Stability of synchronized network of chaotic electromechanical devices with nearest and all-to-all couplings G. S. Mbouna Ngueuteu1, R. Yamapi2,∗ and P. Woafo1 1Laboratory of Modelling and Simulation in Engineering and Biological Physics, Uuniversity of Yaound´ e I, CAMEROON 2 Department of Physics, Faculty of Science, University of Douala , CAMEROON ∗Present Institution: CNR-INFM SUPERMAT Laboratory, Dipartimento di Fisica, Universit´ a degli studi di Salerno, ITALY.

Email:[email protected] International Workshop and Seminar on Bio-inspired Complex Networks in Science and Technology From Topology to Structure and Dynamics, Max Planck Institute for the Physics of Complex Systems, Dresden, Germany, April 14 - May 09, 2008

Abstract In this poster, a shift-invariant set of N mutually (nearest-neighbour or all-to-all) coupled chaotic electromechanical devices is analytically and numerically investigated. The stability of synchronization state is undertaken using the Master Stability Function (MSF) approach. The emanating properties of this method make it possible to have a general study of the network dynamics, and to explain de-synchronization phenomena appearing in the synchronization stability parametric areas.

1

• Synchronization is achieved just after large transitory time, such that one may practically considers that there is no synchronization.

Introduction

Complex networks have provided a challenging framework for the study of synchronization of dynamical units, based on the interplay between complexity in the overall topology and local dynamical properties of the coupled units. In particular, the effects of synchronization in systems of coupled oscillators nowadays provide a unifying framework for different phenomena observed in nature [1,2]. In the study of synchronization, it is important to determine conditions for the stability of the synchronous behavior for a generic network topology with a generic coupling configurations. Consequently, increasingly powerful mathematical methods are being developed to seek the potential conditions for realization of the most interesting form of dynamical behavior that can arise in these networks: synchronization. These methods reverse this centre question by studying when a synchronous state is stable, in terms of coupling scheme or strengths. The Master Stability Function approach was introduced to address this question for arrays of coupled oscillators [3], it was also extended to complex networks of dynamical systems coupled with arbitrary topologies [4-10]. In this poster, one considers the stability of a synchronized network of N chaotic electromechanical devices (EMD). Section II describe the network under nearest-neighbour coupling and all-to-all coupling configurations. The analytical investigation of the stability of synchronization states is tackled in Section III, while Section IV deals with numerical computation of the Master Stability Function (MSF). We end the poster with conclusions.

2

The physical system

Figure 2: (a) Each electromechanical moving coil transducer and (b) each coupling elements link x¨i + γ1x˙ i + xi + β1x3i + β2x5i + λ1dotyi = E0 cos wt + Q1

i = 1, 2, ..., N

y¨i + γ2y˙ i + w22yi − λ2x˙ i = 0

i = 1, 2, ..., N

(2)

The voltage Vi depends on the type of coupling configurations.

2.1

N X

i = 1, 2, ..., N

The interest devoted to these systems is due to their possible applications in electromechanical engineering (As we have recently observed in ref.[11]). • In industry, these synchronized networks of EMD operate in parallel mode for common loading in assembly line works and in manufacturing processes: i.e a series of perforation devices or a series of macro/micro-EMD uses for cutting, drilling and other machining work. • The chaotic state is particularly of interest when the network of mutually coupled chaotic EMD is used for industrial operations such as the mixing of different liquids, chemicals or powders.

2.3

(7)

Secondly, the unstable synchronization (US) domain in which Λ ≤ 0. Two different dynamical states can develop themselves in the network. • Clusters synchronization sub-domain: certain transverse modes are stable while other ones are unstable ( λk < 0, for some values of k). • Spatiotemporal chaos sub-domain: all these modes are unstable (λk > 0, ∀k),

Chaotic state in the electromechanical unit

Chaos appeared in the EMD with the parameters: β1 = 0.9; β2 = 0.1; γ1 = 0.2; γ2 = 0.1; λ1 = 0.01; λ2 = 0.05; w2 = 1.2 : w = 0.85; E0 = 18.4. Fig. 4 shows the Poincar´e cross section in the space [11].

Figure 6: (a) The MSF in the parametric plane (N, Q1). (b) The corresponding stability: (bold line) stability threshold Λ = 0; the line displays the isoline Λ = −0.02, the dark domain is the CS area (obtained numerically). For Q2 = 0.02.

Figure 3: The Poincar´e cross section in the space (x, x). ˙

3

Stability of synchronization

This model is interesting if the resulting dynamical state is stable,depending on Ki, Qi. Introducing uν = x˙ ν , vν = =˙ y˙ ν , the resulting dynamical state is stable when it lies within the synchronization manifold S defined by (x1, u1, y1, v1) = ... = (xN , uN , yN , vN ) = (xs, us, ys, vs). The stability of S is investigate using the MSF approach [2,3]. Therefore, Eqs. (8) can be rewritten in the following single vector form N X ˙ i = F(Xi, t) + (8) X Gij H(Xj), i = 1, 2, ..., N, j

The nearest-neighbour coupling scheme

Figure 5: The stability map in the plane (K1, K2) : (bold line) stability threshold Λ = 0; the line displays the isoline Λ = −0.02, the dark domain is the complete synchronization area (obtained numerically). (a) N=10 and (b) N=20. The parameters used are those on Fig. 4

(x˙ j − x˙ i)

j=1,j6=i

(1)

The electrical part consists of a RLC circuit (with VC,i(qi) = C10 qi + a3qi3 + a5qi5), connected in series with a sinusoidal voltage source e(τ ) = v0 cos Ωtau). Let Vi be the voltage of the ith EMD, Vij be the voltage of the branch coupling between the ith and jth systems, Iij the current crossing this branch, and qij be the coupling capacitor charge. Then, 1 Vi = L¨ qi + Rq˙i + qi + a3qi3 + a5qi5 + lB z˙iv0 cos Ωτ, C0

(xj − xi) + Q2

j=1,j6=i

The network of N coupled identical EMD with the nearest-neighbour and all-to-all coupling configurations is shown, resp. on Figs. 1 and 2. Each EMD presented on Fig. 3 consists of the mechanical part coupled magnetically to the electrical part (see Ref. [11] for more information). According to the second Newton law of dynamics, the mechanical part of each em is described by z¨i + λz˙i + kzi − lB q˙i = 0,

N X

P Gij depend to the coupling scheme. Due to ( j Gij = 0) and to the fact that H is the same for all units, S is an invariant set. Therefore, stability of the synchronous state Xs=[xs, us, ys, vs]T , reduces to taking care of the system’s dynamical properties along directions in phase’s space that are transverse to the synchronization manifold N according to X ²˙i = DXs²i + DH(Xs)²j , (9)

4.2

The all-to-all coupling scheme

Fig. 6 Λ(N, Q1) and the stability map in the (N, Q1) plane. In the all-to-all coupling scheme, all the transverse modes degenerate into only one. We find theoretically that the network of electromechanical systems will be able to exhibit only two main dynamical states; • spatiotemporal chaos when the degenerated eigenmode is unstable, • complete synchronization when this mode is stable. The MSF’s behaviour (see Fig. 6(a)) once more enables us to explain the presence of the no synchronization sub-domain appearing in the stable synchronization domain of Fig. 6(b). According to this figure, Λ admits a peak whose width is spread out in this sub-domain

j=1

Figure 0: The schema of the nearest-neighbour mutually coupled electromechanical systems The Kirchhoff’s voltage and current laws imply that Vi =

1 1 0 0 (q − q ) + R ( q ˙ − q ˙ ) + (q − q ) + R (q˙i − q˙i−1) i−1 i i−1 i i i+1 0 0 C C

(3)

Therefore, the network of N nearest-neighbour coupled em is described by x¨i + γ1x˙ i + xi + β1x3i + β2x5i + λ1y˙ i = K1(xi+1 − 2xi + xi−1) + K2(x˙ i+1 − 2x˙ i + x˙ i−1) i = 1, 2, ..., N y¨i + γ2yi + w22yi − λ2x˙ i = 0 (4)

2.2

The all-to-all coupling scheme

where ²i is the small deviation from S. γi and ω ¯ i are eigenvaluesP and orthonormal eigenN −1 vectors of G. The arbitrary state ²k can be written as ²k = N ¯ k ηk (t), with k=0 ω ηk (t) = (ηk1(t), ηk2(t), ηk3(t), ηk4(t)). One finally obtains the following set of N variational equations (10) η˙ k = [DF(Xs) + γk E]ηk , k = 0, 1, 2, ..., N − 1. © ª 2 kπ Where γk = −4 sin N for the nearest-neighbour coupling [3] and γ0 = 0, γk = −N (k 6= 0) for the all-to-all coupling [4]. S is stable if all the Lyapunov exponents λmax < 0[3]. The MSF Λ, gives informak tion about the general dynamics of the network. Nevertheless, the universal stability standard Λ < 0, implies that any deviation from S should decay and vanish as time goes on.

4

Numerical simulations and results

• Analytically, we compute the Master Stability Function, • Numerical Results: we assume that complete synchronization is achieved in the network for |xi(t) − xj (t)| −3 (11) < h = 10 , ∀(i, j) 2 (N − 1)

Figure 7: The stability map in the parametric plane (Q1, Q2): (bold line) stability threshold Λ = 0, the dark domain is the complete synchronization area (obtained numerically). (a) N = 10 and (b) N = 30.

5

Conclusion

In this poster communication, the stability of the synchronization states in the network of coupled chaotic EMD have studied. The MSF approach has shown to be powerful in the stability of the synchronization process and consequently in the determination of the coupling strengths range which give place to certain synchronous states: spatiotemporal chaos; cluster synchronization and complete synchronization. This method has been necessary to shed light on the explanation of the de-synchronization phenomena that occur in stable synchronization areas. Numerical procedures have confirmed the semi-numerical results of the analytical method.

References

Figure 1: The schema of the all-to-all mutually coupled electromechanical systems The Kirchhoff’s voltage law, applied to the ith em, yields the following equation Vi + Vi1 + Vi2 + ... + ViN = 0,

(5)

where Vij = C10 qij + R0Iij . A rather obvious handling of Kirchhoff’s current law shows that Iij = Ii − Ij , and therefore qij = qi − qj . In this case, Eq. (5) yields N N X 1 X (q˙j − q˙i), Vi = 0 (qj − qi) + R0 C j=1,j6=i

(6)

j=1,j6=i

Equations (2) and (6) lead to the equation describing the electrical part of each EMD of the network. Therefore, the shift-invariant set of N nearest-neighbour coupled em is described by the following set of coupled non-dimensional equations

Figure 4: (a) The MSF in the plane (N, K1): The region of stable synchronization is the region under the plane Λ = 0 displayed by the grid and the solid line is the stability threshold. (b) The stability map: (both line) stability threshold Λ = 0; the dark domain is the CS area (obtained numerically). For K2 = 1.

4.1

The nearest-neighbour coupling scheme

Fig. 5(a) presents Λ(N, K1)’s variations and Fig. 5(b) the stability map in the plane (N, K1). We can toughly distinguish two domains in that plane. Firstly, one has the stable synchronization (SS) domain in which Λ < 0. Numerically, synchronization condition does not always guarantee the realization of synchronous states. Two situations can be observed during numerical simulations: • An intermittent synchronization is observed, (possible sustainable chaos t %),

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