[
Abstract
We investigate the stability of synchronization in networks of dynamical systems with strongly delayed connections. We obtain strict conditions for synchronization of periodic and equilibrium solutions. In particular, we show the existence of a critical coupling strength , depending only on the network structure, isolated dynamics and coupling function, such that for large delay and coupling strength , the network possesses stable synchronization. The critical coupling can be chosen independently of the delay for the case of equilibria, while for the periodic solution, depends essentially on the delay and vanishes as the delay increases. We observe that, for random networks, the synchronization interval is maximal when the network is close to the connectivity threshold. We also derive scaling of the coupling parameter that allows for a synchronization of large networks for different network topologies.
Synchronization in Networks with Strongly Delayed Couplings]Synchronization in Networks with Strongly Delayed Couplings Daniel M. N. Maia et al.] \subjclass34D06, 34K08, 34K13.
Daniel M. N. Maia^{1}^{1}1Current address: Department of Mathematics and Statistics, State University of Rio Grande do Norte – UERN, MossoróRN, 59610210 Brazil., Elbert E. N. Macau
Associate Laboratory of Applied Computing and Mathematics  LAC
National Institute for Space Research  INPE
Av. dos Astronautas, 1758, São José dos CamposSP, 12227010, Brazil
Tiago Pereira
Institute of Mathematics and Computer Sciences  ICMC
University of São Paulo  USP
Av. do Trabalhador SãoCarlense 400, São CarlosSP, 13566590, Brazil
Serhiy Yanchuk
Institute of Mathematics, Technical University of Berlin
Strasse des 17. Juni 136, 10623 Berlin, Germany
(Communicated by the associate editor name)
1 Introduction
Coupled systems and networks with timedelayed interactions emerge in different fields including laser dynamics [21, 32, 11, 10, 36, 40], neural networks [39, 15, 20, 7], traffic systems [28], and others. Time delays in these systems are caused by finite signal propagation or finite reaction times. In many cases, especially in optoelectronics, delays are larger than the other time scales of the system, and they play a major role for system’s dynamics [36, 40]. For neural systems, large delays may emerge as a lump effect of a signal propagation along feedforward chains of neurons [24].
When the interaction among oscillators is diffusive, the network possesses a synchronous subspace, where all subsystems constituting the network behave identically. The stability of the synchronous subspace plays a major role in applications. For instance, neural synchronization is known to be involved in brain functioning [34], synchronization of lasers is used for communication purposes [8, 2].
Therefore, the stability of synchronization for strongly selffeedback delayed interactions has been addressed in various contexts and generalities in a mixture of analytical and numerical techniques [22, 13].
Here, we aim at studying how stability of the synchronization manifold depends on the dynamics of the isolated nodes and network structure. We focus on the case where the isolated dynamics is either periodic or an equilibrium solution. This case is amenable to detailed understanding in the limit of large delays.
Model and discussion of results: We consider a network model with a selffeedback term in the coupling. This kind of coupling is widely studied, for example, in semiconductors lasers models [36, 18], where the selffeedback interaction is generated by the optical feedback. We consider the following system of identical oscillators with timedelayed interactions [18, 13, 22, 37, 9, 7]
(1.1) 
where , is the coupling strength, is the adjacency matrix with the elements if and there is a link from to and otherwise. The function describes the uncoupled dynamics and the coupling function , , describes interactions between the nodes. More details and assumptions on the model (1.1) is given in Sec. 2.
We consider the timedelay as a large parameter. Physically, this means that the interaction time is much larger than the typical time scale of an uncoupled system, which often occurs in, e.g. coupled laser systems [36, 40].
The uncoupled units of the network have an attractor, which is either an equilibrium or periodic orbit. And, we obtain conditions for synchronization and desynchronization which depends on the network dynamics.

For the equilibrium, we show that with strong enough delay, there is a critical coupling parameter , which depends on the dynamics, coupling function and network structure, such that the equilibrium solution in the network remains stable if and only if (see Theorem 3.1). For this case, is independent of the delay.

Synchronization of stable periodic orbits is in sharp contrast to fixed point case. It is shown to be always desynchronized for long enough timedelay. However, for long but finite delay the synchronization of periodic orbits are attained for an interval that is shrinking as the delay grows to infinity, that is, for this case the critical coupling parameter depends also on the delay, , and as . See Theorem 5.1 for the precise statement.
Fixing the vector field , the coupling function and the delay , we can study how the network structure affects the critical coupling . Our results elucidate this dependence. For instance, we will show that

For BarabásiAlbert scale free network, .

For ErdösRéniy (ER) random networks, .
This shows that having a large number of connections is detrimental, and close to percolation threshold is optimal for synchronization. This is in contrast to the nondelayed case [25, 29] – best synchronization scenario happens for homogeneous networks and the large the degrees the better.
Here we use the theory of functional differential equations [17, 35], elements of the spectral theory of graphs [12, 3, 6], as well as a spectral theory for delay differential equations with strong delays [38, 23, 33, 40]. Some important ideas from Ref. [13] on the synchronization of strongly delayed networks are used as well.
The manuscript is organized as follows: In Sec. 2, we give some assumption on the considered model. In Sec. 3 we state the main results, in particular, a special case for the result of synchronization of periodic orbits. In Section 4 we give a description of the spectrum for the variational equation and prove the main results with illustrations given in Sec. 6. In Sec. 7 we discuss the relevance of the structure of the network to the synchronization condition and in Sec. 7.1 we discuss how scaling properties of the coupling parameter allow synchronization.
2 Assumptions on the network model
The initial conditions of (1.1) are history functions and we assume that is continuous for all . The state space where the solutions and initial conditions lives is the space of continuous functions endowed with the maximum norm in defined by where is some norm in .
The function is of class , . Assume that the solutions of the system are uniformly bounded, i.e., the overall system is dissipative. So the solution operator is eventually compact and thus we can apply the spectral mapping theorem. Thus, we can obtain detailed information by studying the linearized system.
The following assumptions for the vector field restrict the model (1.1) to the synchronization scenarios that we are interested in.
Assumption 2.1.
The uncoupled system possesses an exponentially locally stable equilibrium .
Assumption 2.2.
The uncoupled system possesses a locally exponentially stable periodic solution .
The coupling function , is assumed to be bounded and differentiable, is the Jacobian matrix at and .
Due to the diffusive nature of the coupling, the coupling term vanishes identically if the states of all oscillators are identical. This ensures that the synchronized state for all persists for all coupling strengths . Let be the solution specified in Assumptions 2.1 or 2.2 and a tubular neighborhood which belongs to the basin of attraction of . The set
is called the synchronization manifold. The local stability of determines the stability of the synchronization, and it depends, in particular, on spectral properties of Laplacian matrix defined as , where is the adjacency matrix and is the diagonal matrix of the network’s degrees. We assume that the matrix is diagonalizable.
3 Main Results
If the uncoupled system has stable equilibrium, then the coupled system possesses the stable equilibrium at , and the equilibrium solution remains stable for small and small delay by the principle of linearization.
Our result concerns the limit of large delay. In this setting, the principle of linearization does not apply, and the spectral theory of DDE offers the tools to tackle this problem.
Theorem 3.1, gives the interval of the coupling parameter where the equilibrium solution of (1.1) is locally exponentially stable for large delays. More specifically, it provides a critical coupling parameter such that the destabilization occurs for . Moreover, does not depend on the delay.
Theorem 3.1 (Persistence of equilibrium).
Consider system (1.1) and Assumption 2.1. Then there is and a constant such that for and
(3.1) 
and , the equilibrium solution remains locally exponentially stable. Here, is the spectral radius of the Laplacian matrix .
If the condition is fulfilled, then there exists such that the equilibrium solution is unstable for .
As follows from Theorem 3.1, the condition is the strict destabilization value for , while the value does not depend on time delay . The proof of Theorem 3.1 is presented in Sec. 4. The interval of the coupling strength will be called the synchronization window.
We also show, in Sec. 6.3, that the characteristic time in which the trajectories of (1.1) approach the synchronous (equilibrium) solution scales as
(3.2) 
for and large delay. In Sec. 6.3 we derive the estimate (3.2) and illustrate it with an example.
In the case of the synchronous periodic solutions, the situation is subtle. In particular, one can show that the periodic orbits will be desynchronized with an increasing delay, however, it is possible to synchronize them for any finite timedelay.
The generic condition for synchronization of periodic orbits is a codimension one condition. This condition is technical and it will be discussed at length later in Sec. 5.2. To avoid overly technical statements we first give our result for the special case when the isolated dynamics is given by a StuartLandau (SL) system. The network model reads
(3.3) 
where .
The uncoupled SL system is retrieved if . Note that it has one equilibrium at the origin, which is asymptotically stable for , and a stable periodic orbit for , which emerges from the Hopf bifurcation at . This model is illustrative since it represents a normal form for a Hopf bifurcation.
Theorem 3.2 (Synchronization of periodic SL system).
Consider the system (3.3) with , and . Let the condition
be fulfilled. Then, there is such that for each there is such that the synchronous periodic solution is locally exponentially stable

for if or

for if .
Moreover, as .
The Fig. 6.5 will give the numerically computed stability region from Theorem 3.2. The full statement will be given Sec. 5.2, Theorem 5.1.
Remark 3.1.
Also, roughly speaking, the critical coupling parameter in Theorem 3.2, is written as .
The main consequences of the theorems 3.1 and 3.2 for different networks are presented as corollaries below. From the point of view of the network structure, the stability of the synchronization manifold is related to the spectral radius of the Laplacian matrix . Therefore, fixing , and large enough, one can study the effect of the changes in the network to the synchronization.
We give the asymptotic behavior of the synchronization window for two important examples of complex networks, namely, heterogeneous (e.g. BarabásiAlbert (BA) scalefree network), and homogeneous network (e.g., ErdösRéniy (ER) random graph). More details about BA and ER networks and the proofs of the corollaries that follow are given in Sec. 7.
Corollary 3.1 (Synchronization window for large BA networks).
Corollary 3.2 (Synchronization window for large ER networks).
Consider the model (1.1) and its assumptions with a connected ER network with nodes. Then, for sufficiently large , the length of the synchronization window scales as . Furthermore, the synchronization window is maximal near the percolation threshold.
4 Preliminaries: variational equation and spectral theory
In this section, we introduce the relevant elements to proof our main results.
4.1 Dynamics near the synchronization manifold
In order to study the local stability of the synchronization manifold we linearize (1.1) setting with small. We get
(4.1) 
where with the degree of the node and if and otherwise. Here, is the Jacobian matrix of the vector field along and is the Jacobian of at .
Using we rewrite (4.1) as
where is the Kronecker product and is the identity matrix. As is diagonalizable we use , with , and the change of coordinates to get the block diagonal equation
(4.2) 
Then, each block of (4.2) is given by
(4.3) 
Note that and the variational equation corresponding to is
which describes the perturbations within the synchronization manifold. So, the interest concentrates now on the spectrum of (4.3) for .
It is shown in [42, 33, 23, 40] that the spectrum of (4.3) with large delay consists of two parts which we introduce here following Ref. [33]. We consider the following two cases.

Case 1: the synchronized solution is an equilibrium of the uncoupled system and then does not depend on .

Case 2: is a periodic solution of the uncoupled system, implying that .
4.2 Spectral Theory
Omitting the index and denoting , Eq. (4.3) can be rewritten as
(4.4) 
In a general case, the Floquet theory can be used to study the stability of (4.4) [17]. Using the following Floquetlike ansatz
(4.5) 
where is a nontrivial periodic function, and are Floquet exponent and Floquet multipliers respectively, one can obtain a delay differential equation (DDE) for
(4.6) 
As the function is periodic we can rewrite in terms of a fraction of the period so that the large parameter appears only as a parameter in in Eq. (4.6):
(4.7) 
where . The system (4.7) is said to have a nontrivial periodic solution when it posses a Floquet multiplier equals to .
Let be the monodromy operator of (4.7). Then, and is an eigenvector of associated with the eigenvalue (trivial Floquet multiplier), that is, . This leads to a characteristic equation [33]
(4.8) 
The solutions of forms the whole spectrum of (4.4). In the case of (equilibrium solution) then
We will define now certain objects called “instantaneous spectrum”, “strongly unstable spectrum” as well as “asymptotic continuous spectrum”. Strictly speaking, these objects do not belong to the spectrum of the synchronous states, i.e., they are not the Lyapunov exponents of system (4.4). However, they play an important role since the spectrum will be well approximated by the “strongly unstable spectrum” and “asymptotic continuous spectrum” as the delay becomes large. Another advantage of these limiting spectra is that they can be much more easily found or computed in comparison to the actual spectrum of system (4.4), see more details in [42, 41, 38, 23, 33, 40].
Definition 4.1 (Instantaneous spectrum and strongly unstable spectrum).
The set of all for which the linear ODE system
has a nontrivial periodic solution is called the instantaneous spectrum. The subset of those with positive real part is called the strongly unstable spectrum.
Remark 4.1.
In the case when (does not depend on time), the instantaneous spectrum consists of the eigenvalues of .
Definition 4.2 (Asymptotic continuous spectrum).
For any the complex number belongs to the asymptotic continuous spectrum if the DDE
(4.9) 
has a nontrivial periodic solution for some . Here .
Remark 4.2.
If does not depend on time, the asymptotic continuous spectrum can be determined from the following equation
(4.10) 
As follows from [42, 41, 38, 19, 23, 33, 40], the spectrum of generic linear delay system (4.4) converges to either the instantaneous spectrum or to the curves in the complex plane , where is defined from the asymptotic continuous spectrum (note the division by ). The second part of the spectra – consisting of eigenvalues with asymptotically vanishing real parts – approaches a continuous curves asymptotically, while being still discrete for any finite ; for this reason, it is called pseudocontinuous spectrum.
An example of a typical spectrum of the system with long delay is shown in Fig. 6.2. In particular, the distances between neighboring eigenvalues within one curve of the pseudocontinuous spectrum scale as for large delay, and in the limit of they vanish and the eigenvalues fill the curve [33].
The union of the strongly unstable spectrum, , with the asymptotic continuous spectrum forms the approximation of the whole spectrum of Eq. (4.3), given that the instantaneous spectrum does not contain eigenvalues with zero real parts and some nondegeneracy conditions are fulfilled [23, 33]. Moreover, if the set of the strongly unstable spectrum is empty (), and the asymptotic continuous spectrum is entirely contained on the left side of the complex plane () then the trivial solution of Eq. (4.3) is exponentially asymptotically stable.
5 Proof of the main theorems
In this section we use the given description of the spectrum of a general linear DDE to prove Theorems 3.1 (persistence of equilibrium solution), and 5.1 (synchronization of periodic orbit).
The result obtained in Theorem 3.1 could also be proved by using LyapunovKravoskii functionals [16]. However, we use the approach of studying the spectrum for large delay not only because it is powerful, but it also serves as a preparation to prove the result of synchronization of periodic solutions.
5.1 Persistence of equilibrium solution: Theorem 3.1
In this section we specify the stability condition for the case when system (1.1) possesses a synchronous solution i.e. , and the Jacobian is a constant matrix. The corresponding characteristic equation is
(5.1) 
The real parts of the eigenvalues determine the stability, and, as it was discussed above, the whole spectrum converges to either the strongly unstable spectrum or pseudocontinuous spectrum. Lemma 5.1 gives explicit dependence of the asymptotic continuous spectrum on the coupling strength .
Lemma 5.1 (Ref.[42]).
Consider the linear delay differential equation
(5.2) 
with and . Then, the asymptotic continuous spectrum for (5.2) is given by the branches
(5.3) 
where and are complex roots of the polynomial
(5.4) 
Proof of Theorem 3.1 (Persistence of equilibrium solution).
The basic idea of the proof is to show that the strongly unstable spectrum is empty and the asymptotic continuous spectrum is stable if and only if the condition (3.1) is satisfied. The rest of the proof elaborates this idea more carefully in details.
The instantaneous spectrum coincides with the spectrum of and, by the Assumption 2.1, the equilibrium solution of the isolated system is asymptotically stable, thus there are no eigenvalues of with positive real parts, hence .
Using Lemma 5.1 for the variational Eq. (4.3). Then, the real part of the asymptotic continuous spectrum is
(5.5) 
The condition is fulfilled if and only if
(5.6) 
The asymptotic continuous spectrum lies strictly on the left side of the complex plane if the inequality (5.6) holds for all , , and . This leads to the condition
(5.7) 
where exists and is always bounded from zero. Indeed, if we assume the opposite, i.e., , then it means that there exist such and that , and hence, implying that belongs to the spectrum of . In this way we arrive at the contradiction to the assumption that the spectrum of has strictly negative real parts. Moreover, since the functions are the roots of the characteristic equation (5.4). The number stands for the spectral radius of the Laplacian matrix . Therefore, under the condition (3.1), the asymptotic continuous spectrum and therefore the zero solution of (4.3) is stable.
Having both and , Theorem 5.2 implies that there exists such that for all the equilibrium is locally exponentially stable under the condition (5.7).
The statement about the instability of the equilibrium follows from the fact that for there exist such , and that , and, hence the real part of the asymptotic continuous spectrum is positive Therefore, for sufficiently large timedelays and for , there will be eigenvalues from the pseudocontinuous spectrum with positive real parts. ∎
With the result of persistence of equilibrium solution proven, we move to the case of periodic synchronous solution.
5.2 Synchronization of periodic orbits
We have previously stated our result on the synchronization of periodic orbits for the SL oscillators (Theorem 3.2), so, we could avoid the discussion about the characteristic equation . Now we present a generalization of the announced result and we prove it in this section.
Theorem 5.1 (Synchronization of periodic orbits for any finite delay).
Consider system (1.1) and Assumption 2.2. Let an additional nondegeneracy assumption be fulfilled, where is defined by (4.8). Then, given fixed large enough, there exists such that one of the two following statements hold:

The synchronous periodic solution is locally exponentially stable for and unstable for .

The synchronous periodic solution is locally exponentially stable for and unstable for .
Moreover, as . That is, for any , there exists such that for all the synchronous periodic solution is unstable.
Remark 5.1.
The codimension 1 condition may lead to isolated points, where the synchronization of the synchronous periodic orbit cannot be achieved (see example in Sec. 6.2).
The proof of Theorem 5.1 rely on the following lemma, adapted from [33] for the case of the stability of a periodic synchronized solution with respect to desynchronized perturbations (transverse to the synchronization manifold).
Lemma 5.2.
The synchronous periodic orbit of system (1.1) with period is exponentially stable with respect to perturbations transverse to the synchronization manifold for all sufficiently large if all of the following conditions hold:

(No strong instability) All elements of the instantaneous spectrum have negative real parts (this implies in particular that the strongly unstable spectrum is empty),

(Simple trivial multiplier) The trivial Floquet exponent is simple.

(Weak stability) the asymptotic continuous spectrum is contained in for all with , being all nontrivial eigenvalues of the Laplacian matrix .
And, it is exponentially unstable with respect to perturbations transverse to the synchronization manifold for sufficiently large if one of the following conditions hold:

(Strong instability) the strongly unstable spectrum is nonempty, or

(Weak instability) a nonempty subset of the asymptotic continuous spectrum has positive real part for nontrivial eigenvalue of the Laplacian matrix .
Proof.
This theorem follows almost directly from Theorem 6 of [33], and we highlight the main features for our setup.
First of all, the existence of the periodic solution does not depend on time delay . As a result, can be considered as a continuous parameter here, instead of the parameter which is an integer part of used in [33]. Hence, the asymptotic statements with respect to are equivalently formulated for in the present theorem.
Secondly, the stability of the periodic solution is determined by the variational equation (4.3) that splits into the part along the synchronization manifold with and the remaining part for the transverse perturbations with nonzero . Hence, the transverse stability is determined by the variational equation (4.6) with . The conditions for the stability SI – SIII guarantee that the spectrum of the corresponding variational equations is stable for large enough . ∎
The linear stability of a synchronous periodic orbit is governed by the linearized equation (4.4) with periodic . In Ref. [33] the authors proved results about the stability of this equation for large . In particular, the existence of a holomorphic function is shown, such that is a Floquet exponent of the linear DDE (4.4) if and only if (see Eq. (4.8)).
The function in this case is analogous to the characteristic equation (5.1) for the case of equilibrium. The main difference of the periodic case is that the function is not given explicitly. The asymptotic continuous spectrum is determined from the equation
(5.8) 
(compare (5.8) and (4.10)). Here we should emphasize that, in contrast to Ref. [33], we explicitly write the parameter in the argument of the function since the dependence on is of interest for this study. Moreover, this parameter can be eliminated from Eq. (5.8) by the transformation
leading to the equation
which is the same as Eq. (5.8) but with . Here we denoted . As a result, the following Lemma holds:
Lemma 5.3.
Remark 5.2.
The main effect of the feedback strength for strong delays is the shift of the asymptotic continuous spectrum by the value .
Proof of Theorem 5.1.
The proof consists on showing that the conditions SI – SIII of Lemma 5.2 are satisfied.
We remark that
since is the characteristic equation of the uncoupled system, which possesses one simple trivial multiplier by Assumption 2.1. Consider now the implicit function problem
(5.9) 
It has a unique smooth solution with provided . As a result, the asymptotic continuous spectrum
(5.10) 
has a singularity at and . Hence, the asymptotic continuous spectrum of (4.4) is singular as well
independently of the value of . Hence, the asymptotic continuous spectrum possesses always an unstable part, which implies instability for large enough values of timedelays.
We know that for the characteristic equation possesses one simple zero root , hence . We find how the real parts of these eigenvalues change if deviates from zero using the implicit function theorem for the equation . Since
we have the unique solution for small and . Here by assumption. Hence
(5.11) 
In particular, this expression shows that for , the periodic solution will be destabilized for positive , and stabilized for negative . For , the stabilization occurs for positive and destabilization for negative. It is important to notice, that the stabilization (or destabilization) occurs for all transverse modes simultaneously, since the eigenvalues of the Laplacian matrix are positive and can admit values , which have the same sign for all , .
∎
6 Example: Coupled StuartLandau systems
Let us consider an example of the ring of coupled StuartLandau (SL) oscillators shown in Fig. 6.1.
The dynamics of a node is given by (3.3). Throughout this section, we use the identity as the coupling function. We rewrite Eq. (3.3), in terms of the Laplacian matrix as
(6.1) 
Figure 6.2 shows the spectrum of two coupled StuartLandau oscillators linearized at the origin and coupling function being the identity.
6.1 Persistence of equilibrium
Firstly, we consider the case (equilibrium solution) so that the eigenvalues of the uncoupled system have negative real parts. The Jacobian matrix at zero has the eigenvalues . Therefore, the strongly unstable spectrum is empty, and the whole spectrum of the zero equilibrium for the network system (4.3) consists only of the asymptoticcontinuous spectrum for long delays.
The laplacian matrix of the network in Fig. 6.1 has spectral radius . In order to compute the value from the condition (3.1) of Theorem 3.1 we compute the functions , using Eq. (5.4):
Hence, we find . Therefore, the synchronization manifold for the considered network is locally exponentially stable if and only if
(6.2) 
for sufficiently long time delay (and unstable if ).
Fig. 6.3 illustrates the convergence of the trajectories to the equilibrium (left panel) for the case when the condition (6.2) is fulfilled and the absence of convergence in the opposite case. The detailed parameter values are given in the figure’s caption. Moreover, in order to compute the synchronization error we compare each solution to a fixed one taking the norm of maximal difference, that is, we measure the synchronization error by .
6.2 Periodic orbit synchronization
Let us consider the case when the synchronous periodic solution exists. In this section we prove Theorem 3.2.
Consider the transformation , then Eq. (6.1) reads, in terms of ,
(6.3) 
Note that the solution is transformed into a family of equilibria. The variational equation of (6.3) (considering real and imaginary parts) around the equilibrium solution , equivalent to (4.3), is
(6.4) 
where are the eigenvalues of . For short, we write Eq. (6.4) as .
The instantaneous spectrum can be computed using Definition 4.1. It consists of the eigenvalues of the nondelayed part of (6.4) and reads . The eigenvalue in the instantaneous spectrum is associated with the trivial Floquet multiplier producing a singularity in the asymptotic spectrum. Then, as predicted by Theorem 5.1, the synchronization cannot be achieved for an arbitrary large delay. However, Theorem 5.1 guarantee the existence of a synchronization interval (or with ) with the decreasing length with increasing .
The asymptotic continuous spectrum can be computed as in the case of equilibrium solution. Using Eq. (4.10), one obtains where are the solutions of
(6.5) 
Note that Eqs. (5.9) and (6.5) are the same. Hence, we write
or, more specifically,
The nondegeneracy condition (assumption in Theorem 5.1) is satisfied since provided , .
The solution of (6.5) reads
(6.6) 
From (6.6), we get and provided , . Therefore, the singularity occurs uniquely at the function .
In order to obtain , as introduced in Eq. (5.11), we compute . Hence, . As , this implies that synchronization occurs for negative when and for positive otherwise (positive is obtained if ).
We can also compute the spectrum of (6.4) by using Eq. (5.1). So, we get the transcendental equation
(6.7) 
with . Eq. (6.7) can be numerically solved.
We compare the solutions of (6.7) to the analytical approximation. The asymptotic spectrum (given in terms of Eq. (6.6)) and the spectrum points (solutions of Eq. (6.7)) are given in Fig. 6.4 with parameters given in the figure’s caption.
From the data of Fig. 6.4 we notice that for , and , stale synchronization is attained for .
Now, in order to have a complete view of the stability domain, we produce a synchronization map in the parameter space . The Fig. 6.5 shows such a color map presenting the values of , where , , is a solution of Eq. 6.7, in the parameter space. Note that and, as predicted by Theorem 5.1, the stable synchronization either occurs for or for with shrinking as grows.
The parameter values of the color map in Fig. 6.5 can be related to more complicated connected network for coupling parameter values in the range . The interchange between stability domains occurs at the values , , where we considered . For such values of time delays no stable synchronization is attained.
6.3 Characteristic time
In this section, we discuss the transient time such that the trajectories enter a small vicinity of the synchronization manifold. In particular, we are interested in the scaling of the transient time with coupling strength and time delay .
The real parts of the eigenvalues of the linearized system can be estimated rigorously, so the characteristic time should be related to the properties of the linearized system. The solutions of Eq. (4.3), in the case of exponential stability, decay as
(6.8) 
with to be the largest real part of the eigenvalues. Hence, we define the characteristic time as
The characteristic time measures how fast the slowest solution approaches . One can write where is the maximum of the real part of the asymptotic continuous spectrum given in Eq. (5.5). It can be computed as
Hence, the characteristic time is
(6.9) 
Clearly, as
7 Synchronization loss versus network structure
Here, we explore the relationship between growing networks (with strongly delayed connection) and its synchronization window in the case of equilibrium. The main results are the corollaries 3.1 and 3.2.
We will concentrate on the behavior of large networks such as BarabásiAlbert scalefree network, ErdösRéniy random network, and some regular graphs, for which the expressions for the Laplacian spectral radius are known.
First, let us take a look at how regular graphs respond to the synchronization (using the network model (1.1) with long delay) in the limit of large network size. Table 7.1 lists spectral radius of the Laplacian matrix of the main regular graphs: complete, ring, star, and path.
Graph  Synchronization window  

Complete  
Ring  or  
Star  
Path 
Using Theorem 3.1, and Table 7.1 we observe that:

For strong delay and a large network size , the synchronization manifold tends to be always unstable in networks of the types Complete or Star provided that the coupling strength does not scale with .

For strong delay and a large network size , the synchronization manifold tends to be stable for a certain interval of the coupling strength in simple networks of the types Ring or Path.
Now, let us consider some complex networks. We relate the synchronization condition to the graph structure for two important examples of complex networks.
Homogeneous networks, characterized by a small disparity in the node degrees. A canonical example is the ErdösRéniy (ER) random network: Starting with nodes the graph is constructed by connecting nodes randomly. Each edge is included in the graph with probability independent from every other edge. If with then the ER network is almost surely connected [5]. We will consider that is chosen so that the ER network is connected.
Heterogeneous networks, where some nodes (called hubs) are highly connected whereas most of the nodes have only a few connections. A canonical example of such networks is the BarabásiAlbert (BA) scalefree network. One way to construct it is to start with two nodes and a single edge that connect them. Then at each step, a new node is created and it is connected with one of the preexisting nodes with a probability proportional to its degree. This process is called preferential attachment.
More details about the construction, structure, and dynamics of such networks can be found in [27, 4]. Illustrations of ErdösRéniy (ER) random networks (homogeneous) and BarabásiAlbert (BA) ScaleFree networks (heterogeneous) can be seen in Figure 7.1.
The response of the BA and ER networks to synchronization under the considered delayed model already stated in Sec. 3 and encoded in corollaries 3.1 and 3.2. The both results says that ER and BA networks tend to not have stable synchronization in the limit of large . But, ER networks, losses stable synchronization at a slow rate compared to BA networks.
Here we prove the both cited corollaries. The proof of Corollary 3.1 is based on the following wellknown results.
Lemma 7.1 (Ref. [3]).
Let denote the largest degree of a undirected network of size . Then the spectral radius of the Laplacian matrix of has the following estimates:
(7.1) 
Lemma 7.2 (Ref. [26]).
Consider a BA undirected network of size and its largest degree. With probability we have
(7.2) 
the limit is almost surely positive and finite.
Proof of Corollary 3.1.
Remark 7.1.
Corollary 3.1 shows that the synchronization manifold tends to be unstable in large and strongly delayed BA (heterogeneous) networks if the coupling strength is not scaled with or .
Lemma 7.3 (Ref. [30]).
Consider an ErdösRéniy network with nodes, connection probability and . Then, the probability that the maximum degree being at most , for some constant , tends to as .
Proof of Corollary 3.2.
For large and using the probability with one get
So, using Lemma 7.1, we end up with
which means that when with a rate of order . The spectral radius is the lowest possible when maintaining the network connected, or, is the lowest possible when the network crosses the connectivity threshold becoming connected. ∎
Although Corollary 3.2 says that the ER random network does not allow the synchronization manifold to be stable in the limit of , the rate in which it happens is slow, making it possible to synchronize very large ER networks with long time delays.
The properties observed for the stability of the synchronization manifold for BA and ER networks with strong delay, that is, large BA networks doesn’t support strong delay interaction and relatively large ER networks supports strong delay interaction, are similar to the persistence of the synchronization when the nondelayed coupling functions are nonidentical [25].
7.1 Synchronization of BA and ER networks when coupling strength is scaled with
As we have seen from the previous subsection, not only simple but also complex networks tend to have an always unstable synchronization manifold with strong delay and large network size .
In many network dynamical models, it is common to normalize the coupling parameter, in our case , in order to preserve certain behaviors and properties of the network, such as synchronization, mean field oscillation, community clustering, etc [1, 31, 14]. When the network size is dynamical, for instance, the size of the network is growing with time, then this normalization becomes even more important.
With this notion in mind, we can see that in the regime of large network size, the coupling parameter should have some natural scaling depending on the network structure. If those scaling are taken into account in the network model then the stability of the synchronization manifold can always be preserved for .
For example, if we know beforehand that we have a large simple network of the type Complete (or Star) then the natural scaling of the coupling parameter would be
We then state here further consequences of the corollaries 3.1 and 3.2 when considering the scaling of the coupling parameter.
Corollary 7.1.
Consider a BA scalefree network with number of nodes equals to and the network model (1.1) with the coupling parameter
Then for any large enough size and long enough delay, there is always a nonempty interval such that any synchronous steady state is locally exponentially stable for all and unstable if (with probability 1). Moreover, the length of this interval converges to a nonzero constant with with probability 1:
Proof.
Any equilibrium solution is locally exponentially stable if and unstable for for some . For BA scalefree networks we know that (see Lemma 7.2). Then, if we rescale the coupling parameter as the new synchronization condition is , which leads to . ∎
Corollary 7.2.
Consider an ER random network with nodes and the network model (1.1) with new coupling parameter scaled as
Then for large enough network size and long enough time delay , there is an interval such that any equilibrium solution is locally exponentially stable if and unstable if (with probability 1). Moreover, the length of the interval converges to a nonzero constant with with probability 1:
Acknowledgments
The work of D.N.M Maia was supported by CNPq grant 233718/20141. E. Macau also acknowledges the support of CNPq and FAPESP2015/501220 T. Pereira acknowledges the support of CEPIDCeMeai FAPESP project 2013/073750 and S.Yanchuk acknowledges the support of the German Research Foundation (DFG) in the framework of the Collaborative Research Center 910. This research is also supported by grant 2015/501220 of Sao Paulo Research Foundation (FAPESP) and DFGIRTG 1740/2. The authors acknowledge valuable discussions with Jan Philipp Pade and Stefan Ruschel.
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Received xxxx 20xx; revised xxxx 20xx.