the last range (white), the oscillators not only synchronize but al- so the mode locking regime with the excitation occurs. In Fig.16, phases for all cases are presented. Calculation of TLEs enabled one to present the synchronization area for any number of coupled oscillators and any set of coupling scheme, but in spite of RLE, less information about the system is attained. 202 P. Perlikowski Fig. 15. Transversal Lyapunov exponents for all cases: λtr2 < 0, λe1 > 0 and λe2 < 0 Fig. 16. Phase φξ1, φξ2, φβ of the first and the second oscillator as well as the excitation, respectively; (a) hyperchaos for c1 =0.1, d1 =0.1, d2 =0.0, (b) chaos for c1 =0.1, d1 =0.5, d2 =0.0 and (c) steady state (mode locking) for c1 =0.5, d1 =0.0, d2 =0.0 5. Conclusion In this work, phenomena of synchronization of mechanical oscillators excited byacommon signal, also introducedbyamechanical system,havebeen shown. As a result, detection methods of synchronization in coupled and uncoupled Synchronization of mechanical oscillators excited kinematically 203 oscillators have been presented. The dependence on changes in the spectrum of Lyapunov exponents has also been investigated. It is worthmentioning that theperiodof the forced system tends to theperiodof the excitation.Whenever in the case of chaotic excitation, the response tends to the chaotic behaviour. It has been also found that uncoupled oscillators completely synchronize with each otherwhen themode locking regimewith the excitation takes place. This phenomenon depends only on the inner viscous damping and is independent of dissipative connections between the oscillators. It is sufficient to calculate the spectrum of Lyapunov exponents of the single system to detect it. Acknowledgment This study has been supported by Polish Department for Scientific Research (DBN) under Project N50104431/2919. References 1. Blekhman I., Landa P.S., Rosenblum M.G, 1995, Synchronization and chaotization in interacting dynamical systems,Applied Mechanics Review, 48, 733-752 2. Boccaletti S., Kurths J., Osipov G., Valladares D., Zhou C., 2002, The synchronization of chaotic systems,Physics Reports, 366 3. Gabor D., 1946, Theory of Communication, J. IEE, 93, 429-457 4. KapitaniakT., 1994, Synchronization of chaos using continuous control,Phy- sical Review E, 50, 1642-1644 5. 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RosenblumM.G., PikovskyA.S., Kurths J., 1996,Phase synchronization of chaotic oscillators,Physical Review Letters, 76, 1804-1807 13. Rosenblum M.G., Pikovsky A.S., Kurths J., Schaefer C., Tass P., 2001, Phase synchronization: from theory to data analysis,Handbook of Biolo- gical Physics, 4, 279-321 14. Stefański A., Kapitaniak T., 1996, Steady state locking in coupled chaotic systems,Physics Letters A, 210, 279-282 15. StefańskiA.,KapitaniakT., 2003a, Synchronizationofmechanical systems driven by chaotic or random excitation, Journal of Sound and Vibration, 260, 565-575 16. Stefański A., Kapitaniak T., 2003b, Synchronization of two chaotic oscil- lators via negative feedbackmechanism, Int. Journal of Solids and Structures, 40, 5175-5185 17. Stefański A., Perlikowski P., Kapitaniak T., 2006, Rugged synchroni- zability of coupled oscillators,Physical Review E, 75 Synchronizacja oscylatorów mechanicznych wymuszanych kinematycznie Streszczenie Wartykule pokazano numeryczną analizę dynamiki sprzężonych oscylatorówme- chanicznych wymuszonych kinematycznie. Przedstawiono przegląd najważniejszych metod detekcji synchronizacji, zwracając szczególną uwagę na ich własności. Następ- nie opisano powiązania między widmem wykładników Lapunowa, a różnymi typami synchronizacji. Zbadano odpowiedź układu dla różnych typów sygnałuwymuszające- go (harmonicznego, okresowego i chaotycznego).Udowodniono, że zamykaniemodów pomiędzy sygnałem oscylatora a wymuszeniem zależne jest tylko od tłumienia we- wnętrznego oscylatorów, natomiast jest niezależne od sprzężeń pomiędzy nimi. Manuscript received April 23, 2007; accepted for print October 4, 2007