International Journal of Bifurcation and Chaos, Vol. 14, No. 1 (2004) 171–181 c World Scientific Publishing Company

SYNCHRONIZATION OF THE REGULAR AND CHAOTIC STATES OF ELECTROMECHANICAL DEVICES WITH AND WITHOUT DELAY R. YAMAPI and J. B. CHABI OROU Institut de Math´ematiques et de Sciences Physiques, B.P. 613, Porto-Novo, B´enin P. WOAFO Laboratoire de M´ecanique, Facult´e des Sciences, Universit´e de Yaound´e I, B.P. 812, Yaound´e, Cameroun Received August 19, 2002; Revised October 16, 2002 We consider in this paper the problem of stability and duration of the synchronization process between two electromechanical devices, both in their regular and chaotic states. Stability boundaries are derived through Floquet theory. The influence of the precision on the synchronization time is also analyzed using numerical simulation of the equations of motion. Keywords: Synchronization; electromechanical systems.

two or more devices to work in a synchronized manner are required both in regular and chaotic regimes. In this field, electromechanical devices with natural and created (or introduced) nonlinearities are common and can lead to various types of behavior [Jerrelind & Stensson, 2000; Woafo et al., 1998; Woafo, 2000; Chembo Kouomou & Woafo, 2000; Chedjou et al., 2001]. In this paper, we consider the problem of synchronizing two electromechanical devices both in regular and chaotic states with and without delay. The device is described by a Duffing oscillator coupled to a linear oscillator. Our study uses the continuous feedback scheme of Pyragas [1992]. We use the Floquet theory to derive the stability criteria and the optimal coupling strength of the synchronization process. This theory has been used recently to optimize and derive the duration time of the synchronization of two Duffing oscillators [Chembo Kouomou & Woafo, 2002] as well as that of two Van der Pol oscillators [Woafo & Kraenkel, 2002].

1. Introduction In recent years, the field of nonlinear science has seen a growing interest in the synchronization of nonlinear oscillators both in their regular and chaotic states (see [Kapitaniak, 1996; Lakshmanan & Murali, 1996] and references therein). The idea of synchronizing two chaotic oscillators was presented by Pecora and Carrol [1990] by coupling both oscillators with a common drive signal. Later on, Kapitaniak [1994] showed that one can also synchronize two chaotic oscillators using the continuous feedback scheme developed by Pyragas [1992]. Considerable attention paid to such topics is due to the potential applications of synchronization in communications engineering (using chaos to mask the information bearing signals) [Pecora & Caroll, 1990; Oppenheim et al., 1992; Kocarev et al., 1992; Perez & Cerdeira, 1995], in biology and chemistry [Winfree, 1980; Kuramoto, 1980]. Another field where synchronization is of crucial importance is automation engineering where 171

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The paper is organized as follows. In the next section, we describe the physical model and give the resulting equations of motion. We study analytically in Sec. 3 the stability of the synchronization process and present the behavior of the synchronization time. The delay is also taken into account. We derive the critical values Kcr under which, for a given precision, no synchonization is possible. In Sec. 4, we extend the investigation to the synchronization of two electromechanical devices in a chaotic state. Numerical simulations are used to complement our analytical results. Section 5 is devoted to conclusion.

2. Description of the Physical Model and Statement of the Problem 2.1. Description of the electromechanical device and equations The electromechanical device shown in Fig. 1 is composed of an electrical part (Duffing oscillator) coupled to a mechanical part governed by the linear oscillator. The coupling between both parts is realized through the electromagnetic force due to a permanent magnet which creates a Laplace force in the mechanical part and the Lenz electromotive voltage in the electrical part. The electrical part of the system consists of a resistor R, an inductor L, a condenser C and a sinusoidal voltage source e(τ 0 ) = vo cos Ωτ 0 (vo and Ω being respectively the amplitude and frequency, and τ 0 the time), all connected in series. In our model, the voltage of the condenser is a nonlinear function of the instantaneous electrical charge q. It can be written as follows 1 q + a3 q 3 Vc = C0 where Co is the linear value of C and a3 is a nonlinear coefficient depending on the type of the capacitor in use. This is typical of nonlinear reactance components such as varactor diodes widely used in many areas of electrical engineering to design, for instance, parametric amplifiers, up-converters, mixers, low-power microwave oscillators, etc. [Oksasoglu & Vavriv, 1993]. The mechanical part is composed of a mobile beam which can move along the z axis on both sides. The rod T which has a similar motion is bound to a mobile beam with a spring. To close the description of the model, let

Fig. 1.

The electromechanical device.

us note that other types of nonlinear electrical and mechanical components such as nonlinear inductor, nonlinear resistance or nonlinear spring can also be used. This will lead to other types of electromechanical devices described by a set of differential equations analogous or different to the ones analyzed in this paper. An example of an electromechanical device with a nonlinear resistance with a negative slope and with a nonlinear spring had been studied by Chedjou et al. [2001]. Using the electrical and mechanical laws, and taking into account the contributions of the Laplace force and the Lenz electromotive voltage, it is found that the system is described by the following set of differential equations L¨ q + Rq˙ +

q + a3 q 3 + lB z˙ = vo cos Ωτ 0 Co

m¨ z + λz˙ + kz − lB q˙ = 0 where l is the length of the domain of the interaction between B and the two mobile rods supporting the beam. The dot over a quantity denotes the time derivative. Let us use the dimensionless variables q z x= ; y = ; t = we τ 0 ; Qo l where Qo is a reference charge of the condenser and we2 = β= w=

1 ; LCo

a3 Q2o ; Lwe2

Ω ; we

λ1 =

γ2 =

k ; m

γ1 =

R ; Lwe

l2 B ; LQo we

E0 =

vo ; LQo we2

2 = wm

λ ; mwe

w2 =

wm ; we

λ2 =

BQo mwe

Synchronization of the Regular and Chaotic States of Electromechanical Devices

Then the above two differential equations reduce to the following set of nondimensional differential equations x ¨ + γ1 x˙ + x + βx3 + λ1 y˙ = Eo cos wt y¨ + γ2 y˙ + w22 y − λ2 x˙ = 0

(1)

The model represented in Fig. 1 is widely encountered in various branches of electromechanical engineering. In particular, in its linear version, it describes the well-known electrodynamic loudspeaker [Olson, 1967]. In this case, the sinusoidal signal e(t) represents an incomming pure message. Because of the recent advances in the theory of nonlinear phenomena, it is interesting to consider such an electrodynamic system containing one or various nonlinear components or in the state where one or various of its component react nonlinearly. One such state occurs in the electrodynamic loudspeaker due to the nonlinear character of the diaphragm suspension system resulting in signal distorsion and subharmonics generation. Moreover, the model can serve as servo-command mechanism which can be used for various applications. Here, one would like to take advantage of nonlinear responses of the model in manufacturing processes. Due to the presence of cubic nonlinearity, the electromechanical model presents the classical nonlinear resonance, antiresonance, hysteresis and jump phenomena, and when the amplitude E0 is large enough, the model exhibits complex behaviors such as chaos. In this paper, we use the following set of parameters γ 1 = 0.1; γ2 = 0.30; λ1 = 0.01; λ2 = 0.06; w2 = 1.2; w = 1.3. We use two values for β: β = 0.1 and β = 1.32. The values of E0 are indicated when needed.

2.2. Statement of the problem In the regular regime, for some sets of physical parameters and due to nonlinearity, the response of the system to the external excitation shows what is known as the multi-stability. In this case, the system presents the well-known hysteresis phenomena with two stable harmonic oscillations with different amplitudes. Each harmonic state has its own basin of attraction in the space of initial conditions. Consequently, if two systems are launched with different initial conditions belonging to different basins of attraction, they will finally circulate on different orbits. The objective of the synchronization in this case is to call one of the system (slave) from its orbit to that of the other system (master). Extension

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can be made to include a time delay between the slave and the master. In the chaotic states, the essence is the high sensitivity to initial conditions. Indeed when the system is working in the chaotic regime, a very small difference in the initial conditions will lead to different time histories. The goal of synchronization is to facilitate finally synchronization of the time history of two electromechanical devices with different initial conditions. As noted in the introduction, our aim is to study the stability and derive the characteristics of the synchronization of two of our electromechanical systems. The master system is described by the components x and y while the slave system has the corresponding components u and v. The enslavement is carried out by coupling the slave to the master through the following scheme x ¨ + γ1 x˙ + x + βx3 + λ1 y˙ = E0 cos wt y¨ + γ2 y˙ + w22 y − λ2 x˙ = 0 u ¨ + γ1 u˙ + u + βu3 + λ1 v˙ = E0 cos wt − K(u − xτ )H(t − T0 ) v¨ + γ2 v˙ + w22 v − λ2 u˙ = 0

(2)

where xτ = x(t − τ ), K is the feedback coupling or synchronization coefficient, τ is the time delay, T 0 the onset time of the synchronization and H(x) is the Heaviside function defined by  0 for x < 0 H(x) = 1 for x ≥ 0

3. Synchronization of the Regular States of Two Electromechanical Devices 3.1. Stability and duration of the synchronization process without delay Starting from t = T0 , the system changes its configuration and become physically interesting only so long as the dynamics of the slave devices described by the components u and v are stable. We have to determine the range of K for which the synchronization process is achieved. The stability of the synchronization process is therefore strictly equivalent to the boundedness of εi defined as ε1 (t) = u(t) − x(t) ε2 (t) = v(t) − y(t)

(3)

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The new variables εi are the measure of the relative nearness of the slave to the master. From t ≥ T0 , εi obey

Only the two last solutions of both sets are stable. With the form of the master given in Eq. (5), εi are now described by the following linear parametric equation

(4)

ε¨1 + γ1 ε˙1 + (Ω21 + η cos(2wt − 2φ))ε1 + λ1 ε˙2 = 0 ε¨2 + γ2 ε˙2 + w22 ε2 − λ2 ε˙1 = 0 (7)

ε¨1 + γ1 ε˙1 + (1 + K + 3βx2 )ε1 + λ1 ε˙2 = 0 ε¨2 + γ2 ε˙2 + w22 ε2 − λ2 ε˙1 = 0

assuming that they are small. Synchronization is achieved when εi goes to zero as t increases or is less than a given precision. The behavior of ε i depends on K and on the form of the master (x, y). In the harmonic state, the master time evolution can be described as x = A cos(wt − φ) (5) y = B cos(wt − ψ)

where A and B are the amplitudes while φ and ψ are the phases. Using the harmonic balance method [Hayashi, 1964; Nayfeh & Mook, 1979], we insert Eqs. (5) into the generating Eqs. (1). Then by equating the coefficients of sin wt and cos wt separately (assuming that the terms due to higher frequencies can be neglected), it is found that A and φ are the solutions of the following equation 9 2 6 3 β A + βF A4 + (F 2 + G2 )A2 − Eo2 = 0 16 2     (6)   G −1 φ = tan    F + 3 βA2  4 where D = (w22 − w2 )2 + w2 γ22 F = 1 − w2 −

λ1 λ2 w2 (w22 − w2 ) D

λ1 λ2 γ2 w 3 D Using the Newton–Raphson algorithm, one can find that the corresponding real positive solutions of Eq. (6) are G = γ1 w +

• for β = 0.1 with E0 = 0.5 A1 = 2.74 ,

φ1 = −1.05

A2 = 0.76 ,

φ2 = −0.21

A3 = 3.21 ,

φ3 = 1.59

• and for β = 1.32 with E0 = 0.2 A1 = 0.64 ,

φ1 = −0.47

A2 = 0.34 ,

φ2 = −0.23

A3 = 0.94 ,

φ3 = 1.48

where Ω21 = 1 + K + η 3 η = βA2 2 To discuss the stability of the synchronization process, we use the Floquet theory [Nayfeh & Mook, 1979; Hayashi, 1964]. We set the following rescalings τ1 = wt γ τ  1 1 (8) 2w γ τ  2 1 v(τ1 ) = ε2 exp 2w Equations (7) can be rewritten in the form u(τ1 ) = ε1 exp

d2 u + [δ11 + 2ε11 cos(4τ1 − 2φ)]u dτ12 dv (9) + δ12 exp(−ετ1 )v + c1 exp(−ετ1 ) = 0 dτ1 d2 v du + δ21 exp(ετ1 )u + δ22 v + c2 exp(ετ1 ) = 0 2 dτ1 dτ1 where the new parameters δij , ci and ε11 are given by   o n γ12 1 γ12 1 2 = δ11 = Ω1 − 1 + K + η − 2 w 4 w 2 4   λ2 1 η δ22 = 2 w22 − 2 δ11 = 2 ; w 4 w δ12 =

−λ1 γ2 ; 2w

δ21 =

γ1 λ2 ; 2w

ε=

1 (γ2 − γ1 ) 2w

λ1 −λ2 ; c2 = 2w 2w According to the Floquet theory [Nayfeh & Mook, 1979; Hayashi, 1964], the solutions of Eqs. (9) are c1 =

u(τ1 ) = exp(θ1 τ1 )α(τ1 ) =

n=+∞ X

αn exp(an τ1 )

n=−∞

v(τ1 ) = exp(θ2 τ1 )β(τ1 ) =

n=+∞ X

n=−∞

(10) βn exp(bn τ1 )

Synchronization of the Regular and Chaotic States of Electromechanical Devices

where an = θ1 + 2in, bn = θ2 + 2in, and the functions α(τ1 ) = α(τ1 + π) and β(τ1 ) = β(τ1 + π) replace the Fourier series. The quantities θ 1 and θ2 are two complex numbers, while αn and βn are real constants. Inserting Eqs. (10) in Eqs. (9) gives an infinite algebraic system which may have solutions if and only if the associated Hill’s determinant is set equal to zero. This condition defines the boundary dividing the parameters space in two domains: the stable and the unstable ones. Limiting ourselves to a Hill determinant containing the sixth rows and columns, we find that the boundary separating stability to instability domains are given by ∆(θ1 , θ2 ) = [(δ11 + θ12 )(δ22 + θ22 ) − (δ12 + c1 θ2 )(δ21 + c2 θ1 )] × {−(δ21 + c2 (θ1 + 2i))(δ12 + c1 (θ2 − 2i)) × {(δ11 + (θ1 − 2i)2 )(δ22 + (θ2 − 2i)2 )

− (δ12 + c1 (θ2 − 2i))(δ21 + c2 (θ1 + 2i))} − (δ22 + (θ2 + 2i)2 )(δ11 + (θ1 + 2i)2 )

× (δ12 + c1 (θ2 − 2i))(δ21 + c2 (θ1 + 2i)) + (δ22 + (θ2 − 2i)2 )(δ22 + (θ2 + 2i)2 )

× {(δ11 + (θ1 − 2i)2 )(δ11 + (θ1 + 2i)2 ) − ε211 }}

=0

(11)

with θ1 and θ2 given below. Since, we have  γ1   τ1 α(τ1 ) , ε1 (τ1 ) = exp θ1 − 2w  γ2   τ1 β(τ1 ) , ε2 (τ1 ) = exp θ2 − 2w Floquet theory states that the transition from stability to instability domains (π-periodic transitions) occurs at θ1 = γ1 /2w and θ2 = γ2 /2w. Thus replacing θ1 by γ1 /2w and θ2 by γ2 /2w in Eq. (11), we obtain an equation which helps to determine the range of K in which the synchronization process is stable. Considering the case with β = 0.1, it is found from Eq. (11) that the stability is achieved for K ∈ ] − 2.54; 0[U ]0; 4.18[U ]5.71; +∞[ if the slave comes from orbit A2 to follow the master to the orbit A3 . The results obtained from Eq. (11) are verified by a direct numerical simulation of Eqs. (2) with the fourth order Runge–Kutta algorithm. The master and the slave are initially launched with the initial conditions (x(0), x(0), ˙ y(0), y(0)) ˙ = (0.0, 0.01,

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0.0, 0.02) and (u(0), u(0), ˙ v(0), v(0)) ˙ = (2.0, 0.0, 0.1, 0.3), respectively. These sets of initial conditions lead respectively to periodic oscillations with amplitudes approximately equal to A 2 and A3 , respectively. Here, the slave is forced to come from A3 to A2 . For the case A2 to A3 , we just have to inverse the initial conditions. For each set of initial conditions, the synchronization is launched at T0 = 200 and K is varied until synchronization is achieved. With this numerical procedure, we find that the slave transition from A2 to A3 requires that K ∈]0.0808; 4.25[U ]4.38; +∞[. Despite the presence of a subdomain with K < 0 obtained from the analytical treatment (and which does not exist in the numerical results), we find that for K > 0, the agreement between the analytical and numerical results is quite good. In the case β = 1.32, our analytical treatment shows that for the transition A3 to A2 , we need K ∈]−1.2275; 0[U ]0; +∞[. From the numerical simulation of the differential equations (with the initial conditions (x(0), x(0), ˙ y(0), y(0)) ˙ = (0.0, 0.01, 0.0, 0.02) and (u(0), u(0), ˙ v(0), v(0)) ˙ = (1.1, 0.01, 0.1, 0.3), it is found that the transition A 3 to A2 occurs for K > 0.5. Here, we find that the agreement between the analytical and numerical results is not as good as in the case β = 0.1. This can be understood by the increase of the nonlinearity effects for β = 1.32. Indeed, in this case, the single harmonic response (5) may be questionable. But despite this fact, the analytical treatment gives a good indication on the boundary of K for synchronization to be achieved. The synchronization time is defined as Tsyn = tsyn − T0

(12)

where tsyn is the time instant at which the trajectories of the slave and the master are close enough to be considered as synchronized. Here, synchronization is achieved when the deviation ε 1 obeys the following synchronization criterium ε1 = |x − u| < h ,

∀ t > tsyn

(13)

where h is the synchronization precision or tolerance. Tsyn is plotted versus K in Figs. 2 and 3 for different values of the tolerance h. It is found that near the synchronization boundaries, T syn is very large. But as K moves from the boundaries, T syn decreases quickly and for large K, it attains a limiting small value depending on h.

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Fig. 2. Synchronization time Tsyn versus the coupling coefficient K for transitions A2 to A3 with β = 0.1, E0 = 0.5 and the precision h = 10−10 .

Fig. 3. Synchronization time Tsyn versus K for transitions A3 to A2 for β = 1.32, E0 = 0.2 and different values of the precision: (a) h = 10−10 , (b) h = 3.10−10 and (c) h = 5.10−10 .

3.2. Delayed synchronization (τ 6= 0) The purpose of delayed synchronization is to achieve the convergence of u(t) and v(t) towards x(t − τ ) and y(t − τ ), respectively. This is also an important goal to achieve in the field of electromechanical engineering. In this case ε i would rather

obey ε¨1 + γ1 ε˙1 + [1 + K + 3βx2τ ]ε1 + λ1 ε˙2    wτ  wt − wτ = −2E0 sin sin 2 2

(14)

ε¨2 + γ2 ε˙2 + w22 ε2 − λ2 ε˙1 = 0

εi is now submitted to an external sinusoidal

Synchronization of the Regular and Chaotic States of Electromechanical Devices

Fig. 4.

177

Analytical and numerical Kcr versus τ for the slave transition from A2 to A3 with β = 0.1, E0 = 0.5 and h = 10−2 .

Fig. 5. Analytical and numerical Kcr versus τ for the slave transition from A3 to A2 with β = 1.32, E0 = 0.2 and h = 10−2 .

excitation whose amplitude depends on E 0 and τ . When τ = nT (T = 2π/w is the period of the external excitation and n an integer), the external term vanishes and Eqs. (14) are no more different to Eqs. (7). Therefore, the whole analysis developed in the preceeding section remains valid. For τ 6= nT , the stability domain remains the same as obtained herebefore but the external excitation in-

duces steady state oscillations for ε i and the amplitudes of εi can be very large depending on E0 and τ . When K is varied, Ω1 is modified, and resonances may occur with the external and parametric excitations. The method of multiple time scales demonstrates that Ω1 should be far beyond w if we want to obtain small amplitudes for ε i . We can therefore discard the linear parametric excitation

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Fig. 6.

Fig. 7.

A chaotic phase portrait of the model with E0 = 22.0 and β = 1.32.

Synchronization time Tsyn versus K in the chaotic motion when τ = 0. Initial conditions are as in Fig. 3 and h = 10−4 .

and the variational equations (14) reduces to ε¨1 + γ1 ε˙1 + Ω21 ε1 + λ1 ε˙2  wτ   wτ  = −2E0 sin sin wt − 2 2 2 ε¨2 + γ2 ε˙2 + w2 ε2 − λ2 ε˙1 = 0

(15)

Thus, far from the resonance peaks, the amplitudes of the steady state oscillations of ε i are given

by the following expressions  wτ    2D1 E0 sin 2 cos wt − wτ √ ε1 (t) = 2 − ψ1 Do (16)  wτ    2D1 E0 λ2 w sin 2 cos wt − wτ √ ε2 (t) = 2 − ψ2 D1 Do where D1 = (w22 − w2 )2 + w2 γ22

Synchronization of the Regular and Chaotic States of Electromechanical Devices

The analytical expressions of the critical values Kcr are precisely found by comparing the amplitude of ε1 and the precision h, so that |ε1 | < h. This leads to

Do = [(Ω21 − w2 )D1 − λ1 λ2 w2 (w22 − w2 )]2 + [γ1 wD1 + λ1 λ2 w3 λ2 ]2

Q+ K > Kcr =

s

2D1 E0 sin wτ /2 h

179

2

D1

− (γ1 wD1 + λ1 λ2 γ2 w3 )2

−1−η

(17)

where Q = w2 D1 + λ1 λ2 w2 (w22 − w2 )

In Fig. 4, we have plotted Kcr versus τ for the slave transition from A2 to A3 for β = 0.1 and E0 = 0.5, and in Fig. 5, the same is done for the slave transition from A3 to A2 for β = 1.32 and E0 = 0.2. The synchronization tolerance used is h = 10−2 . The agreement between the analytical and numerical results is fairly good. The maximum value of Kcr occurs at τ = T /2. Let us note that Kcr increases when the tolerance h decreases for a given set of parameters and time delay.

4. Synchronization of the Chaotic States of Two Electromechanical Devices In this section, we extend the analysis of the synchronization process to chaotic states. The

objective of this extension is to find if our analytical procedure can also help to derive the stability boundary of the synchronization of two chaotic electromechanical devices with and without delay. As known for the Duffing equation with a hard spring, chaos appears for high values of the external excitation amplitude E0 [Parlitz & Lauterborn, 1985]. This is also the case in our system. We find that with the set of parameters used in this paper, chaos appears around E0 = 22.0. A chaotic phase portrait of the x component of the electromechanical system is shown in Fig. 6. To use the analytical procedure [namely Eqs. (11) and (17)], we need an approximate expression for the chaotic orbit. This is obviously not possible since a chaotic orbit is aperiodic and is composed of an infinite number of Fourier components. But we can however assume that in the chaotic state, the virtual orbit is that obtained using

Fig. 8. Synchronization time Tsyn versus the delay τ in the chaotic motion with the precision h = 10−1 , K = 200 and E0 = 22.

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Fig. 9. Analytical and numerical Kcr versus τ in the chaotic motion. The initial conditions and the parameters are as in Fig. 2, with the precision h = 10−1 and E0 = 22. Analytical results: lines, and numerical results: lines with dots.

the harmonic balance method. Thus with Eqs. (6), β = 1.32 and E0 = 22.0, the virtual harmonic orbit has the amplitude A = 2.89. With this approximation, Eq. (11) indicates that the two chaotic systems can be synchronized if K > 5.75. From the numerical simulation of Eqs. (2), we find that the chaotic electromechanical systems are synchronized if K ∈ [7.2; 9.8]U [10.8; 14.6]U ]15.6; +∞[. Here we find that the difference between the analytical and the numerical results is large. But this is not due to the basic idea leading to the stability equation (11), but to the fact that with β = 1.32 and E 0 = 22.0, our virtual orbit with amplitude A = 2.89 is a very poor approximation and it is expected that by including more harmonics, the gap between the analytical and numerical results could be reduced. Indeed, for other types of chaotic models where chaos does not need high nonlinear components to appear as in the case of soft Duffing oscillators, one finds good agreement between the analytical and the numerical results [Chembo Kouomou & Woafo, 2002]. In Fig. 7, we have plotted the variation of the synchronization time of two chaotic electromechanical systems with h = 10−4 . In Fig. 8, Tsyn versus τ is also reported while in Fig. 9, K cr versus τ , obtained from the analytical investigation and numerical simulation is also presented. For τ near nT , there is a good agreement between the analytical and numerical results.

5. Conclusion In this paper, we have considered the problem of synchronization without and with delay of two nonlinear electromechanical systems in regular and chaotic states. The electromechanical system consists of a classical electrical Duffing oscillator coupled magnetically to a linear mechanical oscillator. The stability boundaries of the synchronization process has been obtained using the Floquet theory and numerical investigation. The influence of the synchronization precision has also been investigated. It has been found that the minimal feedback coefficient Kcr which enables the synchronization is a periodic function of the synchronization delay between the master and the slave devices. Some interesting agreements have been obtained between the analytical and the numerical results.

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