Synchronization of nonlinearly coupled networks of Chua oscillators
Petro Feketa, Alexander Schaum, Thomas Meurer, Denis Michaelis, Karl-Heinz Ochs
aa r X i v : . [ c s . S Y ] M a r Synchronization of nonlinearly couplednetworks of Chua oscillators ⋆ P. Feketa ∗ A. Schaum ∗ T. Meurer ∗ D. Michaelis ∗∗ K. Ochs ∗∗∗
Chair of Automatic Control, Christian-Albrechts-University Kiel,24148 Kiel, Germany (e-mail: { pf,alsc,tm } @tf.uni-kiel.de). ∗∗ Institute for Digital Communications Systems, Ruhr-UniversityBochum, 44801 Bochum, Germany (e-mail: { dennis.michaelis,karlheinz.ochs } @ruhr-uni-bochum.de) Abstract:
The paper develops new sufficient conditions for synchronization of a network of N nonlinearly coupled Chua oscillators interconnected via the first state coordinate only. Thenonlinear coupling strength is governed by a function residing within a sector, i.e. it is boundedfrom above and below by linear functions. The derived sufficient conditions provide a trade-offbetween the characteristics of the sector and the interconnection topology of the network toguarantee the synchronization of the oscillators. Keywords:
Chua oscillators, synchronization, interconnected systems, chaotic behaviour,nonlinear systems.1. INTRODUCTIONControl and synchronization of nonlinear and chaoticsystems have been intensively studied during the lastdecades (Pecora et al., 1997; Pecora and Carroll, 2015;Azar et al., 2017; Ochs et al., 2018; Raychowdhury et al.,2019). In particular, chaos synchronization has many po-tential applications in secure communication (Yang andChua, 1997; Tse and Lau, 2003; Argyris et al., 2005),laser physics (Ohtsubo, 2002), chemical reactor process (Liet al., 2004), biomedical engineering (Strogatz, 2018). Inthis context, the Chua circuit appeared to be one of themost interesting objects for research since it exhibits ex-tremely rich dynamical behavior and variety of bifurcationphenomena despite its structural simplicity. Investigationof synchronization abilities of coupled Chua oscillatorsmay help to understand complex dynamical phenomenaarising in networks of chaotic systems of more generaltypes (Wu and Chua, 1994).Synchronization of two linearly coupled Chua oscillatorshas been studied in Wu and Chua (1994); Wang and Liu(2006); Bowong and Tewa (2009); Wang et al. (1999);Zheng et al. (2002); Chen et al. (2010, 2012). In Wuand Chua (1994) it is conjectured that synchronizationbetween two chaotic Chua circuits can be achieved byusing the second state as the feedback variable for suf-ficiently large coupling constant. This conjecture has beenanalytically proven in Zheng et al. (2002) and Wang et al.(1999) utilizing Lyapunov-type arguments and a novelobserver design methodology. Using the LaSalle invari-ance principle, a linear low-gain controller that exploitsa single variable feedback (first state coordinate x ) has ⋆ This work has been supported by the Deutsche Forschungsge-meinschaft (DFG) within the research unit FOR 2093: Memristivedevices for neuronal systems (subproject C3: Synchronization ofMemristively Coupled Oscillator Networks – Theory and Emulation). been constructed in Chen et al. (2010). The robustnessof this controller with respect to the perturbations of theparameters of Chua oscillators has been studied in Chenet al. (2012). Practical synchronization of chaotic sys-tems with uncertainties via adaptive coupling mechanismhas been investigated in Bowong and Tewa (2009). Thesynchronization of two identical chaotic and hyperchaoticsystems with different initial conditions has been studiedin Wang and Liu (2006).A graph-spectral approach for the synchronization of anetwork of N ∈ N resistively coupled nonlinear oscillatorshas been proposed in Wu and Chua (1995). The sufficientconditions for synchronization have been derived from theconnectivity graph, which describes how the oscillatorsare connected. An upper bound on the coupling conduc-tance required for synchronization for arbitrary graphs hasbeen obtained. Later, the synchronization of networks of N ∈ N nonlinear dynamical systems based on a stateobserver design approach has been studied in Jiang et al.(2006). Unlike the common diagonally coupling networks,see (Wang and Chen, 2003; L¨u et al., 2004), where full statecoupling is typically needed between two nodes, in Jianget al. (2006) it is suggested that only a scalar couplingsignal is required to achieve network synchronization. Thepresented approach has been applied to the chaos syn-chronization problem in two typical dynamical networkconfigurations: global linear coupling and nearest-neighborlinear coupling, with each node being a modified Chua’scircuit. Finally, the problem of syhcnronizing an arbitrarysubset of the nodes in the oscillatory network at fixedcoupling stregth has been tackled in Gambuzza et al.(2019) by creation of appropriate network of additionalinterconnecting links between oscillators.In the current paper, the sufficient conditions for thesynchronization of the network of N ∈ N Chua oscillatorsinterconnected with the static nonlinear coupling via therst state coordinate only are derived. These conditionsprovide a trade-off between characteristics of the con-nectivity graph and properties of the nonlinear couplingfunction to achieve synchronization.The rest of the paper is organized as follows. In Section 2,the network under consideration is defined and the mainproblem of synchronization is formulated. In Section 3, themain result of the paper that is sufficient conditions for thesynchronization of N ∈ N coupled Chua oscillators withstatic nonlinear coupling satisfying a so-called sector con-dition is derived. Also, numerical examples to illustrate theusage of the derived conditions are provided. In Section 4,a corollary of the main theorem for the case of two linearlycoupled Chua oscillators is discussed and compared withthe existing results in the literature. Finally, short conclu-sion and discussion in Section 5 complete the paper.2. PROBLEM STATEMENTConsider the extended Chua circuit system˙ x = α ( − x + x − f ( x ))+ u, x (0) = x (1a)˙ x = x − x + x , x (0) = x (1b)˙ x = − βx − γx , x (0) = x (1c) y = x (1d)with scalar piecewise linear function f ( x ) = ax + 12 ( b − a )( | x + 1 | − | x − | )and parameters α, β > , γ ≥ a < b <
0. The systemallows for vector-valued formulation˙ x = Ax + b u + f ( x ) , x (0) = x (2a) y = c T x (2b)with the state x ( t ) ∈ R , x T = [ x x x ], externalinput u ( t ) ∈ R , which will be later used to interconnectChua oscillators, and the matrix A and vectors f , b , c given by A = " − α α − − β − γ , f ( x ) = " − αf ( x )00 , b = c = " . Now, consider a network of N ∈ N nodes described by agraph Γ = ( V, E ) with vertex (node) set V and edge set E ,so that | V | = N . Let the associated adjacency matrix begiven by A = { α ij } i,j =1 ,...,N , α ij ∈ { , } with zero maindiagonal.Then, applying output feedback u i = − N X j =1 α ij k ( y i − y j ) , i = 1 , . . . , N with an arbitrary nonlinear locally Lipschitz continuouscoupling function k : R → R , the dynamics of N ∈ N coupled Chua oscillators can be written as˙ x i = Ax i − b N X j =1 α ij k ( y i − y j ) + f ( x i ) , x i (0) = x i y i = c T x i , (3)for i = 1 , . . . , N . For any given ξ ∈ R N let x =( x , . . . , x N ) : R → R N denote a solution to (3) satisfyingthe initial condition x (0) = ξ . The Lipschitz continuity of the right-hand side of (3) guarantees the existence anduniqueness of the solution for any initial value ξ ∈ R N .Associated to this network consider the relative synchro-nization errors with respect to the node 1 e j = x j − x , j = 1 , . . . , N. The relative errors e ij between arbitrary nodes i and j canbe expressed using the relative errors e j and e i e ij = x i − x j = x i − x − ( x j − x ) = e i − e j . Accordingly, instead of analyzing N ( N − relative errors e ij between connected nodes, it is sufficient to considerthe behavior of the N − e j , j = 2 , . . . , N .The problem addressed in the sequel consists in provid-ing sufficient conditions on the system parameters, thenonlinear coupling function k and the network topologywhich ensure the synchronization of N ∈ N coupled Chuaoscillators, i.e., the global convergence of the norms oferrors e j to zero:lim t →∞ k e j ( t ) k = 0 , j = 2 , . . . , N.
3. SYNCHRONIZATION CONDITIONSLet { x } i denote the i -th component of the vector x ∈ R , i = 1 , ,
3. By splitting the state vector according to (cid:20) z i ζ i (cid:21) = { x i } (cid:20) { x i } { x i } (cid:21) , rewrite the dynamics (3) in z − ζ coordinates˙ z i = − αz i − αf ( z i ) + [ α ζ i (4a) − N X j =1 α ij k ( z i − z j )˙ ζ i = (cid:20) (cid:21) z i + (cid:20) − − β − γ (cid:21)| {z } =: A ζ i , (4b)where matrix A is Hurwitz with eigenvalues λ , fulfilling R ( λ , ) = − µ < − µ = − γ R r (1 + γ ) − ( γ + β ) ! , where R ( λ ) denotes the real part of λ ∈ C .Using the notation for the relative errors (cid:20) e i η i (cid:21) = (cid:20) z i − z ζ i − ζ (cid:21) and taking into account that e ≡ η ≡ , thesynchronization error dynamics can be written as˙ e i = − αe i − α ˜ f ( e i ) + [ α η i (5a) − N X j =1 (cid:16) α ij k ( e i − e j ) − α j k ( − e j ) (cid:17) ˙ η i = (cid:20) (cid:21) e i + A η i (5b)with initial conditions e i (0) = e i, and η i (0) = η i, , i = 2 , . . . , N , and˜ f ( e i ) = f ( z + e i ) − f ( z ) . (6)n the following subsection, sufficient conditions for globalasymptotic stability of zero solution to the error dynamicssystem (5) will be derived. For this purpose additionalsector requirement on the coupling function k will beimposed. Coupling k : R → R is continuous oddfunction and there exist two constants k ≥ k ≥ e ∈ R : k ( e ) e ≥ , k ( − e ) = − k ( e ) , k | e | ≤ | k ( e ) | ≤ k | e | . (7)Introduce the function˜ k ( e ) = k ( e ) − k e for all e ∈ R . (8)From (7) and (8) it follows that functions k and ˜ k liein the first-third and the second-fourth quadrant pairsrespectively. Lemma 2.
Let Assumption 1 hold. Then (cid:12)(cid:12)(cid:12) ˜ k ( e i − e j ) + ˜ k ( e j ) (cid:12)(cid:12)(cid:12) ≤ ( k − k ) | e i | + ( k − k ) | e j | (9)for all e i , e j ∈ R . Proof.
The left-hand side of (9) can be rewritten as (cid:12)(cid:12)(cid:12) ˜ k ( e i − e j ) + ˜ k ( e j ) (cid:12)(cid:12)(cid:12) = | k ( e i − e j ) − k ( e i − e j )+ k ( e j ) − k e j | = |− k e i + k ( e j ) − k ( e j − e i ) | Consider the cases of positive and negative signs of theterms e i , e j , and e j − e i correspondingly. First, let e i ≥ e j ≥ e j − e i ≥
0. Then, − k e i + k e j − k ( e j − e i ) ≤ − k e i + k ( e j ) − k ( e j − e i ) ≤ − k e i + k e j − k ( e j − e i ) , which yields | − k e i + k ( e j ) − k ( e j − e i ) | ≤ ( k − k ) | e j | . (10)Let e i ≥ e j ≥ e j − e i <
0. Then, − k e i + k e j − k ( e j − e i ) ≤ − k e i + k ( e j ) − k ( e j − e i ) ≤ − k e i + k e j − k ( e j − e i ) ≤ , which yields | − k e i + k ( e j ) − k ( e j − e i ) | ≤ ( k − k ) | e i | . (11)Let e i ≥ e j < e j − e i <
0. Then, − k e i + k e j − k ( e j − e i ) ≤ − k e i + k ( e j ) − k ( e j − e i ) ≤ − k e i + k e j − k ( e j − e i )which yields | − k e i + k ( e j ) − k ( e j − e i ) |≤ ( k − k ) | e i | + ( k − k ) | e j | . (12)Similarly, one may check that for the rest combinations ofthe signs of e i , e j , and e j − e i one of the inequalities (10),(11), (12) holds. Finally, combining (10), (11), (12), obtainthat (cid:12)(cid:12)(cid:12) ˜ k ( e i − e j ) + ˜ k ( e j ) (cid:12)(cid:12)(cid:12) ≤ ( k − k ) | e i | + ( k − k ) | e j | . This completes the proof. (cid:3)
Denote the degree of node i by κ i = N P j =1 α ij and rewritethe dynamics (5a) as follows: ˙ e i = − αe i − α ˜ f ( e i ) + [ α η i − N X j =1 (cid:16) α ij k ( e i − e j ) + α j k ( e j ) (cid:17) = − αe i − κ i k e i − α ˜ f ( e i ) + [ α η i − N X j =1 (cid:16) α ij (cid:0) k ( e i − e j ) − k ( e i − e j ) (cid:1) + α j k ( e j ) − α ij k e j (cid:17) . (13)Introduce the residual connectivity coefficients with re-spect to the first node α j = α ij + ˜ α ij , ˜ α ij = α j − α ij = , ( i, j ) , (1 , j ) ∈ E, , ( i, j ) / ∈ E, (1 , j ) ∈ E, − , ( i, j ) ∈ E, (1 , j ) / ∈ E, , ( i, j ) , (1 , j ) / ∈ E, respectively. The expression inside the sum of (13) can berewritten as α ij (cid:16) k ( e i − e j ) − k ( e i − e j ) (cid:17) + α j k ( e j ) − α ij k e j = α ij (cid:16) k ( e i − e j ) − k ( e i − e j ) (cid:17) + ( α ij + ˜ α ij ) k ( e j ) − α ij k e j = α ij (cid:16) ˜ k ( e i − e j ) + ˜ k ( e j ) (cid:17) + ˜ α ij k ( e j ) . With these definitions the dynamics of e i can be writtenequivalently as˙ e i = − αe i − κ i k e i − α ˜ f ( e i ) + [ α η i − N X j =1 (cid:16) α ij (cid:0) ˜ k ( e i − e j ) + ˜ k ( e j ) (cid:1) + ˜ α ij k ( e j ) (cid:17) . (14)Following the reasoning in (Schaum, 2018) consider theimplicit solution of the preceding ODEs (5b), (14) givenby e i ( t ) = e − ( α + κ i k ) t e i, + Z t e − ( α + κ i k )( t − τ ) (cid:18) − α ˜ f ( e i ( τ )) + [ α η i ( τ ) − N X j =1 (cid:16) α ij (cid:16) ˜ k ( e i − e j ) + ˜ k ( e j ) (cid:17) + ˜ α ij k ( e j ) (cid:17) (cid:19) dτ η i ( t ) = e A t η i, + Z t e A ( t − τ ) (cid:20) (cid:21) e i ( τ ) dτ. (15)Since function f is Lipschitz continuous with Lipschitzconstant | a | it holds that | ˜ f ( e i ) | = | f ( z + e i ) − f ( z ) | ≤ | a || e i | (16)for any e i ∈ R . Taking norms on both sides of (15), apply-ing the triangle inequality, and accounting for Lemma 2and inequality (16) the estimates e i ( t ) | ≤ e − ( α + κ i k ) t × (cid:18) | e i, | + Z t e ( α + κ i k ) τ (cid:16) α | a || e i ( τ ) | + α k η i ( τ ) k + N X j =1 ( α ij (( k − k ) | e i ( τ ) | + ( k − k ) | e j ( τ ) | )+ | ˜ α ij | k | e j ( τ ) | ) (cid:17) dτ (cid:17) k η i ( t ) k ≤ e − µ t (cid:18) k η i, k + Z t e µ τ | e i ( τ ) | dτ (cid:19) hold. Define the right-hand sides of the preceding inequal-ities as σ i and χ i , i.e., σ i = e − ( α + κ i k ) t × (cid:18) | e i, | + Z t e ( α + κ i k ) τ (cid:16) α | a || e i ( τ ) | + α k η i ( τ ) k + N X j =1 ( α ij (( k − k ) | e i ( τ ) | + ( k − k ) | e j ( τ ) | )+ | ˜ α ij | k | e j ( τ ) | ) (cid:17) dτ (cid:17) χ i = e − µ t (cid:18) k η i, k + Z t e µ τ | e i ( τ ) | dτ (cid:19) so that | e i ( t ) | ≤ σ i ( t ) and k η i ( t ) k ≤ χ i ( t ) for all t ≥ | e i (0) | = σ i (0) and k η i (0) k = χ i (0) for all i = 2 , . . . , N .Since e ( t ) = 0 for all t ≥
0, let σ ( t ) ≡ | e i ( t ) | ≤ σ i ( t ) holds for all i = 1 , . . . , N . The time derivatives of σ i and χ i can be estimated by˙ σ i ( t ) = − ( α + κ i k ) σ i ( t ) + α | a || e i ( t ) | + α k η i ( t ) k + N X j =1 (cid:0) α ij (( k − k ) | e i | + ( k − k ) | e j | )+ | ˜ α ij | k | e j ( t ) | (cid:1) ≤ − ( α − α | a | + κ i k ) σ i ( t ) + αχ i ( t )+ N X j =1 ( | α ij | ( k − k ) + | ˜ α ij | k ) σ j ( t )˙ χ i ( t ) = − µ χ i ( t ) + | e i ( t ) |≤ − µ χ i ( t ) + σ i ( t ) . Taking into account that σ ( t ) ≡
0, the precedingdynamics can be written in vector notation as ddt (cid:20) σ i χ i (cid:21) ≤ (cid:20) − ( α − α | a | + κ i k ) α − µ (cid:21) (cid:20) σ i χ i (cid:21) + N X j =2 ( | α ij | ( k − k ) + | ˜ α ij | k ) σ j . (17)Introducing z = [ σ · · · σ N χ · · · χ N ] T , inequality (17)can be written as˙ z ≤ (cid:20) − ( α − α | a | ) I − k K + ( k − k ) A + k A α II − µ I (cid:21)| {z } =: M z (18)with matrices A = | α | · · · | α N || α | | α N | · · · · · · , A = | α − α | · · · | α N − α N || α − α | | α N − α | · · · · · · , K = diag { κ , . . . , κ N } , and ( N − × ( N − I . Sufficient conditions for the synchroniza-tion of the entire network can be formulated in terms ofthe eigenvalues of the matrix M . Theorem 3.
Let Assumtion 1 hold and matrix M definedin (18) be Hurwitz. Then, the norm of errors between thestates of Chua oscillators (3) converges exponentially tozero.The differential inequality (17) for σ i contains both stabi-lizing and destabilizing terms, which have physical inter-pretation and can be used as guidelines for the couplingdesign. In particular, the stabilizing term becomes largerwith the growth of the lower bound k of the nonlinearcoupling. The destabilizing terms vanish when the lowerbound k approaches the upper bound k and the inter-connection graph is fully connected (i.e. the coefficients˜ α ij are zero). These effects can be reached by choosing thecoupling k with a sufficiently large lower bound k . Consider a fully connected network of N =20 Chua oscillators (3) with parameters α = 15 . β =25 . γ = 0, a = − . b = − . k ( e ) = 3 e + arctan ( e ) for all e ∈ R . (19)The chosen parameters correspond to the chaotic behaviorof each oscillator (Pivka et al., 1994). Oscillators’ trajecto-ries converge to an attractor that has a double scroll shapein three dimensional state space (see Fig. 1).Nonlinear coupling strength (19) satisfies the sector con-dition (7) with constants k = 3 and k = 4. For thechosen parameters and the interconnection coupling (19)the matrix M defined in (18) reads M = (cid:20) − . I . II − . I (cid:21) + (cid:20) ⊤ − I 00 0 (cid:21) , where denotes zero ( N − × ( N − denotes ( N − · · · ⊤ . By directcalculation one may check that all eigenvalues lie in theopen left half-plane R ( λ ( M )) ∈ [ − . , − . . Hence, from Theorem 3 it follows that the oscillators aresynchronized. The state evolution for N = 20 oscillatorsis shown in Fig. 2. x x x Fig. 1. Double scroll attractor for the coupled chaotic Chuaoscillators from Example 1.Fig. 2. Time evolution of N = 20 coupled chaotic Chuaoscillators with interconnection coupling (19).4. SYNCHRONIZATION OF TWO OSCILLATORSIn this section a corollary from Theorem 3 for the case oftwo Chua oscillators connected with linear coupling k ( e ) = ke for all e ∈ R (20)with coupling constant k > k = k = k . Corollary 4.
In the case of two Chua oscillators (3) con-nected via the first state variable with linear coupling (20)the condition for synchronization reads k > α (cid:18) | a | + 1 µ − (cid:19) , (21)i.e., the synchronization emerges if the coupling betweenthe first state of each oscillator is sufficiently strong. Proof.
For the case of two Chua oscillators the matrix M from (18) is a 2 × M = (cid:20) − ( α − α | a | + k ) α − µ (cid:21) . Due to the Routh-Hurwitz criterion, the roots of thecorresponding characteristic polynomial λ + λ ( µ + α − α | a | + k ) + ( α − α | a | + k ) µ − α = 0are in the open left half-plane if and only if (cid:26) µ + α − α | a | + k > , ( α − α | a | + k ) µ − α > , which yields k > α (cid:18) | a | + 1 µ − (cid:19) . This completes the proof. (cid:3)
The problem of the synchronization of two identical Chuaoscillators via the first state variable by linear couplingaddressed in Corollary 4 has been also successfully tackledin Chen et al. (2010, 2012) for the case of γ = 0 byemploying the Lyapunov method. The constraint on thecoupling constant obtained in Chen et al. (2010, 2012)reads as k > α | a | , which is less a conservative condition compared to (21).However, the applicability of Theorem 3 is more generaleven in the case of two oscillators due to the possibility of γ = 0. Besides this, the approach proposed in the presentpaper handles the case of an arbitrary number of Chuaoscillators with an arbitrary interconnection topology andnonlinear coupling functions. Example 2.
Consider two Chua oscillators with param-eters α = 10, β = 15, γ = 0 . a = − . b = − . − µ = − .
12 + R r . − ! = − . . From Corollary 4, the coupling constant k should bechosen larger than k > (cid:18) .
31 + 21 . − (cid:19) ≈ . . The state evolution of both oscillators for k = 21 . N ∈ N Chua os-cillators which are coupled via the first state coordinatewith static nonlinear coupling is studied. Sufficient condi-tions for the synchronization are formulated in terms ofthe eigenvalues of the auxiliary matrix M , whose entriesrepresent the interplay between the parameters of theoscillators, the interconnection topology of the networkand the characteristics of the nonlinear coupling function.The extension of the derived conditions to the class ofdynamically coupled Chua oscillators will allow for anal-ysis of wide classes of memristive networks and, moregenerally, time-varying interconnections which are capa-ble to model the bio-inspired plasticity phenomenon. An-other interesting research direction is the study of multi- x x t -10010 x Fig. 3. Time evolution of two coupled chaotic Chua oscil-lators with interconnection gain k = 21 . Nature , 438(7066), 343.Azar, A.T., Vaidyanathan, S., and Ouannas, A. (2017).
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