Synchronization Phenomena in Oscillator Networks: From Kuramoto and Chua to Chemical Oscillators

Abstract

This chapter addresses the problems of synchronization analysis in various types of oscillator networks. In particular, we derive sufficient conditions for emergence of multi-cluster formations in Kuramoto networks with dynamic coupling, prove the output-feedback synchronization of chaotic behavior in networks of Chua oscillators with nonlinear static coupling, and study the synchronization of complex spatiotemporal patterns in coupled infinite-dimensional reaction-diffusion models of chemical oscillators. The obtained results contribute towards a deeper understanding of the internal organization of oscillator networks, explain the prerequisites for the emergence of patterns of synchrony and justify their stability properties in terms of the dynamical characteristics of oscillators, parameters of couplings, and the interconnection topology of the network. The interplay of these three ingredients is required for the complex and dynamically rich behavior of the network. Theoretical results of the chapter are supplemented with numerical case studies.

1 Introduction

Oscillator networks proved to be natural mathematical models for a variety of complex processes in various areas ranging from biology and social sciences to physics and modern technology [36]. Due to a broad spectrum of application domains, oscillator networks as mathematical objects constitute a wide range of dynamical systems: from paradigmatic phase oscillators to multi-dimensional neuro-inspired models, chaotic systems, and infinite-dimensional chemical oscillators. This together with the possibility of complex dynamical interactions between oscillators and complex interconnection topologies make the proper analysis of such systems challenging and complicates the understanding of the emergence of collective behavior therein. One of the fundamental types of oscillator networks behavior is synchronization that is known to be essential for many purposes. For example, the synchronization of oscillatory signals between different brain regions supports the interaction between working and long-term memory [16]. The phenomena of partial synchronization and multi-clustering are of practical importance for neurophysiological systems [35] and distributed power generation [3, 6].

Motivated by the above discussion, this chapter focuses on three essentially different types of oscillator networks and provides their synchronization analysis. Thus, sufficient conditions for the emergence of multi-cluster formations in Kuramoto networks with dynamic coupling are provided in Sect. 2. This is made by proposing an alternative characterization of multi-cluster behavior in terms of the existence of invariant manifolds for the corresponding error-system [14]. These manifolds are of a special topological structure and their dimensions coincide with the number of clusters in the network. The conditions for the existence and stability of the invariant manifolds interrelate the natural frequencies of oscillators, plasticity characteristics, and the interconnection topology of the network. The proofs are based on the perturbation theory of invariant tori of dynamical systems [37, 38].

Section 3 addresses the problem of the output-feedback synchronization of chaotic behavior generated by the ensembles of Chua oscillators and synchronization of spatiotemporal patterns in the master-slave configuration of two Gray–Scott reaction-diffusion models. The latter one is a prototypic model of complex isothermal autocatalytic reactions that is governed by the pair of coupled partial differential equations. Synchronization analysis of this infinite-dimensional model is carried out employing late-lumping observer design techniques [12] thus relating the synchronization property with the convergence of the observer error dynamics. It is worth noting that every considered setup requires a distinct method for its formal analysis. The usage of the presented theorems is demonstrated in the numerical case studies. A short conclusion in Sect. 4 completes the chapter.

Notation

The following notation will be used throughout the chapter. Let $\mathbb N$, $\mathbb R$, $\mathbb R_{>0}$, $\mathbb R_{\ge 0}$, and $\mathbb C$ denote the sets of natural, real, positive real, non-negative real, and complex numbers, respectively. For given $n,m\in \mathbb N$ let $\mathbb R^n$ and $\mathcal T_m$ denote the n-dimensional Euclidean space and m-dimensional torus, respectively. The one-dimensional torus $\mathcal T_1$ is the one-sphere (circle $\mathcal S_1$). Let $f:\mathcal T_m \rightarrow \mathbb R^n$ be a function of the variable $\varphi =(\varphi _1,\ldots ,\varphi _m)^\top \in \mathcal T_m$ which is continuous and $2\pi$-periodic with respect to each $\varphi _s$, $s=\overline{1,m}$. Finally, $C(\mathcal T_m)$ denotes the space of all such functions f equipped with the norm $|f|_0 = \max _{\varphi \in \mathcal T_m} \left\| f(\varphi )\right\| ,$ where $\left\| \cdot \right\|$ denotes the Euclidean norm in $\mathbb R^n$, i.e., $\left\| f(\varphi )\right\| ^2=\sum _{i=1}^n |f_i(\varphi )|^2$, $|f_i(\varphi )|$ stands for the absolute value of the ith component of f evaluated at $\varphi$. By $C^1(\mathcal T^m)$ we denote the subspace of $C(\mathcal T_m)$ with every $f\in C^1(\mathcal T_m)$ having a continuous partial derivative with respect to each $\varphi _s$, $s=\overline{1,m}$ and $|f|_1 = \max \{|f|_0, |\frac{\partial f}{\partial \varphi _1}|_0, \ldots , |\frac{\partial f}{\partial \varphi _m}|_0 \}.$ For a given set $\mathcal V$, $|\mathcal V|$ denotes the number of elements in $\mathcal V$. ${\text {Re}}\,\lambda (A)$ denotes the set of real parts of all eigenvalues of square matrix A and any set $B<0$ if and only if for any $b\in B$ it holds that $b<0$.

2 Multi-clustering in Networks of Phase Oscillators with Dynamic Coupling

In this section, the phenomenon of multi-clustering in networks of phase oscillators with dynamic coupling is studied. To this end, a general nonlinear model is introduced and the multi-cluster behavior of the network is then characterized in terms of the existence and stability of the corresponding invariant toroidal manifolds. These manifolds have a particular topological structure and their dimension coincides with the number of clusters in the network. Finally, the proposed approach is applied to a Kuramoto network with adaptive coupling and sufficient conditions are derived for the emergence of multi-cluster behavior. These conditions interrelate the dynamic properties of oscillators, the plasticity parameters of the adaptive couplings, and the interconnection topology of the network. A numerical example concludes this section.

2.1 General Nonlinear Model

Let $\mathcal G = (\mathcal V, \mathcal E)$ be the directed graph representing the network of nonlinear oscillators, where $\mathcal V=\{1,\ldots ,N\}$ and $\mathcal E \subseteq \mathcal V \times \mathcal V$ represent the oscillators and their interconnection edges, respectively. Let $A = [a_{ij}]_{(i,j)\in \mathcal V\times \mathcal V}$ be the adjacency matrix of $\mathcal G$, where $a_{ij}=1$ if the edge $(i,j)\in \mathcal E$, and $a_{ij}=0$ when $(i,j)\not \in \mathcal E$. Contrary to many existing results, we do not impose any connectivity assumptions on the interconnection graph. The dynamics of the network is given by

$$\begin{aligned} \dot{\theta }_i&= f_i(\theta _i)+\sum \limits _{j\in \mathcal V} a_{ij} k_{ij} g_{ij}(\theta _i,\theta _j),{} & {} i\in \mathcal V, \end{aligned}$$

(1a)

$$\begin{aligned} \dot{k}_{ij}&= \Gamma _{ij}(k_{ij},\theta _i,\theta _j),{} & {} (i,j)\in \mathcal E, \end{aligned}$$

(1b)

where $\theta _i(t)\in \mathcal S_1$ and $k_{ij}(t)\in \mathbb R$ denote the phase of the oscillator $i\in \mathcal V$ and the coupling strength of the link $(i,j)\in \mathcal E$ at time $t\in \mathbb R_{\ge 0}$, respectively. The function $f_i\in C(\mathcal T_1)$ defines the intrinsic dynamics of the ith oscillator, $i\in \mathcal V$, and $g_{ij}\in C(\mathcal T_2)$, $(i,j)\in \mathcal E$ defines the influence of oscillator j on oscillator i depending on their phases. The dynamics of the coupling strength $k_{ij}$, $(i,j)\in \mathcal E$ is defined by the function $\Gamma _{ij}: \mathbb R\times \mathcal T_2\rightarrow \mathbb R$ such that $\Gamma _{ij}(k,\cdot )\in C(\mathcal T_2)$ for every fixed $k\in \mathbb R$. For example, if $f_i(s)\equiv w_i$, $g_{ij}(s_1,s_2)=\sin (s_2-s_1)$, and $\Gamma _{ij}(k,s_1,s_2)=-k+\cos (s_2-s_1)$, then (4) is a network of Kuramoto oscillators with dynamic Hebbian-type coupling [5, 14, 18]. If $\Gamma _{ij}\equiv 0$ for all $(i,j)\in \mathcal E$, then (1) reduces to (1a) with constant coupling strengths $k_{ij}\in \mathbb R$, $(i,j)\in \mathcal E$, and it defines the network of phase oscillators with static coupling (see e.g., [25] and [26], for the synthesis of electrical circuits for both adaptive and non-adaptive Kuramoto models, respectively).

Definition 1

Let $\mathcal P = \{\mathcal P_1,\ldots , \mathcal P_m\}$ with $m\in \mathbb N$, $1<m\le |\mathcal V|$ be a partition of $\mathcal V$, where $\cup _{i=1}^{m}\mathcal P_i =\mathcal V$ and $\mathcal P_i \cap \mathcal P_j = \varnothing$ if $i\not =j$ for all $i,j\in \mathcal V$. The network exhibits cluster synchronization when the oscillators can be partitioned so that the phases of the oscillators in each cluster evolve identically.

Contrary to the notion of multi-clustering given in Definition 1, the term multi-clustering may also refer to the network’s behavior that is characterized by a partition of nodes into subsets so that the frequencies of oscillators in each subset coincide, see, e.g., [4, 5]. This type of behavior also admits an alternative characterization in terms of the existence of a suitable invariant low-dimensional manifold (see Remark 1 for details).

2.2 Synchronization Invariant Manifolds

In sequel, an alternative characterization of multi-clustering given in Definition 1 will be provided in terms of the existence of synchronization invariant manifolds of the corresponding error-system. To this end, for every cluster $\mathcal P_s$, $s=\overline{1,m}$ we pick an arbitrary oscillator $i_s \in \mathcal P_s$ within the cluster, denote its phase by $\varphi _s:=\theta _{i_s}$, and collect all $\varphi _s$, $s=\overline{1,m}$ into the vector of reference phases $\varphi = (\varphi _1, \ldots ,\varphi _m)^{\top }\in \mathcal T_m$. For every oscillator $i\in \mathcal P_s\setminus \{i_s\}$, let $e_i=\theta _i-\varphi _s$ denote the relative phase difference within the cluster $\mathcal P_s$, $s=\overline{1,m}$. All relative phase differences $e_i$, $i\in \mathcal V \setminus \bigcup _{s=1}^m \{i_s\}$ and all coupling strengths $k_{ij}$, $(i,j)\in \mathcal E$ are collected into the vectors e and k of dimensions $|\mathcal V|-m$ and $|\mathcal E|$, respectively.

In many cases, following the introduced notation, system (1) can be rewritten in the form

$$\begin{aligned} \dot{\varphi }&= F_1(\varphi , e, k), \end{aligned}$$

(2a)

$$\begin{aligned} \dot{e}&= F_2(\varphi , e, k), \end{aligned}$$

(2b)

$$\begin{aligned} \dot{k}&= F_3(\varphi , e, k), \end{aligned}$$

(2c)

with suitably defined functions $F_1, F_2, F_3$ such that $F_i(\cdot , e, k)\in C(\mathcal T_m)$, $i\in \{1,2,3\}$ for every fixed (e, k). For instance, such a transformation is always possible when the interaction between nodes are of a diffusive type, i.e., $g_{ij}(\theta _1, \theta _2) \equiv \tilde{g}_{ij}(\theta _j-\theta _i)$, $\Gamma _{ij}(k_{ij}, \theta _i, \theta _j) \equiv \tilde{\Gamma }_{ij}(k_{ij}, \theta _j-\theta _i)$. System (2) has the same number of equations as (1): Eq. (2a) governs the dynamics of a single selected node within every set $\mathcal P_s$, $s=\overline{1,m}$, Eq. (2b) governs the dynamics of the relative phase differences within every set $\mathcal P_s$, $s=\overline{1,m}$, and Eq. (2c) governs the dynamics of coupling strengths.

We are now in position to introduce an alternative characterization of multi-clustering in terms of the existence of invariant manifolds of (2) of a special topological structure.

Definition 2

We say that system (1) admits multi-clustering defined by the partition $\mathcal P$ if (1) can be rewritten in form of (2) and there exists an m-dimensional invariant toroidal manifold for (2) given by

$$\begin{aligned} \mathcal M = \big \{ (e,k,\varphi ) \in \mathbb R^{|\mathcal V| - m} \times \mathbb R^{|\mathcal E|} \times \mathcal T_m: e = 0, \, k = u(\varphi ), \, \varphi \in \mathcal T_m\big \} \end{aligned}$$

(3)

for some $u=(u_1, \ldots , u_{|\mathcal E|})\in C(\mathcal T_m)$.

The invariant manifold $\mathcal M$ corresponds to the oscillating behavior of coupling strengths k preserving zero phase $e=0$ difference within clusters. Although the function u is periodic with respect to every $\varphi _i$, $i=\overline{1,m}$, the oscillations of the coupling strengths are not necessarily periodic, since they are generated by trajectories on the m-dimensional torus $\mathcal T_m$, which can be, for example, quasi-periodic in time.

The existence of the invariant manifold (3) does not automatically imply the emergence of multi-cluster behavior in (1). It only indicates the possibility of such behavior if the trajectory of the error-system (2) reaches the manifold. The invariance property guarantees that the multi-clustering will be maintained for all times since the moment when the trajectory of (2) reaches the manifold. The emergence of the multi-cluster behavior can be associated to the asymptotic stability of the manifold (3): If the manifold is asymptotically stable then any point from a vicinity of the manifold will eventually converge towards it. In Sect. 2.3, sufficient conditions for the existence and local asymptotic stability of invariant toroidal manifolds of type (3) will be derived for a particular subclass of networks (1), namely, for Kuramoto networks with adaptive coupling. The conditions will provide a trade-off between the natural frequencies of oscillators, plasticity parameters of adaptive couplings, and the interconnection topology of the network.

Remark 1

In the spirit of Definition 2, the frequency multi-clustering can be also characterized in terms of the existence of an invariant toroidal manifold

$$\begin{aligned} \mathcal M_{\text {freq}}= \big \{ (e,k,\varphi ) \in \mathbb R^{|\mathcal V| - m} \times \mathbb R^{|\mathcal E|} \times \mathcal T_m: e = d, \, k = u(\varphi ), \, \varphi \in \mathcal T_m\big \} \end{aligned}$$

for some $u=(u_1, \ldots , u_{|\mathcal E|})\in C(\mathcal T_m)$ and constant $d\in \mathbb R^{|\mathcal V|-m}$. Constant phase difference between any two oscillators within a cluster will guarantee the same frequency for all oscillators that belong to the same cluster. Further extensions and generalizations of manifolds $\mathcal M$ and $\mathcal M_{\text {freq}}$ are possible to cover many other types of collective behavior like, generalized multi-clustering or practical synchronization. These are, however, out of the scope of the current chapter.

2.3 Application to Adaptive Kuramoto Networks

Let $\mathcal G = (\mathcal V, \mathcal E)$ be the directed graph defined in Sect. 2.1 that represents the network of oscillators with the adjacency matrix $A = [a_{ij}]_{i,j = \overline{1,N}}$. Additionally, it is assumed that the graph does not have self-loops, i.e., $a_{ii}=0$ for all $i=\overline{1,N}$. The dynamics of the network is given by [14]

$$\begin{aligned} \begin{array}{rcll} \dot{\theta }_i &{} = &{} w_i+\sum \limits _{j=1}^N a_{ij} k_{ij} \sin (\theta _j-\theta _i), &{} i=\overline{1,N}, \\ \dot{k}_{ij} &{}=&{} -\gamma k_{ij} + \mu _{ij}\Gamma (\theta _j-\theta _i), &{} i,j=\overline{1,N}, \end{array} \end{aligned}$$

(4)

where $w_i\in \mathbb R$ and $\theta _i(t)\in \mathcal T_1$ denote the natural frequency and the phase of the ith oscillator. The dynamics of the coupling strength $k_{ij}(t)\in \mathbb R$ is defined by parameters $\mu _{ij}, \gamma \in \mathbb R_{>0}$ and $\Gamma \in C^1(\mathcal T_{1})$ with $|\Gamma |_1=\delta \in \mathbb R_{>0}$.

We are interested in establishing the relationships between the emergence of multi-cluster behavior in (4) (in the sense of Definition 1) and the dynamical properties of oscillators, the adaptive couplings, and the interconnection topology. To this end, for a given partition $\mathcal P$, let $\mathcal E_{in}$ and $\mathcal E_{out}$ be the subsets of $\mathcal E$ that correspond to the intra-cluster links and inter-cluster links, respectively. The cardinalities of these sets $c_{in} = |\mathcal E_{in}|$ and $c_{out} = |\mathcal E_{out}|$ characterize the interconnection structure of $\mathcal G$ with respect to the partition $\mathcal P$. Additionally, let $w_{min}=\min _{i=\overline{1,N}} |w_i|$ and $w_{max}=\max _{i=\overline{1,N}} |w_i|$ denote the minimal and maximal absolute value of the natural frequencies.

Sufficient conditions for the existence and construction procedure of the invariant toroidal manifolds that correspond to the m-cluster behavior of the network (4) have been proposed in [14] for the case of identical plasticity parameters $\mu _{ij}\equiv \mu \in \mathbb R_{>0}$. However, the mentioned result does not answer the question whether the constructed invariant manifold is (asymptotically) stable. Later, this question has been answered in [13] by proposing sufficient conditions for the asymptotic stability of the invariant manifold for the case when the plasticity parameters $\mu _{ij}$ are different for the intra-cluster and inter-cluster links. Namely, let $\mu _{ij}\equiv \tilde{\mu }\in \mathbb R_{>0}$ if the link (i, j) connects nodes within some cluster $\mathcal P_s$, $s=\overline{1,m}$, and $\mu _{ij}\equiv \mu \in \mathbb R_{>0}$ otherwise. For convenience, system (4) with the chosen set of plasticity parameters will be denoted as $\Sigma (\tilde{\mu }, \mu )$ from now on, where the first argument stands for the plasticity parameter $\mu _{ij}$ of the adaptive link connecting nodes within the same cluster, and the second argument stands for the plasticity parameter $\mu _{ij}$ of the adaptive link connecting nodes belonging to different clusters.

The derivation of the error-system follows the steps introduced in Sect. 2.2. Let the partition $\mathcal P$ be given. For every cluster $\mathcal P_s$, $s=\overline{1,m}$ pick an arbitrary oscillator $i_s \in \mathcal P_s$ and denote its phase and natural frequency by $\varphi _s:=\theta _{i_s}$ and $\bar{w}_s:=w_{i_s}$, respectively. For every oscillator $i\in \mathcal P_s$, let $e_i=\theta _i-\varphi _s$ define the relative phase-error within a given cluster $\mathcal P_s$, $s=\overline{1,m}$. Then, $\Sigma (\tilde{\mu }, \mu )$ can be rewritten in the following form:

$$\begin{aligned} \dot{\varphi }_s&= \bar{w}_s + \sum \limits _{j\in \mathcal P_s}a_{i_sj} k_{i_sj}\sin {e_j} + \sum \limits _{r\not =s}\sum \limits _{j\in \mathcal P_r} a_{i_sj}k_{i_sj}\sin (e_j+\varphi _r-\varphi _s), s=\overline{1,m},\end{aligned}$$

(5a)

$$\begin{aligned} \nonumber \dot{e}_i&= w_i-\bar{w}_s + \sum \limits _{j\in \mathcal P_s}\left[ a_{ij} k_{ij}\sin (e_j-e_i)-a_{i_sj} k_{i_sj}\sin {e_j}\right]&~ \\ \nonumber&\quad +\sum \limits _{r\not =s}\sum \limits _{j\in \mathcal P_r} \left[ a_{ij} k_{ij}\sin (e_j-e_i+\varphi _r-\varphi _s)- a_{i_sj} k_{i_sj}\sin (e_j+\varphi _r-\varphi _s) \right] \\[-3mm]&\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \,\,\forall i\in \mathcal P_s\setminus \{i_s\},\, s=\overline{1,m},\end{aligned}$$

(5b)

$$\begin{aligned} \dot{k}_{ij}&= -\gamma k_{ij} + \mu \Gamma (e_j-e_i+\varphi _r-\varphi _s) \quad \forall i\in \mathcal P_s,\, \forall j\in \mathcal P_r,\, s\not =r, s,r=\overline{1,m},\end{aligned}$$

(5c)

$$\begin{aligned} \dot{k}_{ij}&= -\gamma k_{ij} + \tilde{\mu }\Gamma (e_j-e_i)\, \qquad \qquad \qquad \qquad \qquad \quad \quad \forall i,j\in \mathcal P_s,\, i\mathord {\not =}j,\, s = \overline{1,m}. \end{aligned}$$

(5d)

System (5) is a counterpart of system (2) for the case of Kuramoto networks with adaptive coupling and it has the same number of equations as system (4). Equations (5a) describe the dynamics of m arbitrarily selected oscillators (one from every cluster). Equations (5b) describe the error dynamics within each cluster. Equations (5c) describe the dynamics of the coupling strengths between nodes of different clusters. Finally, (5d) describe the dynamics of the intra-cluster coupling strengths. Let $\varphi = (\varphi _1, \ldots , \varphi _m)^{\top }\in \mathcal T_m$ and $e=(e_{i^1_1},\ldots ,e_{i^m_{n_m}})^\top \in \mathbb R^{N-m}$ be the vectors collecting all cluster phases $\varphi _i$, $i=\overline{1,m}$ and all intra-cluster relative phase errors $e_i$, $i\in \mathcal P_s\setminus \{i_s\}$, $s=\overline{1,m}$, respectively. Similarly, all inter- and intra-cluster coupling strengths are collected into the vectors $k^{inter}\in \mathbb R^{c_{out}}$ and $k^{intra}\in \mathbb R^{c_{in}}$, respectively, and $k=({k^{inter}}^\top , {k^{intra}}^\top )^\top$. Following Definition 2, the multi-cluster behavior in network $\Sigma (\tilde{\mu }, \mu )$ is possible if system (5) possesses an invariant toroidal manifold

$$\begin{aligned} \mathcal M = \{ (e,k,\varphi ) \in \mathbb R^{N - m} \times \mathbb R^{c_{in}+c_{out}} \times \mathcal T_m: e = 0, \, k = u(\varphi ), \, \varphi \in \mathcal T_m\} \end{aligned}$$

(6)

for some $u\in C(\mathcal T_m)$. This invariant manifold corresponds to the oscillating behavior of the coupling strengths k preserving zero phase error e within clusters. In [14], it has been shown that the inter-cluster coupling strengths cannot converge to some constant value say d simultaneously guaranteeing the convergence of the phase errors to zero, i.e., $\{(e,k,\varphi )\in \mathbb R^{N - m} \times \mathbb R^{c_{in}+c_{out}} \times \mathcal T_m:e=0$, $k=d$, $\varphi \in \mathcal T_m\}$ is not an invariant set of (5) for any constant $d\in \mathbb R^{c_{in}+c_{out}}$. Hence, the oscillating behavior of the inter-cluster coupling strengths is necessary for the emergence of multi-cluster formations in (4).

Following the steps of the proof of Theorem 3 from [14], the following result can be obtained.

Theorem 1

(Adapted from [14], Theorem 3) Let the following conditions hold true for system $\Sigma (\tilde{\mu },\mu )$ and a given partition $\mathcal P$:

• (A1)    for any $s=\overline{1,m}$ and for any $i,j\in \mathcal P_s$ it holds that $w_i=w_j$,

• (A2)    for any $s,r=\overline{1,m}$, $s\not =r$ there exist constants $c_{sr}\in \mathbb N$ such that for any $i\in \mathcal P_s$

  $$\sum \limits _{j\in \mathcal P_r} a_{ij}=c_{sr},$$

• (A3)    given $c_{max}:=\max \limits _{s=\overline{1,m}} \sum \limits _{r\not =s}c_{sr}$ it holds that

  $$\begin{aligned} w_{min}-\mu \gamma ^{-1}\delta c_{max}>0 \end{aligned}$$

  (7)

  and

  $$\begin{aligned} 4\frac{\mu }{\gamma ^2} \delta \sqrt{c_{out}} \sum \limits _{\begin{array}{c} s,r=\overline{1,m} \\ s\not = r \end{array}}{c_{sr}} \frac{w_{max}+\mu \gamma ^{-1}\delta c_{max}}{w_{min}-\mu \gamma ^{-1}\delta c_{max}}<1. \end{aligned}$$

  (8)

Then, system (5) has an invariant toroidal manifold $\mathcal M$ that corresponds to the m-cluster behavior of $\Sigma (\tilde{\mu }, \mu )$ defined by the partition $\mathcal P$.

Proof

The proof is based on the perturbation theory of invariant tori for nonlinear extensions of dynamical systems on torus. We refer the interested readers to the monographs [22, 38], which provide the fundamentals of the mathematical theory of multi-frequency oscillations and to the papers [11, 33, 37], which address the persistence of the Green–Samoilenko function of the invariant tori problem under the perturbations of the right-hand side of the corresponding ordinary differential equations. A relation of these concepts to the synchronization analysis of oscillator networks is discussed in [14].    $\square$

Conditions (A1)–(A3) allow for the following interpretation:

• (A1) requires the natural frequencies to be equal within every cluster.

• (A2) requires that the number of incoming links to every node within a given cluster $\mathcal P_s$ from a different cluster $\mathcal P_r$, $r\not =s$ is the same. Condition (A2) restricts only the number of links and does not require any symmetry of the corresponding adjacency matrix. It is worth to highlight that the intra-cluster couplings are generally not required for the emergence of multi-cluster behavior in the network since (A2) restricts only the structure of the inter-cluster connections.

• (A3) establishes the relations between the natural frequencies of the oscillators, plasticity parameters $\mu , \gamma , \delta$ and the inter-cluster interconnection topology. The procedure for verifying this conditions will be demonstrated in Example 1.

In order to derive sufficient conditions for the asymptotic stability of the manifold $\mathcal M$, the following auxiliary notation is introduced : For a given cluster $\mathcal P_s$, $s=\overline{1,m}$, let

• $n_s$ be the number of elements in the set $\mathcal P_s$;

• $\mathcal G_s\subset \mathcal G$ be a subgraph that correspond to the nodes from $\mathcal P_s$ and intra-cluster connections, i.e.,

  $$\mathcal G_s = \{(\mathcal P_s, \mathcal E_s): \mathcal E_s = \mathcal P_s \times \mathcal P_s \cap \mathcal E\};$$

• $A_s$ be the adjacency matrix of $\mathcal G_s$.

To define the residual connectivity of $\mathcal G_s$ with respect to the node $i_s$, the nodes inside each cluster are enumerated according to the rule $\mathcal P_s = \{i^s_1, \ldots , i^s_{k_s}, \ldots , i^s_{n_s}\}$, where $i^s_{k_s}=i_s$, i.e., the selected node $i_s$ has a sequential number $k_s$ in the cluster $\mathcal P_s$. Then, let

• $A_s^{-}$ be an $(n_s-1)\times (n_s-1)$-dimensional matrix constructed from $A_s$ by removing its $k_s$th row and column;

• $\tilde{A}_s$ be the residual adjacency matrix w.r.t. the node $i_s$, i.e.,

$$\begin{aligned} \begin{aligned} \tilde{A}_s = A_s^- - \begin{pmatrix} a_{i_s i^s_1} &{} \ldots &{} a_{i_s i^s_{k_s-1}} &{} a_{i_s i^s_{k_s+1}} &{} \ldots &{} a_{i_s i^s_{n_s}} \\ a_{i_s i^s_1} &{} \ldots &{} a_{i_s i^s_{k_s-1}} &{} a_{i_s i^s_{k_s+1}} &{} \ldots &{} a_{i_s i^s_{n_s}} \\ \vdots &{} ~ &{}\vdots &{} \vdots &{} ~ &{} \vdots \\ a_{i_s i^s_1} &{} \ldots &{} a_{i_s i^s_{k_s-1}} &{} a_{i_s i^s_{k_s+1}} &{} \ldots &{} a_{i_s i^s_{n_s}} \end{pmatrix} \end{aligned} \end{aligned}$$

(9)

• $D_s$ be the degree matrix of $A_s$, i.e., the diagonal matrix with diagonal elements equal to the sum of all elements in the corresponding row of $A_s$, and $D_s^-$ be an $(n_s-1)\times (n_s-1)$-dimensional matrix constructed from $D_s$ by removing its $k_s$th row and column.

Theorem 2

(Adapted from [13], Theorem 2) Let the network $\Sigma (\tilde{\mu }, \mu )$ satisfy conditions (A1), (A2), and (A3) of Theorem 1, i.e., there exist the synchronization invariant toroidal manifold $\mathcal M$ that corresponds to the multi-cluster behavior of the network given by the partition $\mathcal P$. If

• (A4)    for every $s=\overline{1,m}$

  $$\begin{aligned} {\text {sign}}\,\Gamma (0) {\text {Re}}\, \lambda \left( \tilde{A}_s - D_s^- \right) <0, \end{aligned}$$

  (10)

then there exist $\mu _0\le \mu$ such that for all $\nu < \mu _0$ the invariant toroidal manifold that corresponds to the multi-clustering of $\Sigma (\tilde{\mu }, \nu )$ is locally asymptotically stable.

The ’stability-condition’ (A4) depends on the value of the plasticity function $\Gamma$ evaluated at zero (sometimes called learning rule) and the eigenvalues of the matrix that is constructed of those elements of the adjacency matrix A, which correspond to the intra-cluster connections. This makes a crucial contrast to the ’existence-condition’ (A2) that restricts the inter-cluster connectivity of the interconnection graph $\mathcal G$.

The usage of both Theorems 1 and 2 is demonstrated in the following example:

Example 1 [13]

Consider a network of $N=7$ Kuramoto oscillators (4) with adjacency matrix

$$\begin{aligned} A= \left( \begin{array}{ccc|cccc} 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ \hline 0 &{} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 \\ 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 &{} 0 \\ \end{array} \right) , \end{aligned}$$

natural frequencies $w = (0.5, 0.5, 0.5, \sqrt{0.8}, \sqrt{0.8}, \sqrt{0.8}, \sqrt{0.8})^\top$, plasticity parameters $\gamma = 0.2$, $\tilde{\mu }= \mu = 0.001$, Hebbian learning rule $\Gamma (s)=\cos (s)$, and the desired two-cluster partition $\mathcal P = \{1,2,3\} \cup \{4,5,6,7\}$.

Condition (A1) is satisfied thanks to the choice of w. Every node in cluster $\mathcal P_1$ has exactly one incoming link from the nodes of cluster $\mathcal P_2$, and vice versa. Therefore, the resulting interconnection topology satisfies the condition (A2) and the network satisfies (A3) with characteristics $c_{out}=7$, $c_{max}=c_{12}=c_{21}=1$, $\delta = 1$. Indeed,

$$\begin{aligned} w_{min}-\mu \gamma ^{-1}\delta c_{max}= \tfrac{1}{2}-\tfrac{0.001}{0.2}=0.495>0 \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} 4\frac{\mu }{\gamma ^2} \delta \sqrt{c_{out}} \sum \limits _{\begin{array}{c} s,r=\overline{1,m} \\ s\not = r \end{array}}{c_{sr}} \frac{w_{max}+\mu \gamma ^{-1}\delta c_{max}}{w_{min}-\mu \gamma ^{-1}\delta c_{max}} \approx 0.9615 <1. \end{aligned} \end{aligned}$$

All conditions of Theorem 1 are satisfied. Asymptotic stability of the two-cluster formation can be concluded from the condition (A4) of Theorem 2. For this purpose, the nodes 3 and 7 are chosen as $k_1$ and $k_2$ for clusters $\mathcal P_1$ and $\mathcal P_2$, respectively. Since ${\text {sign}}\,\Gamma (0)={\text {sign}}\,\cos (0)=1>0$, (A4) for the cluster $\mathcal P_1$ reads

so that

$$\begin{aligned} {\text {Re}}\, \lambda (\tilde{A}_1-D_1^-) = {\text {Re}}\, \lambda \begin{pmatrix} -2&{} 1\\ -1&{} -1 \end{pmatrix} ={\text {Re}}\,\big (-\tfrac{3}{2}\pm \tfrac{\sqrt{3}}{2}i\big )<0. \end{aligned}$$

For the cluster $\mathcal P_2$, condition (A4) reads

thus

$$\begin{aligned} {\text {Re}}\, \lambda (\tilde{A}_2-D_2^-) = {\text {Re}}\, \lambda \begin{pmatrix} -2&{} 1&{} 0\\ -1&{} -1&{} 1\\ -1&{} 0&{} -1 \end{pmatrix} =\left[ \begin{array}{l} {\text {Re}}\,(-1\pm i)\\ {\text {Re}}\,(-2)\end{array}\right. <0. \end{aligned}$$

All conditions (A1)–(A4) of Theorems 1 and 2 are satisfied. Simulation results are presented in Figs. 1 and 2. In particular, the numerical simulations in Fig. 1 demonstrate the conclusions of Theorem 2 and show that starting from a vicinity of the invariant toroidal manifold, which corresponds to the desired multi-cluster behavior of oscillators, the coupling strengths and phase-errors of the considered Kuramoto network converge to this manifold. Figure 2 depicts two snapshots of the graph $\mathcal G$ at the beginning ($t=0$) and at the end ($t=500$) of simulation.

Fig. 1

Left figure: Evolution of absolute values of the phase-errors $e_i$, $i=\overline{1,N}$ within clusters. All $e_i(t)\xrightarrow []{t \rightarrow \infty } 0$, $i=\overline{1,N}$ that correspond to the asymptotic stability of the invariant toroidal manifold $\mathcal M$ and the emergence of two-cluster formation given by partition $\mathcal P$. Right figure: Evolution of coupling strengths $k_{ij}$. The intra-cluster coupling strengths converge to the constant value, and the inter-cluster couplings exhibit quasiperiodic oscillations. Due to the choice of rationally independent natural frequencies, these oscillations are not periodic in time

Fig. 2

Snapshots of the graph $\mathcal G$ at the beginning (left figure) and at the end of simulation (right figure). Colors of the nodes represent their phases. In the right figure, the blue connections denote intra-cluster links whose coupling strengths converge to a constant value. Light-grey links correspond to the oscillating inter-cluster couplings (see also Fig. 1). Red and orange nodes in the right figure form to two different clusters

3 Synchronization of Complex Dynamics

This section is devoted to the synchronization analysis of complex dynamics. In particular, we address the problem of the synchronization of chaotic behavior generated by ensembles of Chua oscillators and the problem of the master-slave synchronization of spatiotemporal patterns generated by the Gray–Scott reaction-diffusion PDE model. Besides theoretical results on the synchronization, both problems are illustrated in numerical simulations.

3.1 Synchronization of Chaotic Behavior

Synchronization of chaotic systems has been intensively studied during the last decades [2, 27, 31, 32] due to their numerous applications, e.g., in secure communication [1, 41, 48], laser physics [28], and biomedical engineering [40]. In this context, the Chua circuit appeared to be one of the most interesting objects of exploration since it exhibits extremely rich dynamical behavior and variety of bifurcation phenomena despite its structural simplicity. As pointed out in [46], the investigation of synchronizability of coupled Chua oscillators may help to understand complex dynamical phenomena arising in networks of chaotic systems of more general types.

Synchronization of two linearly coupled Chua oscillators has been studied in [7,8,9, 44, 46, 51]. A graph-spectral approach for the synchronization of networks of resistively coupled nonlinear oscillators has been proposed in [47], and the upper bound on the coupling conductance required for synchronization has been obtained therein. In this chapter, we present sufficient conditions for the synchronization of the network of $N\in \mathbb N$ Chua oscillators interconnected with the static nonlinear coupling via the first state coordinate only. Our conditions provide a trade-off between the interconnection topology of the network, properties of nonlinear coupling function, and parameters of the Chua circuits in order to achieve synchronization. In Sect. 3.1.1, the network under consideration is defined and the main problem of synchronization is formulated. Then, we propose sufficient conditions for the synchronization of $N\in \mathbb N$ coupled Chua oscillators with static nonlinear coupling satisfying a so-called sector condition (see Eq. (15)), and provide a numerical example to illustrate the usage of the derived conditions.

3.1.1 Chua Oscillator

We consider the extended Chua circuit

$$\begin{aligned} \dot{\xi }_1&= \alpha [-\xi _1+\xi _2-\phi (\xi _1)]{+ u},&\xi _1(0)=\xi _{10}\end{aligned}$$

(11a)

$$\begin{aligned} \dot{\xi }_2&= \xi _1-\xi _2+\xi _3,&\xi _2(0)=\xi _{20}\end{aligned}$$

(11b)

$$\begin{aligned} \dot{\xi }_3&= -\beta \xi _2-\gamma \xi _3,&\xi _3(0)=\xi _{30}\end{aligned}$$

(11c)

$$\begin{aligned} y&= \xi _1 \end{aligned}$$

(11d)

with scalar piecewise linear function

$$\phi ({s}) = as+\frac{1}{2}[b-a][|s+1|-|s-1|] \quad \forall s \in \mathbb R$$

and parameters $\alpha ,\beta >0,\gamma \ge 0$, $a<b<0$, or in a matrix form as

$$\begin{aligned} \dot{x}&= {P}x+ {{b}u} + {f}({x}),&x(0)=x_0\end{aligned}$$

(12a)

$$\begin{aligned} y&= c^Tx\end{aligned}$$

(12b)

with the state $x{(t)} = \begin{bmatrix}\xi _{1}(t)&\xi _{2}(t)&\xi _{3}(t)\end{bmatrix}^T \in \mathbb R^3$, ${x}_0=\begin{bmatrix}\xi _{10}&\xi _{20}&\xi _{30}\end{bmatrix}^T \in \mathbb R^3$, external input $u(t)\in \mathbb R$, which will be later used to interconnect Chua oscillators, and the matrix ${P}$ and vectors ${f},{b},c$ given by

$$\begin{aligned} {P}=\begin{bmatrix} -\alpha &{} \alpha &{} 0 \\ 1 &{} -1 &{} 1 \\ 0 &{} -\beta &{} -\gamma \end{bmatrix}, \quad {f}({x}) = \begin{bmatrix} -\alpha \phi (\xi _1) \\ 0 \\ 0 \end{bmatrix}, \quad {b} = c= \begin{bmatrix} 1 \\ 0 \\ 0\end{bmatrix}. \end{aligned}$$

Now, consider a network of $N\in \mathbb N$ nodes described by a graph $\mathcal G = (\mathcal V,\mathcal E)$ with node set $\mathcal V$ and edge set $\mathcal E$, so that $|\mathcal V|=N$. Let the associated adjacency matrix be given by $A=\{a_{ij}\}_{i,j=1,\ldots ,N}$ with zero main diagonal. Then, applying the output feedback

$$\begin{aligned} u_i= -\sum _{j=1}^{N} a_{ij}k(y_i-y_j), \quad i=\overline{1,N} \end{aligned}$$

with an arbitrary nonlinear locally Lipschitz continuous coupling function $k:\mathbb R\rightarrow \mathbb R$, the dynamics of $N\in \mathbb N$ coupled Chua oscillators can be written as

$$\begin{aligned} \begin{aligned} \dot{x}_i&= {A}x_i -{b}\sum _{j=1}^N a_{ij}k(y_i-y_j) +{{f}({x_i})},\quad x_i(0)=x_{i 0}\\ y_i&= c^Tx_i, \end{aligned} \end{aligned}$$

(13)

${i}=1,\ldots ,N$. For any given $\eta \in \mathbb R^{3N}$ let $x=(x_1,\ldots ,x_N): \mathbb R \rightarrow \mathbb R^{3N}$ denote a solution to (13) satisfying initial condition $x(0)=\eta$. The Lipschitz continuity of the right-hand side of (3) guarantees the existence and uniqueness of the solution for any initial value $\eta \in \mathbb R^{3N}$.

Associated to this network consider the relative synchronization errors with respect to the node 1

$$\begin{aligned} e_i = x_i-x_1,\quad {i}=\overline{1,N}. \end{aligned}$$

(14)

The relative errors $e_{ij}$ between arbitrary nodes i and j can be expressed using the relative errors $e_i$ and $e_j$

$$\begin{aligned} e_{ij} = x_i-x_j = x_i-x_1 - (x_j-x_1) = e_i-e_j. \end{aligned}$$

Accordingly, instead of analyzing $\frac{N(N-1)}{2}$ relative errors $e_{ij}$ between connected nodes, it is sufficient to consider the behavior of the $N{-1}$ errors $e_i,\, i=\overline{2,N}$.

The problem addressed in the sequel consists in providing sufficient conditions on the system parameters, the nonlinear coupling function k and the network topology which ensure the synchronization of $N\in \mathbb N$ coupled Chua oscillators, i.e., the global convergence of the norms of errors $e_i$ to zero:

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert e_i(t)\Vert =0,\quad i=\overline{2,N}. \end{aligned}$$

3.1.2 Nonlinear Couplings Satisfying a Sector Condition

We consider a special case of Lipschitz continuous odd coupling $k:\mathbb R \rightarrow \mathbb R$ satisfying a sector condition: Let there exist two constants $k_1, k_2\ge 0$ with $k_2 \ge k_1 \ge 0$ such that for all $s\in \mathbb R$:

$$\begin{aligned} k(s)s\ge 0, \quad k(-s)=-k(s), \quad k_1|s|\le |k(s)|\le k_2|s|. \end{aligned}$$

(15)

Additionally, define the following auxiliary matrices

$$\begin{aligned} {\mathcal A}_1 = \begin{bmatrix} 0 &{} |a_{23} | &{} \cdots &{}|a_{2N}|\\ |a_{32} | &{} 0 &{} \ddots &{} |a_{3N}|\\ \vdots &{} \ddots &{} \ddots &{} \vdots \\ |a_{N2} | &{} |a_{N3} | &{} \cdots &{} 0 \end{bmatrix}, \, {\mathcal A}_2 = \begin{bmatrix} 0 &{} |a_{23} -a_{13}| &{} \cdots &{}|a_{2N}- a_{1N}|\\ |a_{32} - a_{12}| &{} 0 &{} \ddots &{} |a_{3N}- a_{1N}|\\ \vdots &{} \ddots &{} \ddots &{} \vdots \\ |a_{N2} -a_{12}| &{} |a_{N3} -a_{13}| &{} \cdots &{} 0 \end{bmatrix} \end{aligned}$$

with zero main diagonals, $(N-1)\times (N-1)$–dimensional identity matrix I, and $K=\text {diag}\{\kappa _2,\ldots ,\kappa _N\}$, where $\kappa _i=\sum _{j=1}^N a_{ij}$ denotes the degree of node $i=\overline{1,N}$. Sufficient conditions for the synchronization of the entire network can be formulated in terms of eigenvalues of a matrix M defined below that interrelates the parameters of oscillators, coupling function, and the interconnection topology.

Theorem 3

(Adapted from [15], Theorem 3) Let matrix

$${M} = \begin{bmatrix} -(\alpha -\alpha |a|) {I} -k_1 {K} + (k_2-k_1){\mathcal {A}}_1 + k_2 {\mathcal {A}}_2 &{} \alpha {I} \\ {I} &{} -\mu _0 {I} \end{bmatrix}$$

be Hurwitz, where

$$\begin{aligned} \mu _0=\frac{1+\gamma }{2} - {\text {Re}}\,\Big (\sqrt{\tfrac{(1+\gamma )^2}{4}-(\gamma +\beta )}\Big ). \end{aligned}$$

Then, the synchronization errors (14) between the states of Chua oscillators (13) with nonlinear coupling k satisfying (15) converge exponentially to zero.

Example 2 [15]

Consider a complete graph hosting $N=20$ identical Chua oscillators (13) in its nodes. The oscillators are connected via the nonlinear coupling

$$\begin{aligned} k(s)=3s+\arctan {s}\quad \text {for all}\quad s\in \mathbb R, \end{aligned}$$

(16)

and the parameters of oscillators are $\alpha = 15.61,\, \beta =25.581, \, \gamma =0,\, a=-1.142, b=-0.715.$ These parameters correspond to the chaotic behavior of each oscillator with the double scroll attractor [34] (see Fig. 3 (left)). The considered nonlinear coupling strength (16) satisfies the sector condition (15) with constants $k_1=3$ and $k_2=4$. For the chosen parameters of the network matrix M from Theorem 3 reads as

where denotes zero $(N-1)\times (N-1)$–matrix, and denotes $(N-1)$-dimensional vector $\begin{pmatrix}1&1&\cdots&1 \end{pmatrix}^\top$. The eigenvalues of M lie in the open left half-plane of the complex plane $\mathbb C$ so that ${\text {Re}}\,(\lambda ({M})) \in [-56.0643, -0.0748].$ Hence, from Theorem 3 we conclude that oscillators achieve synchronization. The state evolution of the oscillators is shown in Fig. 3 (right).

Fig. 3

Left figure: Double scroll attractor for the chaotic Chua oscillator. Right figure: Time evolution of $N=20$ coupled Chua oscillators from the Example 2

3.2 Synchronization of Spatiotemporal Patterns

3.2.1 Gray–Scott Reaction-Diffusion Model

The Gray–Scott model [17, 21], that is a simple prototype for the models of complex isothermal autocatalytic reactions, is governed by the pair of coupled reaction-diffusion equations

$$\begin{aligned} \partial _t a&= D_a\partial _z^2a-ab^2+\alpha (1-a) \end{aligned}$$

(17a)

$$\begin{aligned} \partial _t b&= D_b\partial _z^2b+ab^2-(\alpha +\beta )b \end{aligned}$$

(17b)

with homogeneous Neumann (non-flux) boundary conditions

$$\begin{aligned} \partial _z a(t,0)&= \partial _z a(t,L) = 0, \quad t\ge 0\end{aligned}$$

(17c)

$$\begin{aligned} \partial _z b(t,0)&= \partial _z b(t,L)=0, \quad t\ge 0 \end{aligned}$$

(17d)

and initial conditions given by

$$\begin{aligned} a(0,z)&= a_0(z), \quad z\in \Omega \end{aligned}$$

(17e)

$$\begin{aligned} b(0,z)&= b_0(z), \quad z\in \Omega , \end{aligned}$$

(17f)

where $a(t,z)\in \mathbb R_{\ge 0}$ and $b(t,z)\in \mathbb R_{\ge 0}$ denote the concentrations of two chemical species at time $t\in [0,\infty )$ and at position $z\in \Omega :=(0,L)$, $L>0$, parameters $D_a, D_b, \alpha$, and $\beta$ are positive scalars, initial conditions $a_0, b_0 \in L^2(\Omega )$. The space of square integrable functions $f:\Omega \rightarrow \mathbb R^n$, $n\in \mathbb N$ is denoted by $L^2(\Omega )$ and it is equipped with the norm $\left\| f\right\| =(\int _{\Omega } |f(z)|^2 dz)^{\frac{1}{2}}$, where $|\cdot |$ denotes either the absolute value or the $\mathbb R^n$-distance, depending on whether its argument is scalar or a vector.

System (17) may exhibit a variety of irregular spatiotemporal patterns in response to finite-amplitude perturbations of the steady state $(a,b)=(1,0)$ [10, 21] (see also Fig. 4). These include a diversity of complex dynamical regimes ranging from steady states and stationary periodic solutions to traveling waves, pulse splittings, and spatiotemporal chaotic behavior. Pattern formation capabilities and the underlying mechanisms of the Gray–Scott model in one- and two-dimensional spatial domains have been reported and analyzed by mathematical analysis methods [23, 24, 45], by computer simulations [29, 30, 42, 43], and by experiments [19, 20].

Fig. 4

Typical patterns generated by the Eq. (17) in the interval (0, 2.5) with diffusion coefficients $D_a = 2\cdot 10^{-4}$ and $D_b = 10^{-4}$. Both patterns a and b emerge from the same initial conditions, but for different parameters $\alpha$, $\beta$

The problem of synchronization of spatiotemporal chaotic behavior generated by the Gray–Scott model have been addressed in [49, 50] by means of the impulsive control techniques. In particular, a class of pinning impulsive controllers has been designed to stabilize and synchronize the spatiotemporal chaotic behavior. The current chapter addresses the synchronization problem of spatiotemporal patterns in the master-slave configuration in which the master system is given by the Gray–Scott model (17), and the slave Gray–Scott system is coupled to the original system via a finite number of spatial locations. This coupling enters the slave system dynamics as the in-domain injections of the master system state variables at the respective locations (see Eq. (19) for details). Following the late-lumping approach that relies directly on the original PDE description, we examine the minimal number of coupling connections sufficient for the synchronization of the master-slave configuration and additionally analyze the results in numerical simulations. In the considered setup, the slave system can be seen as a special kind of the point-wise innovation (PWI) observer [39] for the master Gray–Scott model. The synchronization between the master and the slave systems will correspond to the convergence of the observation error of the PWI-estimator to zero and sufficient conditions for this convergence can be concluded using analysis techniques from [12].

3.2.2 Master-Slave Setup and Main Results

Let the master system be given by (17) and be coupled to the slave system via $m+1$ spatial locations at positions $z=\zeta _i\in \bar{\Omega }:=[0,L]$, $i=\overline{0,m}$, $m\in \mathbb N$ so that $\zeta _0=0$, $\zeta _m=L$, and $z_{i}-z_{i-1}=:d_i>0$ for all $i=1,\ldots ,m$. System (17) is then equipped with the outputs

$$\begin{aligned} y^a_i(t) = a(t,\zeta _i), \quad y^b_i(t) = b(t,\zeta _i), \quad t\ge 0, \, i=0,\ldots ,m. \end{aligned}$$

(18)

A particular case of $m=1$ with $d_1=L$ corresponds to the availability of the boundary coupling only, i.e., no in-domain coupling points. The slave system is defined as a copy of (17) with the in-domain injection of the concentrations $y_i^a, y_i^b$ at respective locations $\zeta _i$, $i=1,\ldots ,m$:

$$\begin{aligned} \partial _t \hat{a}&= D_a\partial _z^2 \hat{a} - \hat{a} \hat{b}^2+\alpha (1-\hat{a}) \end{aligned}$$

(19a)

$$\begin{aligned} \partial _t \hat{b}&= D_b\partial _z^2 \hat{b} + \hat{a} \hat{b}^2-(\alpha +\beta ) \hat{b}\end{aligned}$$

(19b)

$$\begin{aligned} \hat{a}(t,\zeta _i)&= y^a_i(t), \quad t\ge 0, \quad i=\overline{0,m}\end{aligned}$$

(19c)

$$\begin{aligned} \hat{b}(t,\zeta _i)&= y^b_i(t), \quad t\ge 0, \quad i=\overline{0,m}\end{aligned}$$

(19d)

$$\begin{aligned} \hat{a}(0,z)&= \hat{a}_0(z), \quad z\in \Omega \end{aligned}$$

(19e)

$$\begin{aligned} \hat{b}(0,z)&= \hat{b}_0(z), \quad z\in \Omega . \end{aligned}$$

(19f)

The main result is given in the following theorem.

Theorem 4

(Adapted from [12], Theorem 1) Let the initial conditions $a_0, b_0 \in L^2(\Omega )$ and $\hat{a}_0, \hat{b}_0 \in L^2(\Omega )$ for both master and slave systems satisfy $0 \le a_0(z), b_0(z), \hat{a}_0(z), \hat{b}_0(z) \le 1$ for all $z\in \Omega$. Then, there exist constants $\sigma _i>0$, $i=\overline{1,m}$ such that the states of the master and slave systems asymptotically synchronize in the $L_2$-sense, i.e.,

$$\lim _{t\rightarrow \infty }\left\| a(t,\cdot )-\hat{a}(t,\cdot ) \right\| =\lim _{t\rightarrow \infty }\left\| b(t,\cdot )-\hat{b}(t,\cdot ) \right\| =0,$$

provided that $d_i\le \sigma _i$, $i=\overline{1,m}$.

Proof

A constructive proof is based on the observer design technique proposed in [12]. The resulting constants $\sigma _i$ provide the largest distance between the spatial coupling locations that enable the global synchronization of two Gray–Scott models, i.e., synchronization for any the initial conditions satisfying Theorem 4.    $\square$

Example 3

The master system (17) is considered in the domain $\Omega =(0,2.5)$ with parameters $D_a = 2\cdot 10^{-4}, D_b = 1\cdot 10^{-4}, \alpha =0.02, \beta =0.047$ and the initial conditions $(a_0,b_0)=(1,0)$ which are perturbed to $(a_0(z), b_0(z))=(0.5,0.25)$ at locations $z\in (0.925, 1.05)$ and $z\in (1.925, 2.05)$ and to $(a_0(z), b_0(z))=(0.25,0.75)$ at locations $z\in (0.25, 0.375)$. The slave system (19) is a copy of (17) with 6 uniformly distributed coupling locations. The initial conditions for the slave system are selected at the steady state $(\hat{a}_0, \hat{b}_0)=(1,0)$. A comparison of the behavior of the master system and the slave one is given in Fig. 5, and the corresponding synchronization errors are provided in Fig. 6. The norms of these errors converge to zero as $t\rightarrow \infty$. Additionally, the synchronization errors for the cases of 3, 6, 11, 51, and 126 uniformly distributed coupling locations are depicted in Fig. 7. These numerical simulations demonstrate that the smaller gaps $d_i$ between the coupling locations lead to smaller observation errors and their faster convergence to 0.

Fig. 5

A comparison of the evolution of concentrations b and $\hat{b}$. Six equidistantly located coupling points are marked with red crosses (right figure)

Fig. 6

Synchronization errors in case of 6 equidistantly located coupling points

Fig. 7

Evolution of the sum of the $L^2$-norms of synchronization errors depending on the number of equidistantly located coupling points for $t\in [0,5000]$ (left figure) and $t\in [0,150]$ (right figure). The slave system driven by 3 coupling points does not converge to the master system, whereas all other setups lead to the convergence of the synchronization error to 0 with the convergence rate increasing with the number of coupling points

4 Conclusion

The chapter provides synchronization analysis of three essentially different types of oscillator networks, namely, (i) Kuramoto networks with adaptive coupling, (ii) networks of Chua oscillators connected via nonlinear static output-feedback term, and (iii) the master-slave configuration of two Gray–Scott reaction-diffusion models. Sufficient conditions have been presented for the emergence of synchronization phenomena in these networks in terms of the dynamical properties of oscillators, characteristics of the couplings, and the interconnection topology of the networks. Theoretical results are supplemented with numerical case studies.