CONTEMPORARY PHYSICS, 2017 VOL. 58, NO. 3, 207–243 https://doi.org/10.1080/00107514.2017.1345844
Synchronisation of chaos and its applications
Deniz Eroglua,b, Jeroen S. W. Lambb and Tiago Pereiraa aInstituto de Ciências Matemáticas e Computação, Universidade de São Paulo, São Carlos, Brazil; bDepartment of Mathematics, Imperial College London, London, UK
ABSTRACT ARTICLE HISTORY Dynamical networks are important models for the behaviour of complex systems, modelling Received 20 March 2016 physical, biological and societal systems, including the brain, food webs, epidemic disease in Accepted 2 June 2017 populations, power grids and many other. Such dynamical networks can exhibit behaviour in KEYWORDS which deterministic chaos, exhibiting unpredictability and disorder, coexists with synchronisation, Synchronisation; a classical paradigm of order. We survey the main theory behind complete, generalised and phase interaction; networks; synchronisation phenomena in simple as well as complex networks and discuss applications to stability; coupled systems secure communications, parameter estimation and the anticipation of chaos.
1. Introduction sure that the lasers synchronise [11–13], as without syn- chronisation destructive interference diminishes power. This survey provides an introduction to the phenomenon Synchronisation can also cause engineering problems. of synchronisation in coupled chaotic dynamical systems. A recent well-publicised example concerns the London Both chaos and synchronisation are important concepts Millennium Bridge, traversing the River Thames. On the in science, from a philosophical as well as a practical point opening day in 2000, the bridge attracted 90,000 visitors, of view. holding up to 2000 visitors on the bridge at the same time. Synchronisation expresses a notion of strong corre- Lateral motion caused by the pedestrians made the bridge lations between coupled systems. In its most elementary lurches to one side, as a result of which the pedestrians and intuitive form, synchronisation refers to the tendency would adjust their rhythm to keep from falling over. In to have the same dynamical behaviour. Scientists also turn, this led to increased oscillations of the bridge due recognise weaker forms of synchronisation, where some to the synchronisation between the bridge’s oscillations key aspects of dynamical behaviour are the same – like and pedestrians’ gait [14–16]. Eventually, the oscillations frequencies – or where coupled dynamical behaviours became so extensive that the bridge was closed for safety satisfy a specific spatiotemporal relationship – like a con- reasons. The bridge was only opened to the public again stant phase lag. after a redesign where dampers were installed to increase Synchronisation is fundamental to our understanding energy dissipation and thereby impede synchronisation of a wide range of natural phenomena, from cosmology between bridge and pedestrians. and natural rhythms like heart beating [1]andhandclap- The above examples of synchronisation in coupled ping [2]tosuperconductors[3]. While synchronisation systems describe a spontaneous transition to order is often beneficial, some pathologies of the brain such because of the interaction. Coupled systems are modelled as Parkinson disease [4,5] and epilepsy [6] are also re- as networks of interacting elements. We often have a lated to this phenomenon. In ecology, synchronisation of detailed understanding about the dynamics of the indi- predators can lead to extinction [7,8] while improving the vidual uncoupled elements. For example, we have reason- quality of synchronised behaviour of prey can increase ably good models for individual superconducting Joseph- the odds to survive [9]. In epidemiology, synchronisation son junctions, heart cells, neurons, lasers and even pedes- in measles outbreaks can cause social catastrophes [10]. trians. In the systems we consider here, the coupling Synchronisation is also relevant to technology. Lasers between elements is assumed to be built up from bilateral form an important example. The stability of a laser gen- interactions between pairs of elements, and a network erally decreases when its power increases. A successful structure indicating which pairs of elements interact with way to create a high-power laser system is by combining each other. First, the way the individuals talk to each many low-power stable lasers. A key challenge is to make
CONTACT Tiago Pereira [email protected], [email protected] © 2017 Informa UK Limited, trading as Taylor & Francis Group 208 D. EROGLU ET AL. other. For example, in the neurons the interaction is 2. Synchronisation between two coupled mediated by synapses and in heart cells by electrical dif- systems fusion. Second, the linking structure describing who is 2.1. Synchronisation of linear systems influencing whom. So, it is the network structure that provides the interaction among individual elements. The Before touching upon more topical and interesting set- collective behaviour emerges from the collaboration and tings in which synchronisation is observed in non-linear competition of many elements mediated by the network systems, as an introduction we consider the elementary structure. example of synchronisation between two linearly coupled Synchronisation can be effectively used to create se- linear systems. Although simple, this example bears the cure communication schemes [17–19]. And it can help main ideas of the general case of synchronisation between developing new technologies. Synchronisation is also two or more nonlinearly coupled systems. used for model calibration, that is, the synchronised We consider two identical linear systems regime between data and equations can reveal the pa- rameter of the equations [20,21]. dxi xi axi, i 1, 2 Chaos in dynamics is one of the scientific revolutions ˙ := dt = = of the twentieth century that has deepened our under- with a a non-zero constant. The solution of these lin- standing of the nature of unpredictability. Initiated by ear differential equations with initial condition x (0) is Henri Poincaré in the late-nineteenth century, the chaos i x (t) eatx (0) so that the ensuing dynamics is simple revolution took off in the 1980s when computers with i = i and all solutions converge exponentially fast to zero if which chaotic dynamics can be studied and illustrated, a<0, or diverge to infinity if a>0unlessx(0) 0. became more widely available. Chaos normally arises = We now consider these linear systems coupled in the when recurrent dynamical behaviour has locally dispers- following way ing characteristics, as measured by a positive Lyapunov exponent. In this review, we will not discuss any details of x1 ax1 α(x2 x1) chaotic dynamics in detail. For a comprehensive mono- ˙ = + − x ax α(x x ) (1) graph on this topic, see for instance [22]. ˙2 = 2 + 1 − 2 At first sight it may appear that the concept of synchro- α nisation, as an expression of order, and the concept of and is called the coupling parameter. chaos, associated with disorder, could not be more distant In the context of this model, we speak of Complete ( ) ( ) from one another. Hence, it was quite a surprise when Synchronisation (CS) if x1 t and x2 t converge to each other as t .Inordertostudythisphenomenon,it physicists realised that coupled chaotic systems also could →∞ spontaneously synchronise [23,24]. Despite many years is natural to consider the new variable of studies into this phenomenon and its applications, z x x . many fundamental problems remain open. This survey := 1 − 2 is meant to provide a concise overview of some of the most important theoretical insights underlying our cur- In terms of this variable, synchronisation corresponds to the fact that limt z(t) 0. As z x1 x2,wefind rent understanding of synchronisation of chaos as well as →∞ = ˙ =˙ − ˙ highlighting some of the many remaining challenges. directly by substitution from Equation (1) that In this review, we will discuss the basic results for z (a 2α)z, synchronisation of chaotic systems. The interaction can ˙ = − make these systems adapt and display a complicated which has the explicit solution z(t) z(0)e(a 2α)t. unpredictable dynamics while behaving in a synchronous = − Hence, we find that limt z(t) 0ifandonlyif manner. Synchronisation in these systems can appear →∞ = a 2α < 0(unlesstheinitialconditionisalreadysyn- in hierarchy depending on the details of the individual − chronised, i.e. z(0) 0). Defining the critical coupling elements and the network structure. We will first discuss = value α as this hierarchy in two coupled chaotic oscillators and latter c a αc (2) generalise to complex networks. The review is organised := 2 as follows. In Section 2 we discuss the synchronisation we thus obtain synchronisation if the coupling parameter scenarios between two coupled oscillators. In Section 3 exceeds the critical value: α > αc. we discuss such applications where the synchronisation We finally note that as the system synchronises, the phenomenon can be used for prediction and parameter coupling term converges to zero and the solution of each estimation. In Section 4 we discuss synchronisation in of the components behaves in accordance with the complex networks. underlying uncoupled linear system: to be precise, CONTEMPORARY PHYSICS 209
( ) x1(0) x2(0) at 2.2. Complete synchronisation of non-linear limt xi t +2 e 0 for i 1, 2. It is →∞ − = = systems important! to note that the sign" of the parameter a here determines a difference between synchronisation to the We now consider two fully diffusively coupled identical trivial equilibrium (if a<0) or to an exponentially grow- non-linear n-dimensional systems ing solution (if a>0). In view of later generalisations, we will go through the x1 f (x1) α H(x2 x1) above analysis again, exploiting more the linear structure ˙ = + − x f (x ) α H(x x ) (6) of the problem so that we can appreciate synchronisation ˙ 2 = 2 + 1 − 2 in terms of spectral properties of the coupling term. where f Rn Rn is in general non-linear and H x1 : → : With x ,(1) can be written as Rn Rn is a smooth coupling function. We assume := x2 → # $ that H(0) 0 so that the synchronisation subspace = x x is invariant for all coupling strengths α. Meaning x aI αL x (3) 1 = 2 ˙ =[ − ] that for any synchronised initial condition the entire where solution remains synchronised: as in the synchronised state the diffusive coupling term vanishes, the dynamics 10 1 1 is identical to that of the uncoupled system (with α I and L − . = = 01 = 11 0). Consequently, the coupling has no influence on the # $ # − $ synchronised motion. In particular, it could be the case L is known as the Laplacian matrix. In Section 4,wewill that the synchronised motion is chaotic, if the uncoupled generalise it to any network. The solution of (3)with systems exhibit such behaviour. initial condition x(0) is We aim to show that if the coupling is sufficiently strong, the system Equation (6)willsynchronisex (t) 1 − ∞ n aI αL t At t n x2(t) 0ast . We consider H I (the identity x(t) e[ − ] x(0), where e A . (4) → →∞ = = := n matrix) then the term reads as n 0 %= ! α H(x x ) α(x x ). To solve (4) we note that since I and L commute, 2 − 1 = 2 − 1
aI αL t aIt αLt e[ − ] e e− To analyse stability, we consider – as before – the = evolution of the difference variable z x x in terms := 1 − 2 aIt at αLt and e e I. In order to evaluate e− ,itisuseful of which the synchronisation subspace is characterised as = to observe that v (1, 1) and v (1, 1) are the z 0: 1 = ∗ 2 = − ∗ = eigenvectors of L for its corresponding eigenvalues λ 1 = 0andλ 2. As v , v is a basis of R2, we may write any z x1 x2 (7) 2 = { 1 2} ˙ = ˙ − ˙ initial condition as x(0) c v c v with c , c R, f (x ) f (x ) 2αz (8) = 1 1 + 2 2 1 2 ∈ = 1 − 2 − so that αLt αλ2t e− x(0) c1v1 c2e− v2 The aim is to identify sufficient conditions for the cou- = + pling parameter α to guarantee that locally near z 0 = and we have limt z(t) 0. To this end, we linearise the →∞ = equations of motion Equation (8)nearz 0. We note aI αL t at (a αλ2)t = x(t) e[ − ] x(0) c1e v1 c2e − v2. (5) to this extent that near x x we obtain by Taylor = = + 1 = 2 expansion that Synchronisation corresponds to the phenomenon that ( ) x t converges to the synchronisation subspace generated f (x2(t)) f (x1(t)) D f (x1(t))(x2(t) x1(t)) (a αλ )t = − − by v1.Thisonlyhappensiflimt c2e − 2 v2 0, 2 a →∞ = O( x1(t) x2(t) ) i.e. if α > λ . Thus in view of (2), we define the critical + ∥ − ∥ 2 f (x (t)) D f (x (t))z(t) O( z(t) 2). coupling value = 1 − 1 + ∥ ∥ a a αc . ( ( )) = λ2 = 2 Here D f x1 t is the derivative (Jacobian matrix of f (x))atx x (t).WeusethistowriteEquation(8) We note that the critical coupling value is expressed = 1 near z 0as in terms of the gap between the lowest eigenvalue 0 = and smallest nonzero (and positive) eigenvalue of the dz 2 D f (x1(t)) 2α I z O( z ). (9) Laplacian L. dt =[ − ] + ∥ ∥ 210 D. EROGLU ET AL.
The linear part of this equation, obtained by ignoring the constant C may still vary per trajectory, which leads the O( z 2) term in Equation (9), is commonly known as to non-uniform synchronisation, implying the potential | | the first variational equation.Itshouldbenotedthatthis of large variation of transit times until synchronisation equation is nonautonomous as it depends explicitly on occurs. For similar phenomena, see also [25,26]. We now the reference solution x1(t). In general it is not easy to proceed to apply the above to the examples of coupled analyse nonautonomous differential equations, not even Lorenz and Rössler systems. linear ones. Fortunately, we are able to achieve insights Lorenz system. The Lorenz system was introduced by without solving this equation because the coupling is Edward Lorenz in 1963 as a simplified model for rather convenient adding an extra damping term αz. atmospheric convection: − To simplify the analysis, we introduce a new variable x σ (y x), 2αt ˙ = − w(t) e z(t) (10) y x(ρ z) y, = ˙ = − − z βz xy, (14) in terms of which the linear part of Equation (9)becomes ˙ = − + precisely the variational equation for the solution x (t) 1 where the three coordinates x, y and z represent the of the uncoupled equation of motion x f (x): ˙ = state of the system and σ , ρ, β are parameters. When σ ρ w( ) α 2αt ( ) 2αt ( ) parameter values are chosen as 10, 28 and t 2 e z t e z t β / = = ˙ = + ˙ α 8 3, the equations display unpredictable (chaotic) 2αw D f (x (t)) 2α I e2 t z = = +[ 1 − ] dynamics. Lorenz used this choice of parameters in his D f (x (t)) w. (11) original paper [27]. We use these parameter values as =[ 1 ] well. Let #(x1(t)) be the fundamental matrix for the The Lorenz equations are dissipative and all trajecto- variational equation, so that any solution of this ries eventually enter the absorbing domain nonautonomous equation can be written as z(t) n = β2ρ2 #(x1(t))z(0). Let λj((x1(t)) j 1 be the set of positive 3 2 2 2 { } = ( x R ρx σ y σ (z 2ρ) < , square roots of the eigenvalues of the symmetric matrix = ∈ : + + − β 1 & − ' #(x1(t))∗#(x1(t)) (where ∗ denotes transpose). Then we define see Appendix C or Ref. [28]. For the classical parameters, 1 σ 10, ρ 28 and β 8/3, inside (,trajectoryaccu- $ max lim λj(x1(t)). (12) := j t = = = →∞ t mulates on the chaotic Lorenz attractor [29], as depicted $ is known as the Lyapunov exponent of the orbit x1(t) in Figure 1(b). Close to the attractor nearby trajectories and it measures the infinitesimal asymptotic divergence diverge. To see this, we simulate two trajectories with 2 rate near this trajectory. We refer to Appendix 7 for more nearly the same initial condition. The initial 10− differ- details about the Lyapunov exponent. ence grows to roughly 102 in a matter of only six cycles, The assertion now is that if the orbit x1(t) has Lya- see Figure 1(a). Using numerical simulations, we estimate $ the maximal divergence rate of nearby trajectories $ punov exponent ,thenthereexistsaconstantC>0 ≈ such that 0.906. w(t) Ce$t. (13) We consider two coupled chaotic Lorenz oscillators, || || ≤ as in Equation (6). We derived above that the critical From Equation (13) and using Equation (10)weobtain coupling αc for synchronisation depends on the Lya- that $ $ ($ 2α)t punov exponent . Using the numerical results for z(t) Ce − . ∥ ∥≤ we obtain $ Hence, αc 0.453. $ = 2 ≈ αc := 2 Simulation confirms that this critical coupling is sharp. is a critical coupling strength for synchronisation, above Indeed, for α 0.4 there is no synchronisation, and = which observe synchronisation. trajectories do not move together, see Figure 2(a). On the A complication with the above analysis, is that the other hand, when α 0.5, the trajectories synchronise, = Lyapunov exponent $ and constant C may depend on see Figure 2(b). the chosen trajectory x1(t). The probabilistic (ergodic) To compare the amount of synchronisation at dif- theory of dynamical systems, which we will not dwell on ferent parameter values, we may consider the average here, asserts that often the Lyapunov exponent is constant deviation from synchronisation during a time-interval of for almost all trajectories on a given attractor. However, length T as CONTEMPORARY PHYSICS 211
Figure 1. Illustration of the chaotic dynamics of the Lorenz system (14) with parameter values σ 10, ρ 28 and β 8/3. (a) Two simulations for the Lorenz system starting from two slightly different initial conditions (x, y, z)= ( 10,= 10, 25) and =(x, y, z) ( 10.01, 10, 25).TheLorenzattractorhasapositiveLyapunovexponentandthetrajectoriesdivergefromeachother.(b)Representation= − ˜ ˜ ˜ = of− the trajectory of (x, y, z) ( 10, 10, 25) in the phase space. The shape of the attractor resembles a butterfly. = −
Figure 2. Comparison of trajectories of two initial conditions for the system of two coupled Lorenz systems. The critical transition coupling is αc 0.453 for the classical parameters. The initial conditions are selected as (x1, y1, z1) (3, 10, 15) and (x2, y2, z2) (10, 15, 25). (a) When≈ α 0.4 < α ,thereisnosynchronisation.(b)Ifα 0.5 > α one observes synchronisation= of trajectories. = = c = c
1 T adds a damping term αz in other words when H I. E x1(t) x2(t) dt (15) = = T ∥ − ∥ Indeed, the damping term in general form is α Hz and (t 0 = in this case the above results can no longer be applied. In Figure 3(a) we present a synchronisation diagram Therefore the synchronisation depends on the coupling where we plot E against the coupling strength α.We function, we here just illustrate the effect on synchroni- observe a good correspondence with the derived value of sation if H is chosen to be αc. The synchronisation error depends on initial condi- tions so that we compute the synchronisation diagram via averaging over some realisations. 100 Examples on different coupling functions. It is worth men- H 000 ,(16) = ⎛ ⎞ tioning that this above analysis works when the coupling 000 ⎝ ⎠ 212 D. EROGLU ET AL.
Figure 3. Synchronisation diagram of two coupled Lorenz systems, (a) with coupling matrix H I and (b) with coupling matrix H = as in Equation (16). When H I,theobservedcriticalcouplingconstantcorrespondstothetheoreticallyderivedvalueαc 0.453. With coupling matrix (16), sychronisation= is observed to set in for coupling strengths larger than 3.75. The synchronisation error∼ E was averaged over 300 realisations. Each realisation is simulated by a fourth-order Runge–Kutta scheme∼ for 2000 s with 0.01 time step.
implying that the coupling arises only via the first coordinate x.Thecorrespondingsynchronisationdia- gram shows that the critical coupling αc for x-coupling increases as a result, see Figure 3(b). Importantly, when H does not commute with the Jacobian matrix along the trajectory, we cannot use the ansatz of Equation (10). In that case we need a different approach to derive the critical coupling, which will be discussed in Section 4 that deals with synchronisation in complex networks. Rössler System. As a final example, we consider a system introduced by Otto Rössler in 1976:
x y z, ˙ = − − y x ay, ˙ = + z b z(x c),(17) ˙ = + − Figure 4. The synchronisation diagram of two coupled Rössler systems with the coupling function H I. The theoretical critical coupling constant α 0.0355 indeed= corresponds to where a, b and c denote parameters. c ≈ We consider two coupled Rössler systems with iden- the numerically observed one. The synchronisation error E was averaged over 300 realisations. Each realisation is simulated by a tity coupling function H I and coupling parameter = fourth-order Runge–Kutta scheme for 2000 s with 0.01 time step. α, as in Equation (6). We consider parameter values a 0.2, b 0.2 and c 5.7. We numerically find that = = = the corresponding attractor has a Lyapunov exponent 2.2.1. CS in driven systems $ 0.071. Hence, the expected critical coupling for ≈ synchronisation is Another possibility is that we use certain sets of variables to drive a subsystem. For appropriate choices, we can $ observe synchronisation [30]. We illustrate this scheme αc 0.0355 (18) = λ2 ≈ in the Lorenz system where x-component can be driving signal of another identical system Figure 5. This is in excellent agreement with the numerical results In this scheme, we consider the variable x for the is shown in the synchronisation diagram Figure 4. master the same as in the slave. That is, the x-variable of CONTEMPORARY PHYSICS 213
and along solutions of the subsystem we obtain V ˙ = ) ) ) ) , after some manipulation we obtain y ˙ y + z ˙ z )2 β)2. V˙ y z Figure 5. Master–Slave configuration where, x-variable is made = − − identical to the response and thereby it drives the response ) ) subsystem. Since V is positive and V˙ negative y and z will con- verge to zero. So, the slave subsystem will have the same dynamics as the master.
2.3. Phase synchronisation If there are small mismatches between the systems another type of synchronisation can appear for very small coupling strengths: Phase Synchronisation (PS) – which corresponds to a locking of phases of chaotic oscillators
mφ (t) nφ (t)
(a)
(b)
Figure 8. For a weak coupling constant: Although there is no Figure 7. Projection of the Rössler attractor on x y plane for synchrony for the difference of the amplitudes Equation (15)(top a 0.165. − = panel), there is a tendency towards to phase synchronisation while the coupling constant α increases Equation (24)(bottom We focus on coupled two nonidentical Rössler oscil- panel). The synchronisation error E and frequency mismatch )( were averaged over 300 realisations. Each realisation is simulated lators, the equation is given by by a fourth-order Runge–Kutta scheme for 2000 s with 0.01 time step. x ω y z α(x x ) ˙1,2 = − 1,2 1,2 − 1,2 + 2,1 − 1,2 y1,2 ω1,2x1,2 ay1,2 ˙ = + When phase synchronisation occurs φ (t) φ (t) C, ( ) | 1 − 2 | ≤ z1,2 b z1,2 x1,2 c (21) the average frequency is the same )( 0. The phase ˙ = + − = difference will not be tend to a constant as the phase where a 0.165, b 0.2andc 10 are the constants of = = = nature of the amplitudes acts as a noise in the phases the Rössler system. ω is the mismatch parameter to make causing mismatches. The comparison of the amplitude the oscillators nonidentical and given as ω ω ) 1,2 = 0 ± difference (Equation (15)) and the phase (Equation (25)) where ω 0.97 and ) 0.02. α is the coupling 0 = = is given in Figure 8. If we increase the coupling constant constant, the system is coupled over x components (x- α. coupling). For certain values of the parameter a,the An approximate theory of phase synchronisation can projection of the attractor on x y plane resembles a − be obtained by averaging [43]. We write the model limit cycle and the trajectories rotates around the origin Equation (21) in terms of the phase Equations (22)as (see Figure 7), and phase and amplitudes are given by
y1,2 x1,2y1,2 y1,2x1,2 φ arctan (22) φ1,2 ˙ − ˙ (26) 1,2 = ˙ = A2 #x1,2 $ In this form, using polar coordinates we have x 1,2 = A1,2 cos φ1,2 and y1,2 A1,2 sin φ1,2, and using this rep- A x2 y2 . (23) = 1,2 = 1,2 + 1,2 resentation in Equation (26), we obtain - We consider the phase on the lift (growing in time with- z1,2 out taking the mod). φ1,2 ω1,2 a sin φ1,2 cos φ1,2 sin φ1,2 ˙ = + + A1,2 To gain insight on the adjustment of rhythm leading to A2,1 phase synchronisation, we analyse the average frequen- α cos φ2,1 sin φ1,2 cos φ1,2 sin φ1,2 − A − cies defined as # 1,2 $ (27) φ1,2(T) φ1,2(0) (1,2 lim − . (24) = T The idea now is that since the mismatch is small, →∞ T both phases behave nearly the same. So we can split the And the frequency mismatch is dynamics of the phases as an overall increasing trend ω0t and a slow phase dynamics θ.Thissplitisveryclearin )( ( ( (25) Figure 14. So, we write = 2 − 1 CONTEMPORARY PHYSICS 215
φ ω t θ , 1,2 = 0 + 1,2 To obtain an equation for θ (simpler than the one for φ) we use the fact that θ is a slow variable. That is, while ω0t grows a lot θ is nearly constant. Hence, we will average out the contribution of ω0t. So we average the phases over ω t over a period 2π and keep θ fixed. After some 0 ω0 1,2 laborious manipulation we obtain
d α A2 A1 (θ θ ) 2) sin (θ θ ) (28) 1 − 2 = − + 1 − 2 dt 2 #A1 A2 $
Both amplitudes A1,2 depend on time and display a Figure 9. Scheme of the auxiliary system approach for the chaotic behaviour. Lets assume for a moment that they generalised synchronisation. Originally we only have the system are constant. Then for the phase locking of the Rössler in the dashed line box which is master–slave system as in Section d a systems, (θ1 θ2) 0, the equation has a stable fixed 2.2.1. Then we add an auxiliary (helper) system y .IfthereisCS dt − = s a a,s point, between y and y ,thentheGSoccursbetweenx and y . 4)A1A2 θ1 θ2 arcsin . (29) − = α(A2 A2) 1 + 2 is, the solutions of, say x can be mapped into solutions This fixed point only exists when the argument of the of y. arcsin has modulus less than 1. Therefore, we obtain the y ψ(x) = critical transition coupling where ψ is a function from the phase space of the system x to the phase space of system y.Whenthishappens αc 2). ≈ we have generalised synchronisation between these two ψ For the given parameters () 0.02) we find α 0.04, systems. CS is a particular case of GS when is the = c ≈ in agreement with the numerical analysis Figure 8.The identity. chaotic behaviour of the amplitudes leads to fluctuations To detect a functional relation between two systems of the phases around the stable fixed point, and so the in generalised synchronisation, Rulkov and co-workers phases different will not be identically zero. Close to the proposed a technique called mutual false nearest neigh- critical coupling strength the frequencies exhibit a critical bours [47]. The main idea is the see how nearby points are 1/2 mapped under the dynamics. By studying the properties behaviour )( α αc as observed in Figure 8. ∝ | − | of nearby points one can infer the existence of the map- ping ψ.Here,wefocusonanotherapproachthatturns 2.4. Generalised synchronisation the GS problem into a CS problem. This is the auxiliary system approach [48,49]. The master system drives the When the interacting systems are different, either because slave system and an auxiliary system (copy of the slave). of a large parameter mismatch or the systems have dis- If the two copies of the slave exhibit CS then the master tinct dynamics, these two can still exhibit synchronisation and slave are in GS [48,49]. An illustration of this scheme in a generalised sense. Generalised Synchronisation (GS) can be found in Figure 9. can be observed in mutually coupled systems as well as Necessary conditions for the occurrence of GS for the unidirectionally coupled system [30,44–46]. Surprisingly, system given by Equation (31)isintroducedbyKocarev GS can be a mapped to a complete synchronisation (CS) and Parlitz as following: for all (x , y ) B, where x and problem! 0 0 ∈ 0 y are states for the master–slave systems at time t 0 Here we will focus on the dynamics of unidirectionally 0 = and B is the basin where all the trajectories approach to a coupled systems. The master x and the slave y systems manifold coupled as Mψ (x, y) y ψ(x) . ={ : = } x f (x) ˙ = If Mψ is attractive, different trajectories of the slave sys- y g( y, h(x)) (30) tem will converge to the trajectory lying in M and it ˙ = is determined only by x.Inotherwords,ifthemaster where x Rn, y Rm and h(x) is the coupling. For drives a slave ys and an auxiliary (copy of slave) ya sys- ∈ ∈ 0 0 certain coupling strengths, the dynamics of system y is tems simultaneously, the driven ones must be completely totally determined by the dynamics of system x.That synchronised ys , ya B we have ∀ 0 0 ∈ y 216 D. EROGLU ET AL.
driver. Now consider two copies of the slaves y1 and y2. Because we know the system will exhibit GS when the copies of the slaves synchronise, we introduce a variable z y y .ThesystemwillundergoGSwhenz 0. = 1 − 2 → Differentiating we obtain
z U(t)z α Hz,(32) ˙ = − where we used the mean value theorem [52] to express
U(t)z g( y (t)) g( y (t) z(t)) = 1 − 1 + 1 DG( y (t) s y (t))ds z(t) = 1 + 2 #(0 $ Notice that for the difference z the driving term Hx Figure 10. Generalised synchronisation: averaged over 300 realisations, time = 4000 and time step = 0.01. disappears as it is common for both copies of the slave y1 and y2. The only part of the coupling remaining is the term α Hz, which adds an extra damping and provides lim ys ya 0. − t ∥ t − t ∥ = dissipation. The trivial solution of z is globally stable if →∞ the coupling is large enough. Indeed, we can construct a Example: Consider two identical Rössler systems Lyapunov function for z.Indeed,consider (Equation (17)) with the parameters (a 0.2, b 0.2 = = and c 5.7) are driven by a Lorenz system (Equation 1 = V(z) z, z , (14)) with the classical parameters (ρ 28, σ 10 = 2⟨ ⟩ = = and β 8/3) via x-components. We used the auxiliary = system approach the detect the critical coupling for GS. and differentiating the Lyapunov function along the ( ) Indeed, numerical results showed that α 0.12 as solution z t of Equation (32)weobtain c ≈ seen in the synchronisation diagram Figure 10. For given dV(z(t)) α 0.06 CS is not observed between the slave systems z(t), z(t) (33) = dt = ⟨˙ ⟩ thereforethereisnoGSbetweenmasterandslavesystems ( Dg αλ (H)) y 2 (34) as well Figure 11(a). For a coupling constant larger than ≤ ∥ ∥− min ∥ ∥ the critical one α 0.2 the slave and the auxiliary system = where λ ( ) is the minimum eigenvalue of .Inthis display CS. Hence GS can be observed between master– min H H last passage, we used the Cauchy Schwartz inequality slave system Figure 11(b). U(t)z, z U(t)z z U z 2,andnoticed |⟨ ⟩| ≤∥ ∥∥ ∥≤∥ ∥∥ ∥ that U Dg .1 We also used the fact that H is 2.4.1. Generalised synchronisation between ∥ ∥≤∥ ∥ positive Hz, z λ (H) z 2. Hence, for diffusively coupled oscillators ⟨ ⟩≥ min ∥ ∥ To gain some insight on this, we will use some ideas put Dg forward by [50,51]. This approach allows us to obtain a α > αc ∥ ∥ = λ (H) analytical understanding of the critical coupling associ- min ated with the transition to GS. Consider a master–slave the derivative of the Lyapunov function is negative and system diffusively coupled. every solution of the system sinks to zero. What did we learn? When α > αc we have GS. Any two x f (x) ˙ = trajectories of the slave y1 and y2 will converge towards y g( y) α H(x y) (31) the same asymptotic state. This happens because the cou- ˙ = + − pling terms add extra dissipation. The convergence rate ηt where H is a positive definite matrix. We can write the is exponential y (t) y (t) Ke− because V ηV ∥ 1 − 2 ∥≤ ˙ ≤ slave equation as and η is uniform on the trajectories y1, y2 and x.Asa conclusion, there a function ψ such that the dynamics of y g( y) α Hx the slave can be as y ψ(x). ˙ = ¯ + = In the literature, a typical way to estimate whether one where g( y) g( y) α Hy. The equation then splits has GS is to compute the Lyapunov exponents of the ¯ = − into contributions solely coming from the slave and the slaves. Since we are assuming that the uncoupled systems CONTEMPORARY PHYSICS 217
Figure 11. Two simulations for the generalised coupling scheme: a Lorenz system x drives two Rössler systems ys,a. The critical transition coupling is α 0.12. (a) For the coupling constant α 0.06, there is no synchronisation since α < α (b) for α 0.2synchronisation c ≈ = c = is obtained since α > αc. are chaotic, for α 0 the slave will have a positive = Lyapunov exponents. As we increase α the maximum Lyapunov exponent may become negative for a value αc. We use this αc as an estimate for the critical coupling for GS.
2.5. Summary of synchronisation types We discuss the three cases commonly found in applica- tions. A schematic representation of the cases is found in Figure 12. Diagram of synchronisation types for diffusively Figure 12. coupled systems. The horizontal axis depicts the mismatch Complete synchronisation in identical systems. If the between the isolated dynamics ( f 1 and f 2) and the vertical isolated dynamics are identical, f f and diffusively axis the coupling constant. The diagram shows the typical 1 = 2 coupled, hence, the subspace x x is invariant under balance between mismatch and coupling strength and to achieve 1 = 2 a certain kind of synchronisation. Complete synchronisation Equation (6). Indeed, the coupling vanishes and both (CS) occurs for identical chaotic systems ( f 1 f 2) and systems will oscillate in unison for all coupling strengths large enough coupling strengths. Phase synchronisation= (PS) is α and all times. Such collective motion is called complete observed between slightly different systems for small coupling synchronisation (CS). The question is whether CS is at- strengths. Generalised synchronisation (GS) is a result of master– tractive, that is, if the oscillators state are nearly the same slave system and can occur for large mismatch parameters or even between distinct systems when the coupling strength large x (0) x (0), will they synchronise? Meaning that 1 ≈ 2 enough.
lim x1(t) x2(t) 0. t ∥ − ∥ = proportional to the mismatch f 1 f 2, as illustrated in →∞ − Figure 12. Further details were given in Section 2.3. See Figure 13 for an illustration. In Section 2.2,theCS Generalised synchronisation in master–slave configu- was discussed in detail. rations. If the vector fields are different f f and 1 = Phase synchronisation (PS) when f f .Inthis f g, the systems can synchronise, but in a gener- 1 ≈ 2 2 = situation the subspace x x is not invariant. And each alised sense. We consider systems coupled in a master– 1 = 2 system will have its own frequency given by their phase slave configuration. For certain coupling strengths, the dynamics φ1,2. For small coupling strengths the phases dynamics of the master x can determine the dynamics of can be locked φ φ while the amplitudes remain the slave y,thatis y ψ(x),seeFigure15. This is called 1 ≈ 2 = uncorrelated Figure 14. This phenomenon is called phase Generalised Synchronisation (GS). Further details for GS synchronisation. Typically, the critical coupling for PS is was given in Section 2.4. 218 D. EROGLU ET AL.
Figure 13. Illustration of complete synchronisation. (a) If the Figure 15. Illustration of generalised synchronisation. If the coupling strength is large enough (α > α ), the systems converge c coupling strength is large enough, a functional relationship to invariant synchronisation manifold (x1 x2), (b) otherwise = ( y ψ(x)) exhibits between the dynamical variables x1 and (α < αc), they diverge hence no synchronisation. = x2.(a)Ifα > αc, the generalised synchronisation is observed (b) otherwise α < αc, there is no generalised synchronisation.
Given this complexity, many researchers thought it is unlikely one could possibly synchronise two chaotic systems.Howcouldasystemwithexponentialdivergence of nearby trajectories have a state were trajectories come together while keeping their chaotic nature? That seemed paradoxal. Chaos and synchronisation should not come together. This view was proven wrong in the late eighties. In fact, we have come to think it as rather natural. Funny enough, before this view was accepted synchronisation of chaos had to be rediscovered a few times. Back in the eighties, Fujisaka and Yamada had the first results on synchronisation of chaos [23,58,59]. They publish it in Japan, but their results went fairly unno- ticed in the west. Just two years later mathematicians Figure 14. Illustration of phase synchronisation for two coupled and physicists from Nizhny Novgorod exposed many of slightly different and chaotic systems ( f 1 f 2). The evolution of the phase differences between the systems̸= for two different the concepts necessary for analysing synchronous chaos coupling constants α < αc and α > αc. [24]. This paper is now famous, but back then it also went largely unnoticed. Only some years later the study of synchronisation 2.6. Historical notes of chaos had its boom, largely as a result of the works Studies on synchronisation dates back to Christiaan Huy- by Pecora and Carroll [30]. Lou Pecora and co-workers gens who studied coupled pendulums. In this case, the went systematically tackling two coupled systems and pendulums are periodic and have distinct frequencies, then moved on to study chaotic systems coupled on pe- but due to interaction they adjust their rhythm. In the sev- riodic lattices [60,61]. These early results were relying enties, thanks to the works of Winfree [53]and on ideas from Nuclear physics to diagonalise the lattice Kuramoto [37] the area experienced a boom. In early and stability theory (the Lyapunov methods) to analyse 2000’s many excellent books and reviews were devoted synchronisation. to this subject [1,33,34,54–56]. The nineties proved prolific for synchronisation! Two Chaotic synchronisation on the other hand is younger. groups proposed an extension of synchronisation, the To begin with, the establishment and full acceptance so-called generalised synchronisation [47–49,62,63]. of the chaotic nature of dynamics is fairly new [57]. Generalised synchronisation only asked for a functional The role of chaos in nature was object of intense debate relationship between the states, that is, the dynamics of in the seventies when Ruelle and Takens proposed that one system is fully determined by the dynamics of the turbulence was generated by chaos. other. Also in the mid nineties, Rosenblum, Pikovsky and The chaotic dynamics can be fairly complicated. Typi- Kurthsputforwardtheconceptofchaoticphasesynchro- cally, the evolution never repeats itself, nearby points drift nisation. Here two nearly identical chaotic oscillators can apart exponentially fast, but in the long run the dynamics have their phase difference bounded while the amplitudes return arbitrarily close to its initial state. Such dynamics remain uncorrelated [31,32]. is so unpredictable that modern approach tackles it from A few years down the road, Pecora and Carroll were a probabilistic perspective. able to generalise their approach to undirected networks CONTEMPORARY PHYSICS 219 of diffusively coupled systems [64]. They also wrote a review about their approach [65]. These results open the door to the understanding of the role of the linking struc- ture on the stability of synchronisation. Barahona and Pecora [66] showed that small-world networks are easier to globally synchronise than regular networks. Motter and coworkers [67–70] showed that heterogeneity in the network structure may hinder global synchronisation. On the other hand, Pereira showed such heterogene- ity may enhanced synchronisation of highly connected Figure 16. Illustration of the message masking on bearing signal nodes [71]. scheme. Transmitter generates a chaotic signal x(t) and add the message m(t) on it. This combined two signals s(t) x(t) m(t) is sent to the receiver and both systems synchronise.= Discarding+ 3. Applications the synced signal x from the s, the message m m is restored. r r ∼ In this section, we discuss the role of synchronisation Since the synchronisation of chaotic systems is exponen- phenomena in various applications including secure com- tially stable for such system [30,73] under low ampli- munication approaches, parameter estimation of a model tude of noise the synchronisation (coherence) still occurs. from data and prediction. Then the chaotic signal xr can be obtained from Equation (35). Therefore the message can be regenerated by m(t) = s(t) x (t) (Figure 16). 3.1. Secure communication based on complete − r synchronisation As a message we use the signal
The first approach is to send an analog message [72]. The sin (1.2π sin2 (t)) m(t) 0.1 cos (10π cos (0.9t)) ξ key idea is the following: the sender adds the message = π sin2 (t) + m(t) on a chaotic signal x(t) and generate a new signal s(t) m(t) x(t) (Figure 16). The assumption is that where ξ is a white noise Figure 17(a). We attach this = + the amplitude of x(t) is much larger than the amplitude message on x-component of the Lorenz system Equation of m(t). This method is called the masking information (14) with parameters σ 16.0, ρ 45.2 and β 4.0 = = = on bearing signals. Because chaotic signals are noise-like then the information is masked on bearing signal s Figure and broadband (have many frequencies), it is difficult to 17(b). By synchronisation we restore the message m r ∼ read the message. One could then retrieve the message m (Figure 17(c)). This secure communication application using synchronisation. is also experimentally demonstrated using Chua’s circuits Masking of messages on bearing signals does not re- [74,75]andLorenz-likecircuit[76]. quire encryption. Here is the keystone is selection of the The second approach is the modulation of the pa- transmitter and receiver systems such that they synchro- rameters for the digital communication. In this case, the nise. They are also assumed to be identical (this means message m(t) only carries binary-valued signals. The set- that the receiver knows the parameters of the transmit- up is similar to the masking approach but the message is ter). One can retrieve the message if the parameters are included in the transmitter parameters. The transmitter known. So, the parameters play a role of encryption key. system has an adjustable parameter σ (t) σ δm(t) a = + We illustrate this communication scheme using the such that we can tune the system into synchronisation Lorenz system Equation (14). The Lorenz system has the when m(t) 0andoutsynchronisationwhenm(t) 1 = = particularity that it divided into subsystems (x, z) and Figure 18.Weretrievethemessagem(t) by the synchro- (y, z).Wecanusethevariablesx or y of a subsystem nisation and desynchronisation pattern. to drive the other subsystem. In this driving setting, the The dynamics of transmitter and the receiver is given synchronisation between driver and slave is exponentially by stable provided that the parameters σ , ρ and β are identi- x σ (y x ) cal. Here, we have chosen x component to act as a driver. ˙1 = a 1 − 1 ( ) y x (ρ z ) y The message s t drives the receiver system as ˙1 = 1 − 1 − 1 z1 βz1 x1y1 σ ( ) ˙ = − + xr yr xr x σ (y x ) ˙ = − ˙2 = 2 − 2 y s(ρ z ) y ˙r = − r − r y2 x1(ρ z2) y2 β . ˙ = − − zr zr syr (35) z βz x y ˙ = − + ˙2 = − 2 + 1 2 220 D. EROGLU ET AL.
(a) (a)
(b) (b)
(c) (c)
Figure 17. The masking an analog message on bearing Figure 19. Manipulating the parameter of a transmitter allows chaotic signal. (a) The low-amplitude message, (b) the message digital secure communication. In this application 10 time units embedded into high-amplitude chaotic signal and (c) restored are used for a single digit. (a) Digital message (0101010101), (b) message from the transmitted signal. the synchronisation error and (c) restored message.
synchronisation error E varies according to this change (Figure 19(b)). Due to change in the E, the message is restored (Figure 19(c)). There are more communication applications using the synchronisation mechanism e.g. using hyperchaotic systems [61,63,78,79] or volume-preserving maps [80]. The common idea of all these given approaches is the CS phenomena, negative conditional Lyapunov exponent between the systems are needed to exhibit of the synchro- nisation [30,73]. Figure 18. A secure communication scheme: hiding a digital message on a chaotic signal. Changing a parameter of transmitter 3.2. Secure communication based on phase causes different level of synchronisation errors between the transmitter and the receiver. The amplitude of the error E brings synchronisation the message out. Security is an important issue for communication approaches. As might be expected, some methods were where σ σ δm(t) is the adjustable parameter. Again improved and reported to break the CS-based communi- a = + the key aspect is having mismatch between the trans- cation schemes [81–84]. Then more secure communica- mitter and the receiver does not allow systems to syn- tion scheme demonstrated by means the PS [85]. chronise. Modulating the parameter σa by the message The scheme based on the PS possesses three chaotic m(t) we can produce different levels of synchronisation Rössler systems (x1,2,3). The transmitter of the scheme errors. Choosing the parameters of the transmitter and consists of two weakly coupled identical systems x1 and the receiver identical gives the synchronisation error E x over their x-components Equation (36) and the re- ∼ 2 0(CS),thiscanbeassignedbinary0bym(t) 0. The ceiver x has slightly different dynamics. In this case, we = 3 large mismatch δ causes a certain amount of synchroni- couple the systems with using their phases Equation (36) sation error E>0, this can be assigned binary signal 1 as presented in Ref. [86]. The phase definition for Rössler by m(t) 1. Then the digital communication can be set system is given by Equation (22). The mean of two sys- = between sender and receiver [76,77]. tems’ phases φ1 and φ2 in transmitter can be used as a Using same parameters as in the previous applica- spontaneous phase signal φm to couple the third system as tion (σ 16.0, ρ 45.2andβ 4.0), we illustrate in Equation (36). As distinct from the CS-based schemes, = = = this digital communication. For this example, a digit of we have three systems and the reason behind these to the message is set for 10 time units and the message is improve the security. The return maps of the phase φm is 0101010101 (Figure 19(a)). For each message time, we way more complex than φ1 (or φ2), this makes to break change σ from σ to σ δ and other way round. Then the dynamics not trivial [85]. a + CONTEMPORARY PHYSICS 221
(a)
(b)
(c)
Figure 20. Asecurecommunicationschemebyphase synchronisation: hiding a digital message on a chaotic signal. (d)
We illustrate this application by
Figure 21. An illustration for the secure communication by phase x (ω )ω)y z ϵ(x x ) ˙1,2 = − + 1,2 − 1,2 + 2,1 − 1,2 synchronisation: hiding a digital message on a chaotic signal. y1,2 x1,2 ay1,2 ˙ = + to deal. The phase locking can be still preservable under z b z (x c) ˙1,2 = + 1,2 1,2 − effect of a certain level of noise (Figure 21). x y z α(r cos φ x ) ˙3 = − 3 − 3 + 3 m − 3 y x ay ˙3 = 3 + 3 3.3. Parameter estimation and prediction z3 b z3(x3 c) (36) ˙ = + − Now we have data and we want to learn about the system where constant ω 1andstandardparametersofthe that generated the data. Thus we will be able to predict = Rössler system a 0.15, b 0.2andc 10.0. Coupling the future behaviours and critical transitions. The deter- = 3 = = mining equations of the system are known however the constants ϵ 5 10− is between x1 and x2,andα is = × parameters are not. The goal is to find these unknown between the transmitter and the receiver. r3 is the ampli- tude of the response system given by Equation (23). )ω parameters with using synchronisation phenomena. So, is the adjustable mismatch parameter, for this illustration we blend the data with equations. The data are then used we select to drive the equations. If they and coupled in a proper way (Section 2.2.1), the equations can synchronise with data. 0.01 if bit digit 1 The key point is the following: if the parameters of slave )ω = = 0.01 if bit digit 0. system are identical with the master whose produced & − = driving signal, then the CS exhibits (synchronisation er- ror E 0) otherwise no CS (synchronisation error E> Similar to digital communication by the CS (see = Section 3.1), the modulation of parameters in the trans- 0). Therefore, it is possible to estimate the parameters by a strategy to minimise the synchronisation error E 0 mitter would hide a binary message m(t) on φm.ThePS → such as POWELL technique [87]. will exhibit between φm and φ3.Duetothechangesonthe adjustable control parameters )ω,thephasedifference We assumed that we have a limited data and we want to predict the future of the system. After the parameters between φm and φ3 varies. In other words, the phases are locked on different phase shifts. The message can be are estimated, the state of the synchronised slave matches retrieved from different phase locking values (Figure 20). the data. Because the solution of the equations is then the Because of the weak coupling, the CS never occurs same as the data, we can use the equations to predict Figure 21(a). Every 10 time unit we switch )ω parameter future dynamics. to create a digital message m(t) (010101...) Figure 21(b). The second approach is to estimate a slave system’s The hidden message on chaotic signal can be restored parameters of a master–slave system. In this case, the dynamics of master and slave is distinct. We assume that from the receiver using the phase difference between φm we have two given data-sets: one of them from master and φ3.Ifthemessagedigitis0,thenthephasedifference oscillates most time below 0, otherwise above 0 Figure system and the other one is from the slave. The governing 21(c). Therefore it is easy to restore associated message dynamics of the master–slave system is given m∗ Figure 21(d). x f (x) In real world examples, it is almost not possible to ˙ = create identical systems, and the noise is always an issue y g( y) h(x, y) ˙ = + 222 D. EROGLU ET AL.
(a)
(b)
(c)
(d)
Figure 22. Only two data-sets are known from a master–slave system, into the dashed rectangle, without any info about the parameters. An auxiliary system is driven by the data from Figure 23. Illustration of parameter estimation. Standard the master and measures the amplitude difference between the parameters of Lorenz system. auxiliary and the data from the slave system. If the difference is 0, then there is a CS that means the parameters of the slave and the auxiliary are identical. where x and y are the states of master and slave systems respectively. We aim to estimate the parameters of g. Here we cannot use y data to drive another g system directly as in previous approach since y is driven by x.If we know that master–slave system is in the GS and the Figure 24. Scheme of anticipating synchronisation with memory coupling function h is known, then we can apply the aux- in the driver systems. iliary system approach which is the master system drives an auxiliary (copy of the slave) z Figure 22. We expect that the CS exhibits between the slave y and auxiliary z method allows us to estimate the parameters of the slave systems if the parameters are identical. system (Figure 23). Using the GS idea, the problem turned into the CS problem. From now on, minimising the synchronisation 3.4. Chaos anticipation error E 0 technique can be used to estimate the → unknown parameters. Similar to the previous approach, Chaos is unpredictable but synchronisation can help pre- the future of the system can be predicted as well. dicting the state of a chaotic system ahead of time. Antic- ipating synchronisation (AS) is a good approach for the Example: Consider a Lorenz system with classical pa- future prediction since the slave system synchronises with rameters is driven by a Rössler system. We have only two the upcoming states of the master system at time t τ data sets, x -component of the Rössler (Equation (17)) + 1 where τ is a time delay. The occurrence of AS depends on and y -component of the Lorenz (Equation (14)). Then 1 the coupling constant α.Therefore,itisnotdependenton we drive an auxiliary system z by x as 1 isolated dynamics or time delay τ and regarding to type of desired application higher dimensional chaotic systems z σ (z z ) α(x z ) ˙1 = e 2 − 1 + 1 − 1 can be used for an arbitrary time delay. This anticipation z z (ρ z ) z ˙2 = 1 e − 3 − 2 of chaos can be used or is used in applications such z3 βez3 z1z2. as semiconductor lasers with optical feedback, secure ˙ = − + communications [88]. The goal is to find the parameters of slave system. The Consider two chaotic systems in a master–slave in- spontaneous synchronisation error is teraction and the master has a certain delay τ feedback Figure 24. Because of the internal delay feedbacks, it may E(t) z1 y1 . (37) well happen that the master x and slave y synchronise = ∥ − ∥ but with some time delay Adjusting the parameters of z(σe, ρe, βe) we minimise the simultaneous error E(t) by Powell’s algorithm [87]. This x(t) y(t τ) = − CONTEMPORARY PHYSICS 223
Figure 25. Anticipating chaotic synchronisation. In the beginning, the systems are not in harmony. After a transient time both systems are getting into a time-delay τ synchronisation. For this illustration τ 2. (a) time series of the systems (b) phase space and synchronisation = manifold of the system. The red circle is the initial condition for the trajectory of (x, yτ ).
When this happens we have delay-differential equations for the parameters α 1, = µ 20 and τ 2. The simulation starts from a random = = y(t) x(t τ). initial condition. After enough transient time t,themas- = + ter x and the slave y systems synchronise with a time delay Hence, although the system x is fully chaotic, we can (τ)Figure25(a). The transient time can be observed from precisely predict its future state from the system y. In the phase space of x and yτ . The initial condition is given other words the slave system anticipates the master sys- by a red circle in Figure 25(b), the trajectory converges to tem. This kind of synchronisation is called anticipated the synchronisation manifold (x yτ ). = synchronisation (AS). Example: The AS can occur for higher dimensional Example: We consider two coupled Ikeda equations, chaotic system. For such cases the critical coupling con- stant can be positive (αc > 0). The AS can be obtained x αx µ sin xτ ˙ = − − without delayed state in the master system, that is, with- y αy µ sin x (38) out memory in the master system. The scheme of this ˙ = − − model is given in Figure 26.Wecandemonstratethis We use the notation yτ y(t τ).Theschemeofthe case with Rössler system, the governing equations are = − system is given in Figure 24.Thesynchronisationerror given by for delayed system is given by,
x1 x2 x3 z x yτ ˙ = − − = − x x ax ˙2 = 1 + 2 and to show that synchronisation is attractive we analyse x b x (x c) ˙3 = + 3 1 − the first variational equation y y y α(x y τ ) ˙1 = − 2 − 3 + 1 − 1, y2 y1 ay2 z x yτ ˙ = + ˙ =˙− ˙ y3 b y3(y1 c) (40) αx µ sin xτ ( αyτ µ sin xτ ) ˙ = + − = − − − − − αz. (39) = − where the parameters are a 0.15, b 0.2andc 10. α = = = The solution is z(t) z e t and for α > 0thesynchro- We simulate Equation (40) for no AS (α < αc)Figure27 = 0 − nisation is globally exponentially stable. and AS (α > αc)Figure27 cases. In this memoryless AS To illustrate AS, we simulate Equation (38)witha approach, the synchronisation is also dependent on time τ fourth-order Runge–Kutta integrator for the delay . 224 D. EROGLU ET AL.
of pacemaker cells. Each cell when isolated has its own rhythm, but when put together these cells interact and behave in unison to deliver the strong electrical pulse that make our heart beat [1]. The dynamics of the pacemaker cells are modelled by phase oscillators φi with distinct frequencies ωi and the coupling function is a simple sine function H(φ φ ) sin (φ φ ) [94,98]. Figure 26. Scheme of anticipating synchronisation without 1 − 2 = 1 − 2 memory in the driver systems. Laser Arrays.Laserscanbearrangedarraysorcomplex networks. In this case, one is interested in the electrical 4. Synchronisation in complex networks field dynamics. Such electrical field is govern by equations with interaction akin to diffusion Ref. [99]. So, the ap- Synchronisation is commonly found in networks of nat- proach presented here can be extrapolated to such lasers ural and mankind-made systems such as neural dynam- under slight changes. ics [89–91], cardiovascular systems [92–94], power grids In fact, when we considering periodic oscillators2 the [95], superconducting Joseph junctions [3]. The theory above model is a normal form for the networked sys- we presented in the previous chapters can be generalised tem. That is, the isolated system has a periodic expo- to understand certain aspects of synchronisation in these nentially attracting orbit, we couple the system, and in complex systems. the weak coupling regime, the amplitudes will change We will focus on diffusively interacting identical os- only slightly but the phases can change by large amounts. cillators, so the dynamics of the coupled system reads So the dynamics can be described only in terms of the as phases. The phase description will again fit in our Equation (41). N dxi Our synchronisation results given in the previous sec- f (xi) α Aij H(xj) H(xi) ,(41) dt = + [ − ] tions are exponentially stable. In other words, if once j 1 %= the trajectories are into the synchronisation subset, the where f Rn Rn describes the dynamics of the solution is robust and persistent to the perturbations. : → For N coupled nonidentical systems ( f f isolated system, α is the overall coupling strength, N is 1 ̸= 2 ̸=···̸= the number of oscillators, H Rn Rn is the cou- f N ), complete synchronisation is not possible. However, : → pling function. Finally, Aij dictates who is interacting because of exponentially stable solutions, highly coher- with whom. A 1ifi receives a connection from j ent state around the synchronisation subset can be still ij = and 0 otherwise. The matrix A (the dimension is N N) observed [100,101]. × provides the network linking structure and it is called adjacency matrix. This matrix will play an clear role in 4.1. Interactions in terms of Laplacian the analysis. Because of the diffusive nature of the interaction, it is pos- The network dynamics of diffusively coupled system sible to represent the coupling in terms of the Laplacian in Equation (41) models many physical systems. A few matrix .Indeed, specific examples are: L Electronic Circuits with Resistive interaction. Electric cir- N cuits, e.g. Chua, Roessler-like, Lorenz-like, can be cou- A H(x ) H(x ) ij[ j − i ] pled over their resistors then Equation (41)modelsthis j 1 %= system [30]. Another case, only one electric circuit can be N N driven by an external signal as master–slave system (for A H(x ) H(x ) A = ij j − i ij details, see Section 2.2.1)[96]. j 1 j 1 %= %= Neuron Networks with Electrical Coupling.Inbrainnet- N work, f can be the isolated neuron dynamics modelled by A H(x ) k H(x ) = ij j − i i differential equations with chaotic or periodic behaviour j 1 %= and having different time-scales, that is, the isolated can N have burst and single regime [97]andH the electrical (A δ k )H(x ) = ij − ij i j synapses H(x x ) (x x ,0,0). j 1 1 − 2 = 1 − 2 %= Stable Biological System. In biological systems when the isolated dynamics has a stable periodic motion then typ- where k N A is the degree of the ith node, δ is i = j 1 ij ij ically one can rephrase the network dynamics in terms the Kronecker delta,= and recalling that L δ k A . ij = ij i − ij of our model. For instance, the heart consists of millions we obtain CONTEMPORARY PHYSICS 225
Figure 27. Anticipating synchronisation does not obtain for (α < αc), otherwise (α > αc)theASexhibits.Forthisillustrationthedelayis selected as τ 0.2. (a) time series of the systems where α < α and (b) phase space. (c) time series of the systems where α > α and = c c (d) phase space and synchronisation manifold of the system. The red circle is the initial condition for the trajectory of (x, yτ ).
N dxi trajectories, without coupling, any small perturbation on f (xi) α Lij H(xj). (42) dt = − the globally synchronised motion will grow exponentially j 1 %= fast, and lead to distinct behaviour between the node dynamics. We wish to address the local stability of the Our results will depend on this representation and on the globally synchronised state. That is, if all trajectories start spectral properties of L. close together would become synchronised Notice that
lim xi(t) xj(t) 0 x (t) x (t) x (t) s(t), t ∥ − ∥ = 1 = 2 =···= N = →∞ is an invariant state for all coupling strength α,because The goal of the remaining exposition is to answer this the Laplacian is zero row sum. When α 0theoscillators question. Before, we continue with the analysis, we will = are decoupled, and Equation (52) describes N copies of briefly review some examples and constructions of graphs the same oscillator with distinct initial conditions. Since, and discuss the relevant aspects necessary to tackle for the chaotic behaviour leads to a divergence of nearby problem. 226 D. EROGLU ET AL.
4.2. Relation to other types of synchronisation We will focus on the transition to complete synchroni- sation, which is mainly related to Section 2.2.Thisisno severe restriction as in certain scenarios other types of synchronisation can often be formulated in terms of our model. Extension to Phase Synchronisation. As we discussed in the introduction of Section 4,ifthedynamicsof f is pe- Figure 28. A network visualisation with eight nodes and ten links. riodic then we can introduce a phase variable which will follow our main system of equations Equation (41) as the phase reduction tells us that generically the interaction four for Figure 30(b), three for Figure 30(c) and (d), and between phases are diffusive. Moreover, because our re- two for Figure 30(e). Therefore the shortest path length, sults will yield robust transition to synchronisation, if the also called geodesic path, between these two red nodes is oscillators are nearly identical the phase synchronisation illustrated in Figure 30(e). will persist. The network diameter d is the longest length of the short- Extension to Generalised Synchronisation. Roughly speak- est path between all possible pairs of nodes. In order to ing a form of generalised synchronisation in networks is computethediameterofagraph,firstwefindtheshortest the so-called pinning control, where one tries to control path between each pair of nodes. The longest length of all the behaviour of synchronised trajectories by driving the these geodesic paths is the diameter of the graph. If there network with external nodes. One extends the network to is an isolated node (a node without any connections) or include the driver node. Therefore, the theory necessary disconnected network components, then the diameter of to tackle this problem is the same as presented here. The the network is infinite. A network of finite diameter is main question is how to connect the driver nodes in such called connected (Figure 29(a)), otherwise disconnected a way that control is effective. (Figure 29(b)). A network is directed if the links transmit the information towards only associated direction Figure 29(c). If the 4.3. Complex networks graph is directed then there are two connected nodes say, u and v,suchthatu reachable from v, but v is not A network, also called graph G in mathematical literature, reachable from u.SeeFigure29(c) for an illustration. is a set of N elements, called nodes (or vertices), con- A network is weighted if links have different importance nected by a set of Mlinks(or edges) Figure 28.Networks from each other or the links may be carry different amount represent interacting elements and are all around from of information. Such graphs are called weighted networks biological systems, e.g. swarm of fireflies, food webs or Figure 29(d). Moreover, a network may have self-loops, brain networks, to mankind made systems, e.g. the world that is, a node can affect itself as well Figure 29(d). wide web, power grids, transportation networks or social A graph can be described by its adjacency matrix A networks. with N N elements A .TheadjacencymatrixA encodes × ij A network is called simple if the nodes do not have self- the topological information, and is defined as A 1ifi ij = loops (i.e. nodes have connections to themselves). An and j are connected, otherwise 0. Therefore, the adjacency illustration of a simple network is found in Figure 29(a). matrix of an undirected network is symmetric, A A . ij = ji Anonsimplenetwork,therefore,containingconnections The degree ki of the ith node is the number of edges is depicted in Figure 29(d). We need a bit of technology belongs to the node, defined as from graph theory to make sense of our networks. A few basic notions are as follows k A . i = ij Anetworkisundirectedif there is no distinction between j the two nodes associated with each link Figure 29(a). % A path in a network isaroute(sequenceofedges)between connected nodes without repeating i.e. a path can visit a The Laplacian matrix L is another way to represent the node only once. The length of a path is the number of network, defined as links in the path. See further details in Refs. [102,103]. k if i j In Figure 30 we illustrate the paths of two selected (red) i = L 1ifi and j are connected (43) nodes in a network. Between two red nodes there are ij ⎧ = − . five different paths and each path is given in subplots of ⎨ 0otherwise Figure 30. The length of paths are five for Figure 30(a), ⎩ CONTEMPORARY PHYSICS 227
(a) (b) (c) (d)
Figure 29. Visualisation of network types: (a) an unweighted simple network, (b) a disconnected network, (c) a directed network, (d) a weighted network with self-loops.
(a) (b) (c) (d) (e)
Figure 30. Visualisation of paths (red dashed) between two selected nodes (red) in a network: path length of (a) is five, (b) is four, (c) and (d) are three and (e) is two. Therefore the shortest path length for these two red nodes is two.
There is a direct relationship between the Laplacian L that is, as N tends to infinity, the probability that a graph and the adjacency matrix A. In a compact form it reads on n vertices is connected tends to 1. The degree is pretty homogeneous, almost surely every node has the same Lij δijki Aij expected degree [105]. = − Small World network is a random graph model which where δ is the Kronecker delta, which is 1 if i j possesses the small-world properties; i.e. the average path ij = and 0 otherwise. We demonstrate some example network length is short and clustering is large. The network is sketches with their adjacency A and Laplacian L matrices generated from a 2k regular graph, each link of the graph is rewired with a probability p.Thereforeifp 0thenthe in Figure 31. = there is no rewiring and the graph is 2k regular. For p 1 The networks we encounter in real applications have = then each link is rewired i.e. the graph is approaching to a wilder connection structure. Typical examples are cor- kN Erdös–Rényi network with p N .Thesmall-world tical networks, the Internet, power grids and metabolic = 2 2 networks [103,104]. These networks don’t have a regular properties come true between 0
Figure 31. Various network examples with six nodes. Their adjacency A and Laplacian L matrices.
Random networks serve as a proxy to many applica- and 0 with probability 1 p , where − ij tions as well as a surrogate. There are many nice ways to 1 construct random network ρ . = N Configuration Model is a random network model created i 1 wi by the degree distribution P(k). If the degree distribution = To ensure that p 1itassumedthat. w w(N) is of a graph is known, then the nodes with associated num- ij ≤ = ber of links are known however the connection structure chosen so that between nodes is unknown. The nodes can be drawn )2ρ 1. (44) ≤ with their stubs (half links) Figure 34(a), then randomly The degree k A of the ith is a random variable. these stubs can be linked and two stubs create a proper i = j ij An interesting property of this model is that under this link Figure 34(b). This process is a random matching, . construction w is the expected value of k ,thatis, obviously different network structures can arise from this i i random process. Ew(k ) Ew(A ) w Expected degrees Fix a network of N nodes and consider i = ij = i j avector % w (w , w , ..., w ), = 1 2 N So, the different to the configuration model is that we In this model, each link Aij between nodes i and j is do not fix the node degree, but rather obtain the degree an independent Bernoulli variable taking value 1 with probabilistically. This model also have many desirable success probability concentration properties in the large N limit.
p w w ρ, ij = i j CONTEMPORARY PHYSICS 229
Figure 32. Watts–Strogatz random network approach.
Figure 33. Some examples of complex networks.
Figure 34. Configuration model.
4.4. Spectral properties of the Laplacian spread. In particular, the smallest nonzero eigenvalue of L will determine the synchronisation properties of the The eigenvalues and eigenvectors of A and L tell us a network. Since the graph is undirected the matrix L is lot about the network structure. The eigenvalues of L symmetric its eigenvalues are real, and L has a complete for instance are related to how well connected is the set of orthonormal eigenvectors [109]. The next result graph and how fast a random walk on the graph could characterises important properties of the Laplacian 230 D. EROGLU ET AL.
Theorem 1: Let G be an undirected network and L its on an undirected and connected network. Assume that the associated Laplacian. Then: H is a positive definite linear operator. The reason of the assumption will appear in Step 5 of the theorem. Then, (a) L has only real eigenvalues, there is 4 4( f , H) such that for any (b) 0 is an eigenvalue and a corresponding eigenvector = is 1 (1, 1, ...,1) , where stands for the trans- = ∗ ∗ 4 α > , pose. λ (c) L is positive semidefinite, its eigenvalues enumer- 2 ated in increasing order and repeated according to the global synchronisation is uniformly asymptotically their multiplicity satisfy stable. Moreover, the transient to the globally synchronised behaviour is given the spectral gap λ2, that is, for any i and 0 λ λ λ = 1 ≤ 2 ≤ ···≤ N j (αλ2 4)t xi(t) xj(t) Ce− − , (d) The multiplicity of 0 as an eigenvalue of L equals ∥ − ∥≤ the number of connect components of G. where C is a constant. The above result relates the threshold coupling for Therefore, λ2 is bounded away from zero whenever the network is connected. The smallest non-zero eigen- synchronisation in contributions coming solely from dy- value is known as algebraic connectivity, and it is often namics 4, and network structure λ2. called the Fiedler value. The spectrum of the Laplacian Definition 1: [Synchronisation Threshold] We call is also related to some other topological invariants. One 4( f , H) of the most interesting connections is its relation to the αc( f , H, G) diameter, size and degrees. = λ2(G) Theorem 2: Let G be a simple network of size N and L the critical synchronisation coupling. its associated Laplacian. Then: Therefore, for a fixed node dynamics f and coupling 4 (1) [110] λ2 H we can analyse how distinct network facilitates or in- ≥ Nd hibits global synchronisation. To continue our discussion N we need the following (2) [111] λ2 k1 ≤ N 1 Definition 2: [Better Synchronisable]We say that the − We will not present the proof of the Theorem here, network G1 is more synchronisable than G2 if for fixed f however, they can be found in references we provide in and H we have the theorem. We suggest the reader to see further bounds on the spectrum of the Laplacian in Ref. [112]. Also αc(G1) < αc(G2) Ref. [113] presents many applications of the Laplacian eigenvalues to diverse problems. One of the main goals Likewise, we say that G1 has better synchronisability than in spectral graph theory is to obtain better bounds by G2. having access to further information on the graphs. For a fixed network size, the magnitude of λ reflects 2 5.1. Which networks synchronise best how well connected is graph. In this setting, the coupling function H is positive defi- 5. Stability of synchronised solutions nite and the network is undirected, the synchronisability depends only on the spectral gap. Using the previous We will state two results on network synchronisation. study on the properties of various networks presented in The first assumes that the coupling function H is linear Section 4.4.WeconsidernetworksofN nodes then and the network is undirected. This assumption facilitates A complete graph is the most synchronisable. In fact, the discussion of the main ideas. Then, we will discuss • α 1/N. So, the larger the graph the less coupling the case where the coupling is non-linear. Basically, the c ≈ results are the same with an additional complication as strength is necessary to synchronise the network. Apathor ring are poorly synchronisable. For these the latter involves the theory of Lyapunov exponents. • 2 networks, αc N . Theorem 3: Consider the diffusively coupled network ≈ 2k-regular graphs share the same properties for also model • N poorly synchronisable when k N. ≪ x f (x ) α L H(x ), Erdös–Renyi graphs the synchronisation properties i = i − ij j • j 1 depend only on the mean degree α 1/ k . %= c ≈ ⟨ ⟩ CONTEMPORARY PHYSICS 231
Small world networks are better than regular but Av λ v and Bu µ u then • i = i i j = j j worse than random graphs. In the limit of large A B(vi uj) λiµjvi uj graphs α 1/s where s is the fraction of added ⊗ ⊗ = ⊗ c ≈ random links. In general, adding links to a network Since we are assuming that L is undirected and H favours synchronisation. N is positive definite the eigenvectors vi i 1 of L form a { } = basis of RN . Likewise, the eigenvectors of H form a basis of Rm. This implies that the eigenvectors of L H form ⊗ 5.2. Proof of the stable synchronisation abasisofRNn. Using this fact we can represent X as
Now we present the proof of Theorem 3.Weomitsome N details that are not relevant for the understanding of the X v y = i ⊗ i proof. A full discussion of the proof can be found in i 1 %= [116]. We will show that whenever the nodes start close together they tend to the same future dynamics, that is, where yi is the coordinates of X in the Kronecker basis. For sake of simplicity we call y s, and remember that xi(t) xj(t) 0, for any i and j. For pedagogical 1 = ∥ − ∥→ v 1 is an eigenvector. Hence purposes we split the proof into six main steps. 1 = Step 1: Kronecker Form. We have N coupled equa- X 1 s U, tions each has dimension n.Becauseofthenicestructure = ⊗ + of the interaction we can use the Kronecker product to where write them as a single block. Given two matrices A N p q r s ∈ R × and B R × ,theKroneckerProductofthema- U v y . ∈ = i ⊗ i trices A and B and defined as the matrix i 2 %= In this way we split the contribution in the direction of A11 B A1q B the global synchronisation and U,whichaccountsforthe . ···. . A B . .. . , contribution of the transversal. Note that if U converges ⊗ = ⎛ . . ⎞ A A to zero then the system completely synchronise, that is X ⎜ p1 B pq B ⎟ ··· converges to 1 s which clearly implies that ⎝ ⎠ ⊗ we introduce the following notation x x s 1 =···= N = X col(x1, ..., xN ), = Thegoalthenistoobtainconditionssothat U 0. ∥ ∥→ Step 3: Variational equations for the Transversal where col denotes the vector formed by stacking the Modes. The equation of motion in terms of the Laplacian columns vectors xi into a single vector. Similarly modes decomposition reads
F(X) col( f (x1), ..., f (xN )). ds dU = 1 F(1 s U) ⊗ dt + dt = ⊗ + Then Equation (52) can be arranged into a compact form α(L H) 1 s U , − ⊗ ⊗ + dX 2 3 F(X) α(L H)X,(45)We assume that U is small and perform a Taylor expan- dt = − ⊗ sion about the synchronisation manifold. where is the Kronecker product. The easiest way to ⊗ F(1 s U) F(1 s) DF(1 s)U R(U), check that this is correct is to compute the ith block of ⊗ + = ⊗ + ⊗ + dimension n and compare with the equation for the ith where R(U) is the Taylor remainder R(U) O( U 2). ∥ ∥ = ∥ ∥ node. Using the Kronecker product properties Equation (46) Step 2: Transversal Laplacian Eigenmodes. The and the fact that L1 0,togetherwith Kronecker product has many nice properties such as = ds 1 (1 ) 1 ( ) (A B)(C D) AC BD. (46) F s f s ⊗ ⊗ = ⊗ ⊗ dt = ⊗ = ⊗ And this holds whenever the matrix multiplication make and we have sense. A nice consequence of the multiplication result in dU DF(1 s) α(L H) U R(U) (47) Kronecker form is that if dt =[ ⊗ − ⊗ ] + 232 D. EROGLU ET AL. and likewise formula. So, first notice that
κ Ht κλmin(H) DF(1 s)U I D f (s) U, u(t) e− u0 u(t) K u0 e− ⊗ =[ N ⊗ ] = ⇒∥ ∥≤ ∥ ∥ therefore, the first variational equation for the transversal where λmin(H) is the smallest eigenvalue of H. So, by the modes reads variation of constants formulate t dU κ Ht κ H(t τ) ( ) α . u(t) e− u0 e− − D f (s(τ))dτ IN D f s L I U (48) = + dt =[ ⊗ − ⊗ ] (0 Step 4: Decoupling of Transversal Modes. Instead taking norms of analysing the full set of equations, we can do much t better by projecting the equation into subspace W κλmin(H)(t τ) i = u(t) K u0 e− − D f (s(τ)) dτ span v I . Let P RN Rn W be a projection ∥ ∥ = ∥ ∥ + 0 ∥ ∥ { i ⊗ } i : ⊗ → i ( operator given by P v v∗ I, it follows that P is an i = i i ⊗ i where K is a constant and defining M sup D f (s(t)) orthogonal projection since v ’s are orthonormal. Using f = t ∥ ∥ i by Gronwal inequality Equation (48) and the identity Equation (46)weobtain ( κλ (H) M )t u(t) K u e − min + f . dU ∥ ∥ = 1∥ 0∥ Pi viv∗ D f (s) α(viv∗ L) I U,(49) dt =[ i ⊗ − i ⊗ ] The trivial solution will be exponentially stable when v v∗ D f (s) αλ (v v∗) I U (50) =[ i i ⊗ − i i i ⊗ ] Mf where in the last passage we used that the network is κλmin(H) Mf < 0 κ > 4 − + ⇒ = λmin(H) undirected implying v L λ v∗.Usingandthefact i ∗ = i i that v∗v δ , where is δ the Kronecker delta, we have Recall that taking κ αλ > 4 we are stabilising j i = ij ij = i N the equation for the ith block. But, once we stabilise the that (vivi∗ I)U j 2 viδij yi. Moreover, since Pi ⊗ = = ⊗ ( ) second block all blocks will be stable (because λ2 is the does not depend on time PiU˙ Pi˙U . = smallest nonzero eigenvalue) N N d yi viδij viδij D f (s) αλi I y αλN αλ3 αλ2 4 ⊗ dt = ⊗ [ − ] i ≥ ···≥ ≥ ≥ j 2 j 2 %= %= Hence, the stability condition so that all blocks have the nonzero coefficients give the dynamics in Wi.Hence, exponentially stable trivial solution is 4 d yi α D f (s) αλi I yi > dt =[ − ] λ2
All blocks have the same form which are different only Step 6: Norm Estimates and Nonlinearities. Using by λi,theith eigenvalue of L. We can write all the blocks the bounds for the blocks it is easy to obtain a bound for in a parametric form the norm of the evolution operator. Indeed, note that
du N N (t) ,(51) K u U 2 vi yi vi yi 2 dt = ∥ ∥ = 6 ⊗ 6 ≤ ∥ ∥∥ ∥ 6 i 2 62 i 2 6%= 6 %= where 6 6 Therefore, 6 6 K(t) D f (s(t)) κ I = − N with κ R. Hence if κ αλ we have the equation ηi(t s) i U v K e− − y (s) ∈ = ∥ ∥2 ≤ ∥ i∥ i ∥ i ∥ for the ith block. This is just the same type of equation i 2 we encounter before in the example of the two coupled %= oscillators, see Equation (9). Now using that e ηi(t s) e (αλi 4)(t s),so − − ≤ − − − Step 5. Stability.BecauseH is positive definite we can η(t s) first solve the homogeneous equation u κ Hu.This U(t) K e− − ˙ = − ∥ ∥2 ≤ 2 equation has an globally attracting trivial solution. Then, we incorporate D f in terms of the variation of constants with η αλ 4 for any t s. = 2 − ≥ CONTEMPORARY PHYSICS 233
Because the trivial solution is exponentially stable we obtain (uniformly in s(t)) by the principle of linearisation, we conclude that the nonlinearities coming Taylor remain- g(x , x ) G(s)(ξ ξ ) R(ξ , ξ ) j i = j − i + i j der does not affect the stability of the trivial solution, which correspond to the global synchronisation. where G(s) D g(s, s) and R(ξ , ξ ) contains quadratic = 1 i j The claim about the transient is straightforward, be- terms. So, the first variational equation about the syn- cause all norms are equivalent in finite dimensions we chronisation manifold can take N η(t s) ξ i D f (s(t))ξ i α Aij G(s)(ξ j ξ i) (53) X(t) 1 s(t) K3e− − U(s) ˙ = + − ∥ − ⊗ ∥∞ ≤ ∥ ∥∞ j 1 %= η(t s) N implying that maxi xi(t) s(t) 2 K3e− − U(s) ∥ − ∥ ≤ ∥ ∥∞ D f (s(t))ξ αG(s) L ξ . (54) and in virtue of the triangular inequality = i − ij j j 1 %=
xi(t) xj(t) xi(t) s(t) xi(t) s(t) ∥ − ∥∞ ≤∥ − ∥∞ + ∥ − ∥∞ Now we can perform the same steps as before. In fact, Steps 1–4 remain unchanged. The change difference and using the previous bound, we concluding the is Step 5, which concerns the stability of the modes. proof. ! Performing all the steps 1 to 4 we obtain the parametric equation for the modes 6. General diffusive coupling and master u D f (s(t)) κG(s(t)) u stability function ˙ =[ − ] Until now we have considered linear coupling functions And we can no longer apply the trick of using the coupling which are positive definite. This assumption can be re- function H to solve the equation. Here, G(s(t)) depends laxed and thereby we are generalise our previous results. on time and this generality tackling the stability of the The statement will then become rather technical and will trivial solution is challenging. be beyond the scope of our review. So, here we will discuss The main idea is to fix κ compute the maximum Lya- the main ideas but will not give much details on the punov exponent $(κ) as technical issues. Consider the function g Rn Rn n : × → $(κ) R . We say that g is diffusive if u(t) Ce t ∥ ∥≤ g(x, x) 0andg(x, y) g( y, x) = = − see Appendix. The map
Hence, we can extend the model to a general diffusive κ $(κ) (55) coupling 5→ is called master stability function. Notice that if $(κ) < 0 N 1 2 when κ (αc , αc ) then u 0. xi f (xi) α Aij g(xj, xj). (52) ∈ ∥ ∥→ ˙ = + The stability condition then become j 1 %= α1 αλ αλ α2 We perform the analysis close to synchronisation x c 2 N c i = ≤ ≤ ···≤ ≤ s ξ i so + Or
g(xj, xj) g(s ξ j, s ξ i) 2 = + + α λN c (56) g(s, s) D1 g(s, s)ξ j D2 g(s, s)ξ i 1 = + + αc ≥ λ2 This is a well studied condition. Much energy has been but because the coupling is diffusive devoted to study the master stability function Equation (55), see e.g. [117]. D g(s, s) D g(s, s) 2 = − 1 234 D. EROGLU ET AL.
6.1. Examples of master stability functions 6.2. Synchronisation conditions and synchronisation loss Now let us consider coupled Rössler systems which are coupled through their x-coordinates: In Section 5.1,wediscussedthesynchronisabilityofnet- work types when there is only one critical threshold in N 2 other words αc . This case is true when H is a x y z α A (x x ) →∞ ˙i = − i − i + ij j − i positive matrix. In the general case, there are two finite j 1 %= critical couplings Equation (56)andthemasterstability y x ay ˙i = i + i function has a finite stability region between these critical zi b zi(xi c). (57) points (as discussed in the above section). Now we discuss ˙ = + − this synchronisation scenario for various network types. In order to compute $max(κ),wefindthatD f and To quantify the synchronisability, we will use the network DH are given by properties given in Table 1. 2k-regular graphs – Diameter driven synchronisation loss. 0 1 1 When k N,themeangeodesicdistance(shortest − − ≪ D f (s) 1 a 0 and path length) between nodes are increasing very fast as = ⎛ ⎞ z 0 x c N increasing. Hence, while the diameter of the network ∗ ∗ − ⎝ ⎠ is increasing d 2N/k, speed of information exchange 100 = DH H 000 , (58) between the nodes is decreasing drastically. In this case, = = ⎛ ⎞ 000 the complete synchronisation is not stable and it is visible from the Laplacian spectrum of the graph. For N ⎝ ⎠ →∞ x and z are the components of s.Theconstantsarea and k N, the extremal eigenvalues given in Table 1 can = ≪ 0.2, b 0.2andc 5.7. be extended to Taylor expansion and major role playing = = To compute $(κ), we first simulate the isolated dy- part can be rewritten as namics s f (s) and obtain the trajectory s(t), then ˙ = 4π 2(k 1 )3 we feed this trajectory to u D f (s(t)) κ H u and 2 N ˙ λ2 + and λN 2k κ =[ − ] = 3N2 = N 1 then for each estimate the maximal Lyapunov exponent − $(κ).TheresultisdepictedinFigure35. The ratio between the largest λN and second smallest Stability region where $ is negative is bounded be- eigenvalue λ2 of the Laplacian is not growing in the same tween α1 0.13 and α2 4.4. So, if the network is a c ≈ c ≈ scale, complete graph then λ2 λN N,sothenetwork λN =···= = N2/k2 synchronisation when λ2 ≈ and the condition in Equation (56) is never satisfied. α2 α1 ER graphs – Optimal Synchronisation.Inthiscase,the c > α > c N N extremal eigenvalues λ2 and λN of the Laplacian matrix increase in the same scale. The diameter of the network increases very slowly d log N when the network size N ≈ increases Table 1. Therefore the synchronisation is stable for any scale of size. BA networks – Heterogeneity driven synchronisation loss. When the network is too heterogeneous the complete synchronisation is unstable, this is because the extremal eigenvalues λ2 and λN grow in different scales and the condition in Equation (56)isnevermet.Forinstance, consider a BA network. Then, the eigenvalues satisfy
λ m and λ m N1/2 2 ≈ 0 N ≈ 0
where m0 is the mean degree. Hence, the eigenration becomes λN N1/2 λ2 ≈ Figure 35. Master stability function for x-coupled Rössler this should be compared to the stability interval given by attractors on a network structure. the master stability function. Lets consider the example CONTEMPORARY PHYSICS 235 ] ] ] ] ] 56 56 115 114 103 [ [ [ [ [ Reference is odd is even is odd N is even ) N ) N k if N if N N Np N d ⌉ log log k log 2if log log ( 2 2log ⌉ ( )/ k )/ O 2if 1 2 1 / / N + N + ⌈ N N ( ( ⌈ .Forrandomnetworks,themeandegreeis 33 )) 1 ( . Consider the degree distribution for the SF network 1 p O 1 − γ 11 kO + 2 k N 1 − min 2 0 k ∼ )( N m p ∼ − 1 )) ( 1 ( and random networks in Figure 1 11 2 0 k O 2 k m − − mm m 2 31 + max k ∼ ∼ N 1 ( Cp + : ) : 1 ) 1 − " N N − ) " ( N 1 N ) π ( 1 − 3 ) N π N − 1 1 ) ( N N m N 1 1 − π ( 2 + γ π ) ! + k 2 Np pN N N ! 422 N 2 NN k 0 − λ ( sin 2 ∼ N 9 ( m m sin ( 9 ∼ sin 2ln sin − − 1 1 + + k 2 k 2 ∼ Cp + 8 8 N π ) N π " 3 1 ) ) $ 1 3 1 N π + 1 − π N 2 π N + 2 k γ 2 ( 2 k # sin ( is the degree distribution of the network. The randomness of the networks arises via probability 2 2 7 Np mean degree. Examples of nonrandom networks are depicted in Figure sin ( − N ) λ 7 1 k ∼ m sin ! ( 2cos sin P 0 − − m − 1 1 ∼ + + k where k 2 ) k nodes with ∼ ( N kP k is constant. . C )2 . = m γ m − Network of k = ) k -regular 2 ( k ER network StarHypercube (Q 1 P SF network 2 Table 1. given by Network Complete Ring 2 SW network 236 D. EROGLU ET AL. in the above section with the Rössler Equation (58). The would yield E O(N 1/2). For complete networks, = − master stability function gives (as seen in Figure 35)an the interaction and synchronisation yields E O(N 1) = − stability interval α2/α1 34. The stability conditions which is a large improvement over the naive application c c ≈ Equation (56) reads as of the Central Limit theorem. In certain sense, this shows that interacting maybe better then isolation. N1/2 < 34 Nonidentical Coupling Functions. In many applications the coupling function are not identical and has fluctuat- So, when the BA network is large enough it is not possible ing components [121]. Consider an undirected networks to synchronise the system. In particular the critical system of identical oscillators and coupling function size to be able to synchronisation a network of Rössler as in Equation (58) is, therefore, H (x x , t) H(x x ) P (x x , t) ij i − j = i − j + ij i − j 3 N 10 . where Pij(xi xj, t) η.Inthiscase,thenetwork ≈ ∥ − ∥≤ structure will play a major on the size of perturbation η. 6.2.1. Extensions If the network is random and the degree distribution ho- There are a few extensions of the model. Here we discuss mogeneous, then even for large perturbations δ synchro- afewdirections. nisation is stable. If the network is heterogeneous degrees such as Barabasi–Albert then typically δ O(N β ) is Directed Networks. The major problem considering di- c = − rected networks is that they may not be diagonalisable. the critical perturbation size. If δ > δc synchronisation So, the decoupling of transversal modes by projection is becomes unstable, solely because of the interaction be- a nontrivial steps. There are a few ways to overcome this. tween network structure and perturbations in the cou- The first, using Jordan decomposition of the Laplacian. pling function. The other possibility is to perturb the Laplacian to make Cluster Synchronisation.Accordingtosimilaritiesincou- the eigenvalues simples. This must be done in such a way pled systems, such as symmetries in network topology that the perturbation does not spoil the stability. In both or identical dynamics in a diverse population or equally cases, the stability condition remain unchanged. Only the time-delayed nodes in differently distributed feedbacks, transients may be longer. the partial or cluster synchronisation can emerge. In When the network is nondiagonalisable, small per- order to enlighten the reason of these cluster synchro- turbations in the network may lead to large perturba- nisation cases, many techniques are developed and ex- tions in the eigenvalues (the eigenvalues in this case are perimental observations are analysed [122–127]. not differentiable functions of the perturbations) [118]. The symmetries are easy to detect for some network Moreover, structural improvements in the network may geometries for instance Bethe lattice is a regular graph lead to desynchronisation [119]. which grows from a root (parent) node by p-nodes for Nonidentical Nodes. Here we consider f (x) f (x) ℓ-levels, an example of Bethe lattice given in Figure 36 5→ + ( ) for p 3andℓ 3. The nodes in the same level of ri x, t , where ri is either a perturbation of the vector field = = or a signal playing the role of noise. When ri is very small the Bethe graph, they all symmetric to each other. In synchronisation will persist [120]. For general networks, Figure 36, the levels of the graph are given in the same Bollt and co-workers [100] extended the master stability colour and the cluster synchronisation occurs for each function approach when ri is a perturbation of the vector level. field. Pereira and co-workers [101]studytheeffect of Recently, Pecora et al. put forward that all the symme- general perturbations r δ and the role that the tries in network structures are not visible directly. They ∥ i∥≤ network structure plays in suppressing the fluctuations. developed a computational group theory-based method In the case where H is positive definite and the network to reveal these hidden symmetries and predicted possible is undirected, they showed that the synchronisation error synchronisation patterns [128,129]. Time-Delay Coupling. Simultaneous coupling is not al- 1 ways possible for real world applications in other words E xi xj = n(n 1) ∥ − ∥ some time delays can occur in the interaction process. − ij % Therefore it is important to investigate synchronisabil- behaves as ity and stability of coupled time-delayed systems. The δ necessary conditions for time-delayed synchronisation is E K ≤ αλ α analytically shown by Pyragas [130]. The finding of time- 2 − c For example, if the oscillators where uncoupled and delay synchronisation is used as an application for the ri independent noise then the Central Limit theorem anticipating synchronisation (see Section 3.4). CONTEMPORARY PHYSICS 237
Jeroen S. W. Lamb is a professor of applied mathematics at Imperial College London (UK). He obtained his Masters and PhD degrees in theoretical physics in the Netherlands, from the Radboud University Nijmegen (1990) and the University of Amsterdam (1994), respectively. He has been at Imperial College London since 1999, since 2005 as a full professor. Before that, he held postdoctoral positions at the University of Warwick (UK) and the University of Houston (USA). The main thrust of his research concerns nonlinear dynamical systems, in particular bifurcation theory in the presence of symmetries and/or Hamiltonian structure, also in non-smooth and random settings.
Figure 36. Bethe lattice graph: an example of symmetries in Tiago Pereira is an associated professor network structure. Nodes in the same level of the Bethe graph of mathematics. His interest in this are exactly symmetric to each other. Therefore same colour nodes field started when he was a under- constitute a cluster. graduate student playing with electronic circuits and observed chaotic phase synchronisation. He obtain his PhD from Notes Potsdam University in Germany, and held postdoctoral positions in Berlin, Sao 1. We are using the uniform operator norm U Paulo and was a Leverhulme and Marie ∥ ∥ = sup U(t) . Curie Fellow at Imperial College London. t 0 ∥ ∥ 2. Or Roessler≥ type oscillator where the phases are well defined and the coupling between chaotic amplitudes and phases are small.
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