Synchronization of complex networks with interacting units is an active research topic with applications in many fields such as engineering, computer sciences, neural networks (both biological and not), neuromorphic circuit computing and information security. Arenas et al. (2008) formulated a comprehensive review in 2008, describing the impacts of this field of research. One of the most challenging problem in complex network is the study of spatio–temporal patterns emerging from synchronization phenomena and their relationship with the network topology and interactions' dynamics among the (both oscillatory and/or chaotic) network units (Pecora and Carroll, 1998; Belykh V. N. et al., 2004; Boccaletti et al., 2006).

In literature great attention has been paid to investigate synchronization of complex networks with fixed topology and constant couplings. Only in recent years researchers have begun to focus on time–varying coupling strengths, due to the greater resemblance of this type of networks to real world systems. Different approaches have been adopted to study synchronization properties of these complex dynamical networks. For instance, Wu and Chen (2008) have provided several criteria of global synchronization, in the case of linearly coupled networks with continuous time-varying coupling. Following a complex networked control systems approach, De Lellis et al. (2008) and DeLellis et al. (2009) have considered time-varying adaptive couplings that evolve with respect to the difference among the oscillators. Several other authors (Berner et al., 2019a,b; Menara et al., 2019; Feketa et al., 2020) have focused on cluster synchronization in networks of phase oscillators with adapting coupling based on the dynamical change of the coupling strength between the oscillators. Moreover, Berner et al. (2019a,b) explored how such complex systems behave by “splitting the networks” into different clusters of nonlinear oscillators grouped in such a way that all the oscillators in a poll have the same frequency. For the scope of this work it is also necessary to take into account the dynamics of networks formed by memristors, in particular the work by Ascoli et al. (2015). In their work the authors extensively studied the synchronization dynamics between two memristor-coupled Hindmarsh-Rose oscillatory neural cells through theoretical investigation and numerical verification. On the other hand, by exploiting the mathematical tools from the control theory of switched systems, Stilwell et al. (2006) have studied a network of oscillators with switching topology, in order to model networks where some connections may fail or may create over time. In their work they have demonstrated that complex network can synchronize even when the topology is time-varying and instantaneously disconnected. Their thesis stands, though, on two pillars that serve as a condicio sine qua non: first the different circuits must be able to synchronize in a static time-average fashion; secondly the transition between two different topology must be sufficiently fast. Synchronization criteria in the case of complex network with arbitrary switching topology and strategies for designing switching signals to obtain synchronization have been investigated by Zhao et al. (2009).

A particular class of complex networks with switching topology permits to introduce the concept of blinking complex network (Belykh I. V. et al., 2004; Hasler and Belykh, 2005; Hasler et al., 2013a,b) in which the topology evolves in time according to a stochastic process. In particular, blinking networks have the peculiar feature that, starting by a fixed topology (called pristine network), new "long–range links" between two network units are randomly added according to a fixed probability. Once the new topological configuration is set, it will remain constant for a given amount of time, then again new links are randomly created, and so on. Belykh I. V. et al. (2004) have demonstrated that such time–varying random switching (i.e., blinking) topology induces complete synchronization in the networks. By creating randomly selected couplings, the chance of creating interconnected shortcuts increases, lowering the synchronization threshold sensibly.

The study and the development of blinking complex networks finds its importance especially in boosting functional properties of bio–inspired and neuromorphic circuits acting on the topology of synaptic connections. Neuromorphic circuits made of (1 or 2-d) arrays of silicon neurons exploits its analog nature, among other architectural characteristics, to enhance their computational efficiency and speed, but one of the main drawback is the necessity of a very high number of wired connections between the computational elements (e.g., large numbers of synapses is also used to overcome low resolution in synaptic weights—see also Chicca et al., 2014). Strategies that involve solely the reduction of the required connections, for instance reshaping the network topology adding a sum cell which has inputs and outputs that are not connected to all cells has been exploited by Seiler and Nossek (1993). This approach has permitted to reduce the number of connections for a Winner Take All architecture, however it did not take into account that most of the connections had to cross a wide part of the circuit, thus still rendering it not feasible. A more effective approach to this is to generate long distance connections, activated and deactivated randomly, in such a way that the network computations performance is the same as that of a corresponding locally–connected non–switching (averaged) network. The successful use of blinking/switching networks for solving this kind of issue has been proven, for example in a similar case to the one introduced by Seiler et al., by means of a Winner Take All locally–connected architecture (Devoe and Devoe, 2012). These results highlight the importance of the study of the synchronization dynamics of blinking networks since they can be one representative of the feasibility factors of real world neuromorphic computation.

This work focuses on Blinking Memristor Oscillatory Networks (BMONs) made of memristor–based nonlinear oscillators interacting via blinking couplings. A thorough study of nonlinear dynamics and bifurcation phenomena in memristor oscillators have been recently carried out by means of the Flux-Charge Analysis Method (FCAM). Such circuit methodology permits to show that the state space in the (voltage–current)–domain [i.e., (v−i)-domain] can be decomposed into invariant manifolds, where the dynamical behavior of the memristor oscillator may correspond to different complex attractors. In addition, bifurcation phenomena can be induced either by parameters on a fixed manifold (as in the classical theory of dynamical systems) or by changing the invariant manifold but with fixed parameters (a.k.a. bifurcation without parameters).

The ultimate goal of the manuscript is to show that blinking couplings in complex networks of memristor oscillators induce bifurcations without parameters so that the whole dynamics of a BMON takes place on an invariant manifold where periodic oscillations correspond to synchronized states. In this manuscript analysis of nonlinear dynamics and numerical simulations are focused on a specific BMON in which each oscillator is a third–order dynamical system describing the Chua's circuit with a memristor replacing the nonlinear resistor, but the FCAM provides the theoretical framework to grasp the role of network topology in BMONs including Memristor–based Oscillator Circuit (MOCs) of any order. It is also worth noting that bifurcation without parameters, and then the shift of nonlinear dynamics from one manifold to another, can be obtained by applying suitable charges or flux sources via time-varying voltage and/or current pulses with finite time duration (Corinto and Forti, 2017b). This work shows that BMONs exhibit analogous nonlinear phenomena without any external input. Moreover, we show that for memristor oscillators not having all the same initial conditions it is possible to change the invariant manifold on which the nonlinear dynamics takes place only by acting on the coupling topology. Our numerical investigations of BMONs shows how the creation of time–varying long–range random connections results into switching between invariant manifolds and then into synchronization.

At first, a brief introduction to the memristor oscillators is presented showing the internal dynamics and the numerical formulations of such systems in the flux-charge domain. Secondly it will be shown how the change of the topology of interacting memristor oscillators affects invariant manifolds on which nonlinear dynamics occur. Finally, synchronization phenomena in BMONs are analyzed via numerical simulations.

Let us briefly recall some results obtained in Corinto and Forti (2017a) for a MOC in the flux-charge domain. The nonlinear oscillator consists in a modified version of the well-known Chua's circuit where the nonlinear resistor is replaced by a memristor. The cited paper details and extends the results obtained in Corinto et al. (2011) For further details regarding the circuit model and its analysis please refer to Corinto and Forti (2017a). The State Equations (SEs) in dimensionless form¹ are the following (t≥t₀):

with the initial conditions

where φ_M(t₀) is the initial flux across the memristor, φ_L(t₀) is the initial flux across the inductor, whereas q_C2(t₀) is the initial charges of capacitor 2. Moreover, we have

and

Readers interested in the circuit–theoretic definitions of the state variables, parameters and circuit analysis can refer to the study reported in Corinto and Forti (2017a).

Hereinafter and throughout the paper we set the following initial states and parameters:

As shown in Corinto and Forti (2017a), the state space of the MOC in the (v, i)-domain can be decomposed in infinitely many three-dimensional manifolds M(Q0) defined, for any Q₀ ∈ ℝ as

with

Furthermore, these manifolds are positive invariant for the dynamics of the MOC in the (v, i)–domain. Therefore, at instant t = t₀ the system is on the manifold M(Q0), with Q₀ = Q(w(t₀)) and there it stays. The value of Q(w(t₀)) depends on the memristor initial conditions and the circuit parameters.

Proposition 1. Let us suppose the parameters and initial states as in (4) and φ_M(t₀) ∈ {−1.5, 0, 1.5}. Therefore, Q(w(t₀)) = 0, that is the nonlinear dynamics of the MOC is on the so-called zero-manifold M(0). Thus, every trajectory defined by [x(t), y(t), z(t)] evolves on a three–dimensional space M(0) embedded in ℝ⁴.

Proof. According to the definition of f in (2), it is easy to see that f(x)+xR=f(x)+x=0 for all x ∈ {−1.5, 0, 1.5}. Therefore, by using (4) and the definition of Q in (5), we deduce:

By extending (1), let us consider a one-dimensional array of N diffusively² coupled MOCs described in the flux–charge domain by (i = 1, …, N):

where Ni is the sphere of influence of ith MOC and dik=αRik with d_ik = d_ki. Following the same procedure for X₀ in (3), the parameters (4) permit to derive

Analogously as in the single MOC, it is possible to prove that the state space of the Network of Memristor-based Oscillatory Circuits (NMOCs) in the (v, i)-domain is foliated by an infinite number of 3N–dimensional manifolds defined, for any Q0=(Q0,1,…,Q0,N)∈ℝN as

with w_i = (v_{C1, i}, v_{C2, i}, i_L,i, φ_M,i), wc=(w1,…,wN)T and

From (7) and (8), we easily deduce that

In the next sections we exploit NMOCs described by (6) and (7) in the flux–charge domain to make clear the influence of the couplings d_ik on synchronization phenomena. In particular, by assuming that d_ik describes blinking links added to a pristine complex network (i.e., an underlying networks with fixed topology) then the NMOC acts as a BMON.

Nonlinear dynamics and global attractors of a single MOC described by (1) with X₀ in (3) can be controlled by applying suitable time-varying voltage and/or current pulses with finite time duration (Corinto and Forti, 2017b). The concept of bifurcation without parameters is the crucial tool to enable the programming of nonlinear dynamics via invariant manifolds. For instance, a sequence of period–doubling bifurcations (without parameters) leading from a periodic oscillation to a chaotic attractor can be induced by means of pulses applied to the MOC; each pulse has the effect of shifting the trajectory (due to the change of X₀) from an invariant manifold on which the MOC exhibits a periodic behavior to an invariant manifold with a double–scroll chaotic attractor (Corinto and Forti, 2017b). In other words, bifurcations without parameters make possible to tune nonlinear dynamics of a MOC by selecting via pulses an invariant manifold embedding all the attractors.

This section shows how nonlinear dynamics and invariant manifolds of NMOCs described by (6) and (7) are influenced by the coupling terms d_ik (i.e., by the network topology).

First of all, it is possible to see that when all the memristors have the same initial conditions, the coupling terms d_ik play a role on the dynamics of the network [since they are present on the network Equations (6)], but they do not influence the manifold embedding the dynamics of the whole NMOC.

Proposition 2. Let us consider the NMOC (6). If all the initial conditions of memristors are the same [i.e., φ_M,i(t₀) = φ₀, ∀i = 1, …, N], then the attractors of the NMOC belong to an invariant manifold that is independent on the coupling parameters d_ik.

Proof. It is easy to notice that when the initial conditions of each memristor are the same, that is φ_M,i(t₀) = φ₀ for all i = 1, …, N, then

Moreover, we have

with Q defined as in (5). Therefore, the invariant manifold Mc(Q0) is independent on the coupling.

Thus, in order to induce bifurcations without parameters by means of the couplings and then change the invariant manifold embedding the attractors of the NMOC, at least one initial condition of the memristors has to be different. Without loss of generality, let us suppose that the memristor initial condition in the first oscillator is different with respect to the others. Therefore, we can prove the following result:

Proposition 3. Let us consider the NMOC (6) with the memristor initial conditions:

Let us suppose that the global dynamics of the NMOC evolves on the manifold Mc(Q0), with Q(w_c(t₀)) = Q₀ = (Q_0,1, …, Q_0,N) defined as in (8).

Moreover, let us suppose to add a link between the nodes i₁ and j₁, that are not yet linked. Therefore, we have the following two cases:

Proof. Let us suppose:

and Δφ0=φ0-φ~0.

In this case, the NMOCs Equation (6) become:

with

Moreover, according to (8), for any t≥t₀ the global dynamics of the NMOC (11) evolves on the manifold Mc(Q0), with Q(w_c(t₀)) = Q₀ = (Q_0,1, …, Q_0,N) and

Let us suppose now to add a link between the nodes i₁ and j₁, that are not yet linked. We can have two different cases (see Figure 1):

The next section shows the application of this analysis to the numerical study of synchronization phenomena in NMOCs and BMONs.

This section presents the numerical analysis of spatio–temporal synchronization phenomena in NMOCs in two specific cases of interest: (a) the addition of just one single link in order to make clear the effect of a connection on the manifold shift; (b) the inclusion of blinking interconnections having a given probability to connect two MOCs. As preliminary step, we show the effect of the memristor initial conditions in such NMOC with fixed diffusive topology.

In this paper we have considered a network of memristor nonlinear oscillators in the Flux-Charge Analysis Method (FCAM) framework, in order to show that it is possible to tune nonlinear dynamics of complex networks of memristor–based oscillators by simply adding new links among the oscillators, thus by acting on the blinking coupling topology. This has been done under the hypothesis of different memristor initial conditions. The FCAM permits to interpret the dynamical evolution of the whole blinking network of memristor oscillators as the periodic/chaotic attractors occurring on different invariant manifolds. Numerical simulation show that the synchronization properties can be improved by adding blinking long-range links and consequently by randomly switching of the invariant manifold.

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author/s.

VL and FC supervised and developed the mathematical model described in the paper. JS performed the simulations. FC wrote abstract and section 1. VL and JS wrote sections 3–7. All authors contributed in the analysis of the results and in writing the manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.