Adaptive synchronization in delay-coupled networks of Stuart-Landau oscillators
Anton Selivanov, Judith Lehnert, Thomas Dahms, Philipp Hövel, Alexander L. Fradkov, Eckehard Schöll
aa r X i v : . [ n li n . AO ] J u l Adaptive synchronization in delay-coupled networks of Stuart-Landau oscillators
Anton Selivanov, Judith Lehnert, Thomas Dahms, Philipp H¨ovel, Alexander Fradkov, and Eckehard Sch¨oll ∗ Department of Theoretical Cybernetics, Saint-Petersburg State University, Saint-Petersburg, Russia Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstr. 36, 10623 Berlin, Germany (Dated: November 3, 2018)We consider networks of delay-coupled Stuart-Landau oscillators. In these systems, the couplingphase has been found to be a crucial control parameter. By proper choice of this parameter one canswitch between different synchronous oscillatory states of the network. Applying the speed-gradientmethod, we derive an adaptive algorithm for an automatic adjustment of the coupling phase suchthat a desired state can be selected from an otherwise multistable regime. We propose goal functionsbased on both the difference of the oscillators and a generalized order parameter and demonstratethat the speed-gradient method allows one to find appropriate coupling phases with which differentstates of synchronization, e.g., in-phase oscillation, splay or various cluster states, can be selected.
PACS numbers: 05.45.Xt, 02.30.Yy, 89.75.-kKeywords: synchronization, delay, networks, Hopf normal forms
I. INTRODUCTION
The ability to control nonlinear dynamical systems hasbrought up a wide interdisciplinary area of research thathas evolved rapidly in the last decades [1]. In particular,noninvasive control schemes based on time-delayed feed-back [2–4] have been studied and applied to various sys-tems ranging from biological and chemical applications tophysics and engineering in both theoretical and experi-mental works [5–12]. Here we propose to use adaptivecontrol schemes based on optimizations of cost or goalfunctions [13–15] to find appropriate control parameters.Besides isolated systems, control of dynamics in spatio-temporal systems and on networks has recently gainedmuch interest [16–20]. The existence and control of clus-ter states was studied by Choe et al. [21, 22] in networksof Stuart-Landau oscillators. This Stuart-Landau sys-tem arises naturally as a generic expansion near a Hopfbifurcation and is therefore often used as a paradigm foroscillators. The complex coupling constant that arisesfrom the complex state variables in networks of Stuart-Landau oscillators consists of an amplitude and a phase.Similar coupling phases arise naturally in systems withall-optical coupling [6, 23]. Such phase-dependent cou-plings have also been shown to be important in over-coming the odd-number limitation of time-delay feedbackcontrol [24, 25] and in anticipating chaos synchronization[26]. Furthermore, it was shown in [21, 22] that the valueof the coupling phase is a crucial control parameter inthese systems, and by adjusting this phase one can delib-erately switch between different synchronous oscillatorystates of the network. In order to find an appropriatevalue of the coupling phase one could solve a nonlinearequation that involves the system parameters. However,in practice the exact values of the system parameters areunknown, and analytical conditions can be derived only ∗ corresponding author: [email protected] for special values of the complex phase. An efficient wayto avoid these limitations and find optimal values of thecoupling phase is the use of adaptive control.In this paper, we present an adaptive synchronizationalgorithm for delay-coupled networks of Stuart-Landauoscillators. To find an adequate coupling phase we applythe speed-gradient method [15], which was used previ-ously in various nonlinear control problems, yet not forthe control of dynamics in delay-coupled networks. Bytaking an appropriate goal function we derive an equa-tion for the automatic adjustment of the coupling phasesuch that the goal function is minimized. At the sametime the coupling phase converges to the theoreticallypredicted value. Our goal function is based on the Ku-ramoto order parameter and is able to distinguish thedifferent states of synchrony in the Stuart-Landau net-works irrespectively of the numbering of the nodes.This paper is organized as follows. After this introduc-tion, we describe the model system in Sec. II. Section IIIintroduces the speed-gradient method and its applicationusing the coupling phase in networks of Stuart-Landauoscillators. We present the main results for the control ofin-phase synchronization in Sec. IV, and for cluster andsplay states in Sec. V. Finally, Sec. VI contains someconlclusions. II. MODEL EQUATION
Consider a network of N delay-coupled oscillators˙ z j ( t ) = f [ z j ( t )] + Ke iβ N X n =1 a jn [ z n ( t − τ ) − z j ( t )] (1)with z j = r j e iϕ j ∈ C , j = 1 , . . . , N . The coupling matrix A = { a ij } Ni,j =1 determines the topology of the network.The local dynamics of each element is given by the normalform of a supercritical Hopf bifurcation, also known asStuart-Landau oscillator, f ( z j ) = [ λ + iω − (1 + iγ ) | z j | ] z j (2)with real constants λ, ω = 0, and γ . In Eq. (1), τ is thedelay time. K and β denote the amplitude and phase ofthe complex coupling constant, respectively. Such kindsof networks are used in different areas of nonlinear dy-namics, e.g., to describe neural activities [27].Synchronous in-phase, cluster, and splay states arepossible solutions of Eqs. (1) and (2). They exhibita common amplitude r j ≡ r ,m and phases given by ϕ j = Ω m t + j ∆ ϕ m with a phase shift ∆ ϕ m = 2 πm/N and collective frequency Ω m . The integer m determinesthe specific state: in-phase oscillations correspond to m = 0, while splay and cluster states correspond to m = 1 , . . . , N −
1. The cluster number d , which de-termines how many clusters of oscillators exist, is givenby the least common multiple of m and N divided by m ,and d = N (e.g., m = 1), corresponds to a splay state.The stability of synchronized oscillations in networkscan be determined numerically, for instance, by the mas-ter stability function [28]. This formalism allows a sep-aration of the local dynamics of the individual nodesfrom the network topology. In the case of the Stuart-Landau oscillators it was possible to obtain the Flo-quet exponents of different cluster states analyticallywith this technique [21]. By these means it has beendemonstrated that the unidirectional ring configurationof Stuart-Landau oscillators exhibits in-phase synchrony,splay states, and clustering depending on the choice ofthe control parameter β . For β = 0, there exists mul-tistability of the possible synchronous states in a largeparameter range. However, when tuning the couplingphase to an optimal value β = Ω m τ − πm/N accordingto a particular state m , this synchronous state is monos-table for any values of the coupling strength K and thetime delay τ . The main goal of this paper is to find ade-quate values of β by automatic adaptive adjustment. Forthis purpose, we make use of the speed gradient method[15], which is outlined in the next section. III. SPEED-GRADIENT METHOD
In this section, we briefly review an adaptive controlscheme called speed-gradient (SG) method . Consider ageneral nonlinear dynamical system˙ x = F ( x, u, t ) (3)with state vector x ∈ C n , input (control) variables u ∈ C m , and nonlinear function F . Define a control goallim t →∞ Q ( x ( t ) , t ) = 0 , (4)where Q ( x, t ) ≥ Q = ω ( x, u, t ) is calculated, that is, the speed(rate) at which Q ( x ( t ) , t ) is changing along trajectoriesof Eq. (3): ω ( x, u, t ) = ∂Q ( x, t ) ∂t + [ ∇ x Q ( x, t )] T F ( x, u, t ) . (5) Then we evaluate the gradient of ω ( x, u, t ) with respectto input variables: ∇ u ω ( x, u, t ) = ∇ u [ ∇ x Q ( x, t )] T F ( x, u, t ) . Finally, we set up a differential equation for the inputvariables u dudt = − Γ ∇ u ω ( x, u, t ) , (6)where Γ = Γ T > speed-gradient (SG) algo-rithm , since it suggests to change u proportionally to thegradient of the speed of changing Q .The idea of this algorithm is the following. The term −∇ u ω ( x, u, t ) points to the direction in which the valueof ˙ Q decreases with the highest speed. Therefore, if oneforces the control signal to ”follow” this direction, thevalue of ˙ Q will decrease and finally be negative. When˙ Q <
0, then Q will decrease and, eventually, tend to zero.We shall now apply the speed-gradient method to net-works of Stuart-Landau oscillators. Since the couplingphase β is the crucial parameter that determines sta-bility of the possible in-phase, cluster, and splay states,we use this control parameter as the input variable u .Setting u = β and x = ( z , . . . , z N ), Eq. (1) takesthe form of Eq.(3) with state vector x ∈ C N and in-put variable β ∈ R , and nonlinear function F ( x, β, t ) =[ f ( z ) , . . . , f ( z N )] + Ke iβ [ Ax ( t − τ ) − x ( t )].The SG control equation (6) for the input variable β then becomes dβdt = − Γ ∂∂β ω ( x, β, t ) = − Γ (cid:18) ∂F∂β (cid:19) T ∇ x Q ( x, t ) , (7)where Γ > IV. IN-PHASE SYNCHRONIZATION
To apply the SG method for the selection of in-phasesynchronization we need to find an appropriate goal func-tion Q . It should satisfy the following conditions: thegoal function must be zero for an in-phase synchronousstate and larger than zero for other states. Hence, asimple goal function can be introduced by taking the dis-tance of all oscillator phases to a reference oscillator’sphase ϕ : Q ( x ( t ) , t ) = 12 N X k =2 ( ϕ k − ϕ ) , (8)Taking the gradient of the derivative along the trajecto-ries of the system (1) with local dynamics (2) one canderive an adaptive law of the following form by straight-forward calculation. Using ω ( x, β, t ) = ˙ Q , Eq. (7) be-comes˙ β = − Γ K N X k =2 ( ϕ k − ϕ ) " N X n =1 a kn (cid:18) r n,τ r k cos( β + ϕ n,τ − ϕ k ) − cos β (cid:19) − N X n =1 a n (cid:18) r n,τ r cos( β + ϕ n,τ − ϕ ) − cos β (cid:19) , (9)where we used the abbreviations r n,τ = r n ( t − τ ) and ϕ n,τ = ϕ n ( t − τ ) for notational convenience.Figure 1 presents the results of a numerical simula-tion for a random network with N = 6 nodes and unityrow sum. Throughout this paper we use Γ = 1. Ac-cording to the numerical simulations decreasing Γ willyield a decrease of the speed of convergence. On theother hand, if Γ is too big, undesirable oscillations ap-pear. The model parameters are chosen as in [21]. InFig. 1(a) it can be seen that the absolute values | z j | ofall nodes converge after about 60 time units. Fig. 1(b)shows that the phase differences of the different oscilla-tors approach zero, which corresponds to the in-phasesynchronous state. Fig. 1(c) depicts the evolution of β .The blue dashed line represents the value of the couplingphase β = Ω τ = 0 . π , for which stability was shownanalytically in [21]. It can be seen that the adaptivelyadjusted phase comes close to this value. In other words,even without knowing the exact values of the system pa-rameters, the SG algorithm yields an adequate value of β that stabilizes the target state of in-phase synchroniza- tion. Fig. 1(d) shows that the goal function (8) indeedapproaches zero.Note that the above choice of the goal function Q is notthe only possibility to generate a stable in-phase solution.Let us consider a function based on the order parameter R = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 e iϕ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (10)It is obvious that R = 1 if and only if the state is in-phase synchronized. For other cases we have R < Q = 1 − N N X j =1 e iϕ j N X k =1 e − iϕ k . (11)From ˙ β = − Γ ∂∂β ˙ Q we derive an alternative adaptivelaw:˙ β = Γ 2 KN N X k =1 N X j =1 sin( ϕ k − ϕ j ) N X n =1 a jn ( r n,τ r j cos( β + ϕ n,τ − ϕ j ) − cos β ) . (12)Fig. 2 shows the results of a numerical simulation. Asbefore, the amplitude and phase approach appropriatevalues that lead to in-phase synchronization. This time,however, the obtained value of β does not converge tothe one for which the analytical approach [10] has estab-lished stability of the in-phase oscillation (blue dashedline), but to another limit value. This can be explainedas follows: There exists a whole interval of acceptablevalues of β around the value of the coupling phase forwhich an analytical treatment is possible, such that forany value from this interval an in-phase state is stable.Our SG algorithm finds one of them, depending uponinitial conditions. V. SPLAY AND CLUSTER STATESSTABILIZATION
In this section we will consider unidirectionally coupledrings with N = 6 nodes. That is, the coupling matrix has the following form: A = · · ·
00 0 1 · · · · · ·
11 0 0 · · · Let 1 ≤ m ≤ N −
1. Then d = LCM( m, N ) /m , whereLCM denotes the least common multiple, is the numberof different clusters of a synchronized solution. A splaystate corresponds to d = N while cluster states yield d < N . Using similar arguments as those leading toEq. (8) we could choose a goal function of the followingform: Q = 12 N X j =1 (cid:18) ϕ j − ϕ j +1 − πd (cid:19) (13)with j = j mod N . Figure 1: (Color online) Adaptive control of in-phase oscilla-tions with goal function Eq. (8). (a): absolute values r j = | z j | for j = 1 , ...,
6; (b): phase differences ∆ φ j = ϕ j − ϕ j +1 for j = 1 , ...,
5; (c) temporal evolution of β , blue dashed line: ref-erence value for Ω = 0 .
92; (d): goal function. Parameters: λ = 0 . , ω = 1 , γ = 0, K = 0 . τ = 0 . π , N = 6. Initialconditions for r j and ϕ j are chosen randomly from [0 ,
4] and[0 , π ], respectively. The initial condition for β is zero. The goal function Eq. (13) has a crucial disadvantage:we need to define an ordering of the system nodes. Sincethis is inconvenient for practical applications, we will ex-
Figure 2: (Color online) Adaptive control of in-phase oscil-lations with goal function Eq. (11). (a): absolute values r j = | z j | ; (b): phase differences ∆ φ j = ϕ j − ϕ j +1 ; (c): tem-poral evolution of β , blue dashed line: reference value forΩ = 0 .
92; (d): goal function. Other parameters as in Fig. 1. tend the alternative goal function Eq. (11) such that wecan stabilize splay and cluster states. First of all, notethat the following condition holds for splay and cluster
Figure 3: Schematic diagrams of splay ( d = 6), three-cluster( d = 3), and two-cluster ( d = 2) states in panels (a), (b), and(c), respectively ( N = 6). Each cluster contains the samenumber of nodes. states: N X j =1 e iϕ j = 0 . (14)Indeed, if we have only three nodes and take Q = P j =1 e iϕ j P k =1 e − iϕ k as a goal function, we will ensurestability of a splay state, as we have verified by numericalsimulations. Note that this goal function does not needa fixed ordering of the nodes. Renumbering all nodes ina random way will yield the same goal function. One candefine a generalized order parameter R d = 1 N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X k =1 e diϕ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (15)with d ∈ N . However, if we derive a goal function fromthis order parameter in an analogous way as in Eq. (11),this function will not have a unique minimum at the d -cluster state because R d = 1 holds also for the in-phase state and for other p -cluster states where p are divisorsof d .For example, suppose that the system has six nodes.Then states for which conditions (14) and (15) with R d = 1 for d = 6 hold are schematically depicted inFig. 3(a),(b),(c). In order to distinguish between thesethree cases, let us consider the functions f p ( ϕ ) = 1 N N X j =1 e piϕ j N X k =1 e − piϕ k . (16)A splay state (Fig. 3(a)) yields f = f = f = 0,while in the 3-cluster state displayed in Fig. 3(b) we have f = f = 0, f = 1, and in the 2-cluster-state shownin Fig. 3(c) f = f = 0, f = 1. Hence, we obtain P p f p = 0 if and only if there is a state with d clusters,where the sum is taken over all divisors of d . x,t)Combining all previous results we adopt the followinggoal function: Q = 1 − f d ( ϕ ) + N X p | d, ≤ p 96; (d): goalfunction. Other parameters as in Fig. 1. the complex interval [ − , × [ − i, i ] for each oscillator z j . Figure 7 shows the fraction f c of those realizationsthat asymptotically approach a splay state after applyingthe speed-gradient method. We observe that the speed-gradient method is able to control the splay state in a Figure 5: (Color online) Adaptive control of 2-cluster state( m = 3) with goal function Eq. (17). (a): absolute values r j = | z j | ; (b): phase differences ∆ φ j = ϕ j − ϕ j +1 ; (c): temporalevolution of β , blue dashed line: reference value for Ω = 1 . wide parameter range. The range of possible couplingstrengths K does, however, shrink considerably with in-creasing time delay τ . We conjecture several reasons forthis shrinking. Firstly, multistability of different splayand cluster states is more likely for larger values of K and τ , which narrows down the basin of attraction for Figure 6: (Color online) Adaptive control of 3-cluster state( m = 2 , 4) with goal function Eq. (17). (a): absolute values r j = | z j | ; (b): phase differences ∆ φ j = ϕ j − ϕ j +1 , blue dashedline: reference value for Ω = 1 . 03; (c): temporal evolution of β ; (d): goal function. Other parameters as in Fig. 1. a given state. Secondly, Eq. (18), which describes thedynamics of the coupling phase under the adaptive con-trol, is influenced by the time delay τ . Using large delaytimes, we observe overshoots of the control leading to afailure. 0 1 2 3 4 5K0 p p p p t f c Figure 7: (Color online) Success of the speed-gradient methodin dependence on the coupling parameters K and τ for thesplay state in a unidirectionally coupled ring of N = 4 Stuart-Landau oscillators. Other parameters as in Fig. 1. The colorcode shows the fraction of successful realizations. VI. CONCLUSION We have proposed a novel adaptive method for the con-trol of synchrony on oscillator networks, which combinestime-delayed coupling with the speed gradient method ofcontrol theory. Choosing an appropriate goal function,a desired state of generalized synchrony can be selectedby the self-adaptive automatic adjustment of a controlparameter, i.e., the coupling phase. This goal function,which is based on a generalization of the Kuramoto or-der parameter, vanishes for the desired state, e.g., in-phase, splay, or cluster states, irrespectively of the or-dering of the nodes. By numerical simulations we haveshown that those different states can be stabilized, andthe coupling phase converges to an optimum value. Wehave elaborated on the robustness of the control schemeby investigating the success rates of the algorithm in de-pendence on the coupling parameters, i.e., the couplingstrength and the time delay. In this work, we focused onthe adaptive adjustment of the coupling phase while theother coupling parameters were fixed. The input variable u in Eq. (3) may in general contain all of the couplingparameters. Thus, as a generalization, our method mightbe applied to all coupling parameters including the cou-pling amplitude and the time delay. In this way control ofcluster and splay synchronization might be possible with-out any a priori knowledge of the coupling parameters.Given the paradigmatic nature of the Stuart-Landau os-cillator as a generic model, we expect broad applicability,for instance to synchronization of networks in medicine,chemistry or mechanical engineering. The mean-field na-ture of our goal function makes our approach accessibleeven for very large networks independently of the partic-ular topology. 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