Synchronization in Cartesian-Product Networks of Time-Delay Coupled Systems

4th IFAC26-28, Conference on Analysis August 2015. Tokyo, Japan and Control of Chaotic Systems August 2015. Tokyo, Japan and Control of Chaotic Systems 4th IFAC26-28, Conference on Analysis Available online at www.sciencedirect.com August 26-28, 2015. Tokyo, Japan

ScienceDirect IFAC-PapersOnLine 48-18 in (2015)Cartesian-Product 245–250 Synchronization Synchronization in Cartesian-Product Networks of Time-Delay Coupled Systems Synchronization in Cartesian-Product Networks of Time-Delay Coupled Systems Networks of Time-Delay Coupled Systems ∗ ∗∗ ∗

Carlos Murguia , Jonatan Pe˜ na , Niels Jeurgens , ∗ ∗∗ Carlos Murguia , Jonatan Pe˜ na∗∗∗ , Niels Jeurgens ∗∗ ,∗ Rob H.B. Fey ∗ , ∗∗Toshiki Oguchi ∗∗ , Henk Nijmeijer ∗∗ Carlos Murguia , Jonatan Pe˜ na∗∗∗ ,, Niels Rob H.B. Fey ∗∗ , Toshiki Oguchi HenkJeurgens Nijmeijer∗ ,∗∗ ∗∗∗ ∗ ∗∗∗ , HenkUniversity Nijmeijerof∗ ∗Rob H.B. Fey , Toshiki Oguchi Dept. of Mechanical Engineering, Eindhoven ∗ Dept. ofTechnology, Mechanical Eindhoven, Engineering,the Eindhoven University of Netherlands, ∗ ∗ Dept. of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, the Netherlands, (e-mail: [email protected], [email protected], Technology, Eindhoven, the Netherlands, (e-mail: [email protected], [email protected], [email protected], [email protected].) ∗∗ (e-mail: [email protected], [email protected], [email protected], [email protected].) Dept. of Electronics, Center for Scientific Research and Higher ∗∗ [email protected], Dept. Education of Electronics, Center [email protected].) for ScientificB.C., Research and Higher at Ensenada, Ensenada, Mexico, ∗∗ ∗∗ Dept. of Electronics, Center for Scientific Research and Higher Education at Ensenada, Ensenada, B.C., Mexico, (e-mail: [email protected].) ∗∗∗ at Ensenada, Ensenada, B.C., Mexico, (e-mail: [email protected].) Dept. Education of Mechanical Engineering, Tokyo Metropolitan University, ∗∗∗ (e-mail: [email protected].) Dept. of Mechanical Engineering, Tokyo Metropolitan University, Tokyo, Japan, (e-mail: [email protected].) ∗∗∗ ∗∗∗ Dept. of Tokyo, Mechanical Engineering, Tokyo Metropolitan University, Japan, (e-mail: [email protected].) Tokyo, Japan, (e-mail: [email protected].) Abstract: We present a theoretical result about synchronization in Cartesian-product networks Abstract: present asystems theoretical result about synchronization Cartesian-product networks of coupled We semipassive interconnected through diffusivein time-delay couplings. First, Abstract: We present asystems theoretical result about synchronization in time-delay Cartesian-product networks of coupled semipassive interconnected through First, for two diffusively time-delay coupled systems, we give diffusive sufficient conditionscouplings. that guarantee of coupled semipassive systems couplings. First, for two diffusively systems, through wespace give diffusive sufficienttime-delay conditions that guarantee the existence of a time-delay region S ∗ coupled ininterconnected the parameter (coupling strength vs. time delay), ∗ coupled systems, we give sufficient conditions that guarantee for two diffusively time-delay the existence of a region S in the parameter space (coupling strength vs. time delay), such that if these parameters belong to the region, the two coupled systems synchronize. ∗ ∗ the existence of aregion region thespectral parameter spacethe(coupling strength vs. time such that these parameters belong to the properties region, systems synchronize. Then, usingif this S ∗ Sandin the of two the coupled Cartesian-product, we delay), derive ∗ such that if these parameters belong to the region, the two coupled systems synchronize. using region and the spectral properties of the Cartesian-product, we derive aThen, method to this predict the Svalues of coupling strength and time-delay for which diffusive time∗ ∗ Then, using region andofthe spectralstrength properties of the Cartesian-product, we derive a method to this predict theonSvalues coupling and time-delay for which diffusive timedelay coupled systems Cartesian-product networks synchronize. The obtained theoretical a method to predict the values of coupling strength and time-delay for which diffusive timedelay coupled systems on Cartesian-product networks synchronize. The obtained theoretical results are experimentally validated using an experimental setup built around electronic circuit delay coupled systems on validated Cartesian-product networks synchronize. obtained theoretical results are experimentally using anmodel. experimental setup builtThe around electronic circuit realizations of the Hindmarsh-Rose neuron results are experimentally validated neuron using anmodel. experimental setup built around electronic circuit realizations of the Hindmarsh-Rose © 2015, IFACof(International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. realizations the Hindmarsh-Rose neuron model. 1. INTRODUCTION on network synchronization of identical nonlinear systems 1. INTRODUCTION on network through synchronization identical couplings. nonlinear In systems interacting diffusive of time-delay prac1. INTRODUCTION on network synchronization of identical nonlinear systems interacting through diffusive time-delay couplings. In practical situations, time-delays caused by signal transmission interacting through time-delay couplings. In pracThe emergence of synchronization in networks of coupled tical time-delays caused by signal transmission affectsituations, the behavior ofdiffusive the interconnected systems. Diffusive The emergence of synchronization in networks of coupled dynamical systems is a fascinating topic in various tical situations, time-delays caused by signal transmission affect the behavior of arise the interconnected systems. Diffusive time-delay couplings naturally for interconnected sysThe emergence of synchronization networks of coupled dynamical systems is a fascinating topicphysics, in various scientific disciplines ranging from inbiology, and time-delay affect the behavior of arise the interconnected systems. Diffusive couplings naturally for can interconnected systems since the transmission of signals be expected to dynamical is a fascinating topicphysics, in applicavarious scientific ranging from biology, and tems chemistry disciplines tosystems social networks and technological time-delay couplings arise naturally for can interconnected syssince time. the transmission signals be expected to take some We present aoftheoretical result about synscientific disciplines ranging from biology, physics, and chemistry social networks andittechnological applicatems since time. the signals can be to tions. For to instance, in biology, is well known that take some We present aoftheoretical result about synchronization in transmission Cartesian-product networks of expected semipassive chemistry social applications. For to instance, in biology, ittechnological is well known that take thousands of firefliesnetworks light upand simultaneously, see Strosome time. We present a theoretical result about synin Cartesian-product networkstime-delay of semipassive systems interconnected through diffusive couit is well that chronization tions. For of instance, in biology, thousands fireflies light up simultaneously, see (Hyla Strogatz [2003], and that groups of Japanese treeknown frogs chronization in Cartesian-product networks of semipassive systems through time-delay couplings. Ininterconnected Oguchi et al. [2013], the diffusive authors started studying thousands of fireflies light up simultaneously, see Strogatz [2003], andshow that synchronous groups of Japanese tree systems through diffusive time-delay coujaponica) may behavior in frogs their (Hyla calls, plings. Ininterconnected Oguchi et al. [2013], thedelay-free authors started studying these ideas for both delayed and diffusively cougatz [2003], andal. that groups Japanese tree frogs japonica) may show synchronous behavior in their (Hyla calls, these plings. In Oguchi et al. [2013], the authors started studying see Aihara et [2011]. In of medicine and neuroscience, ideas for both delayed and delay-free diffusively coupled Cartesian-product networks. They present theoretical japonica) show synchronous behavior their calls, these see Aihara et al. [2011]. In medicine and in neuroscience, clusters ofmay synchronized pacemaker neurons regulating ideas for bothresults delayed and delay-free diffusively coupled Cartesian-product networks. They present theoretical and experimental about partial synchronization see al. [2011]. In medicine and neuroscience, clusters of et synchronized pacemaker regulating our Aihara heartbeats, Peskin [1975], and ourneurons circadian rhythm, and pled Cartesian-product networks. They present theoretical experimental results about partial synchronization in such networks. In this manuscript, using the results clusters of synchronized pacemaker regulating our heartbeats, Peskinto[1975], andday-night ourneurons circadian rhythm, and experimental about partial synchronization which is synchronized the 24-h cycle, Czeisler in such networks. In manuscript, the results presented in Steur results andthis Nijmeijer [2010]using and Steur et al. our heartbeats, Peskin [1975], and our circadian rhythm, which synchronized the 24-h day-night cycle, in such networks. In this manuscript, using the results et al. is [1980], are cleartoexamples. There are also Czeisler several presented in Steur and Nijmeijer [2010] and Steur etthat al. [2014], we derive sufficient conditions that guarantee which synchronized the day-night cycle, et al. is [1980], are cleartoexamples. There are also Czeisler several engineering applications; for 24-h instance, synchronization in presented Steur andis Nijmeijer [2010] Steur et al. [2014], we in derive sufficient conditions thatand guarantee that if the interconnection sufficiently strong and the timeet al. [1980], are Dorfler clear examples. arevelocity also several engineering applications; for synchronization in if power networks, andinstance, BulloThere [2012], syn[2014], we derive sufficient conditions that guarantee that the interconnection is sufficiently strong and the timeis sufficiently small the systems in the Cartesianengineering applications; instance, synchronization in delay power networks, Dorfler for and Bullo [2012], velocity synif the is interconnection is sufficiently strong and the timechronization in platoons of vehicles, Ploeg et al. [2014], delay sufficiently small the systems in the Cartesianproduct network asymptotically synchronize. Particularly, power networks, Dorfler and Bullo [2012], velocity synchronization in platoons of vehicles, Ploeg et al. [2014], and synchronization in robotics, where multiple robots delay is sufficiently small the systems in the Cartesianproduct network synchronize. Particularly, using the scalingasymptotically result in Steur et al. [2014] and the chronization in platoons of vehicles, Ploeg al. [2014], and in robotics, where multiple robots carrysynchronization out tasks that cannot be achieved by eta single one spectral product network synchronize. Particularly, using the scalingasymptotically result in Cartesian-product, Steur et al. [2014] and the properties of the we derive and synchronization in robotics, whereexamples multiple carry out tasks that cannot be achieved by a single one spectral Rodriguez and Nijmeijer [2001]. More ofrobots synusing the scaling result in Steur et al. [2014] and the properties of the Cartesian-product, we derive method to predict the values of coupling strength and carry outbehavior tasks that cannot be achieved by can a single one aspectral Rodriguez and Nijmeijer [2001]. More examples synofdiffusive thevalues Cartesian-product, wesystems derive chronous in science and engineering beoffound atime-delay methodproperties to predict the of coupling strength and for which time-delay coupled Rodriguez and Nijmeijer [2001]. More examples syn- a method to predict the values of coupling strength and chronous behavior in science and engineering can beof found in, for instance, Blekhman [1988], Pikovsky et al. [2001], time-delay for which diffusive time-delay coupled synchrosystems on Cartesian-product networks asymptotically chronous behavior in science and engineering can be found in, for instance, Blekhman [1988], Pikovsky et al. [2001], time-delay coupled systems Strogatz [2003], and references therein. time-delay for whichtheoretical diffusive on Cartesian-product networks asymptotically synchronize. The obtained results are supported by in, for instance, Blekhman [1988], Pikovsky et al. [2001], on Cartesian-product networks asymptotically synchroStrogatz and references The obtained are supported by One of the[2003], first technical resultstherein. regarding synchronization nize. experimental results,theoretical which are results performed in an experiStrogatz [2003], and references therein. nize. obtained theoretical are board supported by One of the first technical results results, which are results performed in an realizaexperiof coupled nonlinear systems is regarding presented synchronization in Fujisaka and experimental mentalThe setup built around electronic circuit One of the first technical results regarding synchronization of coupled nonlinear systems is presented in Fujisaka and experimental results, which are performed in an experibuilt around electronic circuit board realizaYamada [1983]. In this paper, the authors show that cou- mental tions ofsetup the Hindmarsh-Rose neuronal model. To account of coupled nonlinear systems isthe presented in Fujisaka and tions Yamada [1983]. In this paper, authorsin show that coumental setup builtdissimilarities around electronic circuit board realizathe Hindmarsh-Rose neuronal model. To circuits, account pled chaotic oscillators may synchronize spite of their for theofinevitable of the individual Yamada [1983]. Intothis paper, the authors that cou- for pled oscillators mayconditions. synchronize inshow spite ofresult, their tions of the Hindmarsh-Rose neuronal model. To account the inevitable dissimilarities of the individual circuits, high chaotic sensitivity initial After this we introduce the notion of practical synchronization, which in spite their for thethat inevitable dissimilarities of the individual circuits, pled chaotic oscillators synchronize high sensitivity to initial After this ofresult, introduce thecircuits notion of practical synchronization, which considerable interest in may theconditions. notion of synchronization of we states the may be called synchronized if the high sensitivity to initial conditions. After this result, considerable interest in the notion of synchronization of we introduce the notion of practical synchronization, if the general nonlinear systems has arisen. This paper focuses states that the circuits may be called synchronized which considerable interest in thehas notion of This synchronization of states that the circuits may be called synchronized if the general nonlinear systems arisen. paper focuses general nonlinear systems has arisen. This paper focuses

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differences between their outputs are sufficiently small on a long finite time interval. The remainder of the paper is organized as follows. In Section 2, we recall some useful definitions. The notion of semipassivity and some basic terminology of graph theory are introduced. The definition of diffusive time-delayed couplings, the problem formulation, and a summary of the main results presented in Steur and Nijmeijer [2010] and Steur et al. [2014] are given in Section 3. Then, the main result is derived in Section 4. Next, in Section 5, the notion of practical synchronization, the experimental setup, and a set of experimental results are presented. Finally, concluding remarks are given in Section 6. 2. PRELIMINARIES Throughout this paper, the following notation is used: the symbol R>0 (R≥0 ) denotes the set of positive (nonnegative) real numbers. The Euclidian norm in Rn is denoted by | · |, |x|2 = xT x, where T denotes transposition. The notation col(x1 , ..., xn ) stands for the column vector composed of the elements x1 , ..., xn . This notation is also used for the case in which the components xi are vectors. The n × n identity matrix is denoted by In . The spectrum of a matrix A is denoted by spec(A). For any two matrices A and B, the notation A ⊗ B (the Kronecker product, see Bollobas [1998]) stands for the matrix composed of submatrices Aij B , where Aij , i = 1, ..., n, j = 1, . . . , m, stands for the ij th entry of the n × m matrix A. The space of continuous functions from X ⊂ Rn to Y ⊂ Rm that are (at least) r ≥ 0 times continuously differentiable is denoted by C r (X , Y). For simplicity of notation, we often suppress the explicit dependence of time t. 2.1 Basic concepts of graph theory We represent the communication topology of a network of k systems by a weighted undirected graph G = (V, E, A), where V = {v1 , v2 , . . . , vk } is the set of nodes, E ⊂ V × V is the ordered set of edges and A = AT ∈ Rk×k is the weighted adjacency matrix. The convention is that system vi receives information from system vj if and only if [vi , vj ] ∈ E. The weighted adjacency matrix encodes the weights that the edges carry. That is, for aij being the ij th entry of the adjacency matrix A, aij is a positive real constant if [vi , vj ] ∈ E and aij = 0 otherwise. Given a node vi ∈ V, we define Ni := {vj ∈ V|[vi , vj ] ∈ E} as the set of its neighbors, i.e., the set of all nodes that have edges pointing to node vi . Throughout this paper, it is assumed that the graphs G are simple, that is, they do not contain any edges of the form [vi , vi ]. In addition, we assume that the graphs G are strongly connected, which means that for any two nodes vi , vj ∈ V, there exists a path (i.e. a sequence of undirected edges) in G that connects them. For a given graph G with k nodes, we define the weighted degree matrix as D(G) := diag(d1 , d2 , . . . , dk ), where the constants di := j∈Ni aij define the total weight of edges pointing to node vi . The Laplacian matrix is defined as L(G) := D(G) − A(G). Note that L(G) ∈ Rk×k is singular by construction, i.e., it has at least one eigenvalue equal to zero. It is well known that the zero eigenvalue of L(G) is simple if L(G) is the weighted Laplacian matrix of a strongly connected graph, see Bollobas [1998] for details. 246

2.2 Cartesian-product graphs The Cartesian-product graph of graphs G1 = (V(G1 ), E(G1 )) and G2 = (V(G2 ), E(G2 )) is denoted by G1 G2 . The set of nodes of G1 G2 is given by V(G1 G2 ) = V(G1 ) × V(G2 ), and any two nodes (v1 , v2 ), (v1 , v2 ) ∈ V(G1 G2 ) are adjacent if v1 = v1 and (v2 , v2 ) ∈ E(G2 ), (or v2 = v2 and (v1 , v1 ) ∈ E(G1 )). An example is depicted in Figure 1. The Cartesian-product preserves connectedness properties of the graphs G1 and G2 , i.e., if both G1 and G2 are strongly connected, then G1 G2 is strongly connected. Lemma 1. [Mesbahi and Egerstedt [2010]]. Let G1 and G2 be two strongly connected graphs with k1 and k2 nodes, respectively. Assume that the constants µi ∈ R≥0 , νj ∈ R≥0 , i = 1, . . . , k1 , j = 1, . . . , k2 are, respectively, the eigenvalues of L(G1 ) and L(G2 ) with corresponding eigenvectors ωi and vj . Then, L(G1 G2 ) = L(G1 ) ⊗ Ik2 + Ik1 ⊗ L(G2 ),

and the eigenvalues of L(G1 G2 ) are µi + νj with corresponding eigenvectors ωi ⊗ vj . 2.3 Semipassive systems Consider the system x˙ = f (x, u), (1a) y = h(x), (1b) with state x ∈ Rn , input u ∈ Rm , output y ∈ Rm , and sufficiently smooth functions f : Rn × Rm → Rn and h : Rn → Rm .

Definition 1. [Pogromsky et al. [1999]]. System (1) is called C r -semipassive if there exists a nonnegative storage function V ∈ C r (Rn , R≥0 ) such that V˙ ≤ y T u − H(x), where the function H ∈ C(Rn , R) is nonnegative outside some ball, i.e., ∃ ϕ ∈ R>0 s.t. |x| ≥ ϕ → H(x) ≥ (|x|), for some continuous nonnegative function (·) defined for |x| ≥ ϕ. If the function H(·) is positive outside some ball, then system (1) is said to be strictly C r -semipassive. Remark 1. System (1) is C r -passive (strictly C r -passive) if it is C r -semipassive (strictly C r -semipassive) with H(·) being positive semidefinite (positive definite). In light of Remark 1 a (strictly) C r -semipassive system behaves like a (strictly) passive system for large |x(t)|. 3. NETWORK SYNCHRONIZATION OF TIME-DELAY COUPLED SYSTEMS Consider k identical nonlinear systems of the form  z˙i = q(zi , yi ), y˙ i = a(zi , yi ) + ui ,

(2)

with i ∈ I := {1, ..., k}, internal state zi ∈ Rn−m , output yi ∈ Rm , state xi := col(zi , yi ), input ui ∈ Rm , and sufficiently smooth functions q : Rn−m ×Rm → Rn−m and a : Rn−m × Rm → Rm . Let the systems be interconnected on a simple strongly connected graph G through full delay couplings of the form  ui (t) = γ aij [yj (t − τ ) − yi (t − τ )], (3) j∈Ni

where γ ∈ R>0 denotes the coupling strength, τ ∈ R≥0 is the induced time-delay, and the constants aij ≥ 0 are

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Figure 1. (a) Strongly connected graphs G1 and G2 with interconnection weights a1 , a2 , a3 ∈ R>0 and b ∈ R>0 , respectively. (b) Cartesian-product graph G1 G2 . the entries of the weighted symmetric adjacency matrix A ∈ Rk×k associated with the graph G.  Without loss of generality, it is assumed that maxi∈I j∈Ni aij = 1. Notice that coupling (3) can be written in matrix form u(t) = −γ (L(G) ⊗ Im ) y(t − τ ), where u := col(u1 , ..., uk ) ∈ Rkm , L(G) ∈ Rk×k denotes the Laplacian matrix of G, and y := col(y1 , ..., yk ) ∈ Rkm . Full delay couplings may arise, for instance, when the systems are interconnected by centralized control laws. Define the stacked state x := col(x1 , ..., xk ) ∈ Rkn . A necessary condition for a network of systems to synchronize is that the synchronization manifold M := {x ∈ Rkn |xi = xj , ∀ i, j ∈ I}, is forward invariant with respect to the dynamics of the coupled systems. We observe that because systems (2) are identical and the full delay coupling (3) vanishes on M, the synchronization manifold is forward invariant under the closed-loop dynamics (2),(3). The interconnected systems (2),(3) are said to be fully synchronized, or simply synchronized, if the synchronization manifold M contains an asymptotically stable subset. 3.1 Boundedness of the solutions To have the synchronization problem well-defined, we require the solutions of the time-delay coupled systems to be bounded on the whole positive time-axis. Using the notion of semipassivity, we present sufficient conditions which ensure the solutions of the coupled systems to be ultimately bounded. Lemma 2. [Steur and Nijmeijer [2010]]. Consider the k identical systems (2) interconnected through the time delay coupling (3) with coupling strength γ ∈ R≥0 and time-delay τ ∈ R≥0 on a simple strongly connected graph G. Assume that each system (2) is strictly C 1 -semipassive with radially unbounded storage function V (xi ) and the functions H(xi ) are such that there exists R ∈ R>0 such that |xi | > R implies that H(xi ) − γ|yi |2 > 0. Let γmax be the largest γ that satisfies the above inequality, then the solutions of the coupled systems (2),(3) are ultimately bounded for γ ∈ [0, γmax ] and any finite τ ∈ R≥0 . 3.2 Network synchronization Next, we present conditions for the synchronization manifold M to contain an asymptotically stable subset. 247

Theorem 1. [Steur and Nijmeijer [2010]]. Consider the k coupled systems (2),(3) on a simple and strongly connected graph G. Assume that the conditions stated in Lemma 2 are satisfied. Additionally, suppose that: (H3.1) There exists a positive definite matrix P ∈ R(n−m)×(n−m) such that the eigenvalues of the symmetric matrix    T ∂q ∂q P (zi , yi ) + (zi , yi ) P, ∂zi ∂zi are uniformly negative and bounded away from zero for all zi ∈ Rn−m and yi ∈ Rm . Then, there exist positive constants γ¯ and χ ¯ such that, ¯ then (a subset of) the if γ ∈ (¯ γ , γmax ] and γτ ≤ χ, synchronization manifold M is globally asymptotically stable for (2),(3). Assumption (H3.1) implies that the zi -subsystem in (2) is an exponentially convergent system with input yi . Such an exponentially convergent system has a unique, exponentially stable steady state solution that depends on the driving input yi , but not on the initial data zi (t0 ). Roughly speaking, an exponentially convergent system “forgets” its initial conditions exponentially fast. It follows that given two exponentially convergent systems defined by the same function q(·), but driven by different inputs yi and yj , if |yi (t) − yj (t)| → 0 as t → ∞, then |zi (t) − zj (t)| → 0 as t → ∞. This implies that if all outputs yi of systems (2) synchronize, then all “internal states” zi synchronize. For a precise definition of convergent systems and many more interesting properties, we refer to Pavlov et al. [2004]. 3.3 Synchronization in relation to the network topology Let L(G) = L(G)T be the Laplacian matrix of a strongly connected undirected graph G with k nodes, and order the eigenvalues of L(G) as 0 = λ1 < λ2 ≤ . . . ≤ λk . In Steur et al. [2014], it is shown that for any network with symmetric Laplacian matrix, the values of coupling strength γ and time-delay τ for which (k > 2) full delay coupled systems (2),(3) synchronize can be determined from the non-zero eigenvalues of L(G) and the values of coupling strength and time-delay for which k = 2 coupled systems (2),(3) synchronize. Particularly, given two coupled systems (2),(3), if the conditions stated Theorem 1 are satisfied, there exists a non-empty set S ∗ := {(γ, τ ) ∈ R>0 × R≥0 |¯ γ ≤ γ ≤ γmax and γτ ≤ χ}, ¯

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such that for any (γ, τ ) ∈ S ∗ the two coupled systems synchronize. We refer to this set S ∗ as the synchronization region of two full delay coupled systems. Theorem 2. [Steur et al. [2014]]. Consider a network of k > 2 coupled semipassive systems (2),(3) on a strongly connected undirected graph G. Assume that the conditions of Theorem 1 are satisfied and let S ∗ be the synchronization region of two full delay coupled systems. Then, the network of k > 2 systems (2),(3) synchronizes if (γ, τ ) ∈ S2 ∩ Sk , where    λj ∗ Sj := , j = 2, k, 2 γ, τ ∈ S with λ2 and λk the smallest non-zero and the largest eigenvalues of the symmetric Laplacian matrix L(G), respectively.

In other words, Theorem 2 states that for any undirected network of full delay coupled semipassive systems, the values of coupling strength and time-delay for which the systems in the network synchronize can be predicted by taking the intersection of S2 and Sk . Note that S2 and Sk are copies of S ∗ that are scaled by the factors λ22 and λ2k over the γ-axis, respectively. 4. MAIN RESULT: SYNCHRONIZATION IN CARTESIAN-PRODUCT NETWORKS In this section, we use the result stated in Theorem 2 and the properties of the Cartesian-product to predict synchronization regions in Cartesian-product networks of coupled semipassive systems. Consider two undirected strongly connected networks Gj , j = 1, 2, with kj nodes and weighted Laplacian matrix L(Gj ) ∈ Rkj ×kj . Let µi and νκ , i = 1, . . . , k1 , κ = 1, . . . , k2 , be the eigenvalues of L(G1 ) and L(G2 ), respectively, with corresponding eigenvectors ωi and vκ . By Lemma 1, the Laplacian matrix of the Cartesian-product network G1 G2 is given by L(G1 G2 ) = L(G1 ) ⊗ Ik2 + Ik1 ⊗ L(G2 ) with eigenvalues µi + νκ , and corresponding eigenvectors ωi ⊗ vκ . Let k = k1 + k2 systems (2) on the Cartesian-product network G1 G2 be interconnected through u(t) = − (Ik1 ⊗ γ2 L(G2 ) + γ1 L(G1 ) ⊗ Ik2 ) y(t − τ ),

(4)

with u := col(u1 , ..., uk1 , . . . , uk1 +k2 ) ∈ R , stacked output y := col(y1 , ..., yk1 , . . . , yk1 +k2 ) ∈ R(k1 +k2 )m , coupling strengths γ1 , γ2 ∈ R>0 , and time-delay τ ∈ R≥0 . (k1 +k2 )m

Corollary 1. Consider the k = k1 + k2 coupled systems (2),(4) with time-delay τ ∈ R≥0 and coupling strengths γ1 , γ2 ∈ R>0 on the Cartesian-product graph G1 G2 . Assume that the conditions of Theorem 1 are satisfied, and let S ∗ be the synchronization region of two full delay coupled systems (2),(3). Then, the coupled systems (2),(4) asymptotically synchronize, if γ2 = αγ1 for some α ∈ R>0 , and (γ1 , τ ) ∈ R2 ∩ Rk , where    λj ∗ , j = 2, k, γ , τ ∈ S Rj := 1 2 λ2 = min(µi + ανκ ), λk = max(µi + ανκ ), i ∈ {1, . . . , k1 }, and κ ∈ {1, . . . , k2 }. Proof: If γ2 = αγ1 , coupling (4) can be written as u = −γ1 (Ik1 ⊗ αL(G2 ) + L(G1 ) ⊗ Ik2 ) y(t − τ ). 248

Therefore, under the conditions of Corollary 1, the closed loop dynamics (2),(4) is equivalent to the dynamics of k = k1 + k2 coupled systems (2),(3) with γ = γ1 , and Laplacian L(G) = (Ik1 ⊗ αL(G2 ) + L(G1 ) ⊗ Ik2 ). From Lemma 1, spec(Ik1 ⊗ αL(G2 ) + L(G1 ) ⊗ Ik2 ) = µi + ανκ with i = 1, . . . , k1 , and κ = 1, . . . , k2 . Finally, the result follows from Theorem 2.  The result of Corollary 1 amounts to the following. The values of γ1 , γ2 , τ ∈ R≥0 , for which the k = k1 + k2 coupled systems (2),(4) on a Cartesian-product network synchronize, can be predicted by taking γ2 = αγ1 for a fixed positive constant α; and then, taking the intersection of R2 and Rk , where, for j = 2, k, Rj is a copy of S ∗ that is scaled by a factor λ2j over the γ-axis, and λ2 and λk denote the smallest nonzero and the largest eigenvalues of the modified Laplacian L(G) = (Ik1 ⊗αL(G2 )+L(G1 )⊗Ik2 ). Finally, we can predict the synchronization region in the extended parameter space (γ1 , γ2 , τ ) ⊂ R≥0 × R≥0 × R≥0 by varying the constant α over the interval (0, ∞). 5. EXPERIMENTAL RESULTS In this section, the theoretical results summarized in Corollary 1 are experimentally validated. The experiments are performed in an experimental setup of timedelay coupled Hindmarsh-Rose (HR) circuits. The HR model, Hindmarsh and Rose [1984], offers a qualitative description of the mechanisms of spike generation in neural cells. Particularly, it models the membrane potential and the transport of ions across the membrane of single neurons. An ith HR neuron model consists of three nonlinear ordinary differential equations of three variables, the membrane potential yi (t) ∈ R, and two internal variables zi,1 (t), zi,2 (t) ∈ R, which represent the fast and slow transport of ions across the membrane. In Steur et al. [2009], it is proved that the Hindmarsh-Rose neuron is strictly C 1 -semipassive. Using Lemma 2, the authors show that for every finite coupling strength γ and time-delay τ , the solutions of time-delay coupled HR neurons are ultimately bounded. Moreover, they also prove that the internal dynamics of the HR neuron satisfies assumption (H3.1) with matrix P = I2 , which implies that its internal dynamics is an exponentially convergent system. Thus, the HR neuron satisfies the assumptions of Theorem 1, and it can be concluded that synchronization in networks of timedelay coupled HR neurons may happen when the coupling strength and time-delay are chosen appropriately. In Neefs et al. [2010], an electronic circuit board with off-the-shelf components (resistors, capacitors, operational amplifiers, and analog voltage multipliers) is designed to mimic the dynamics of the HR neuron. To ensure that the signals z1 (t), z2 (t), and y(t) are in the (linear) operating range of the components of the circuit board, we redefine the variables t, zi,1 (t), zi,2 (t), and yi (t): t → 100t, zi,1 (t) → 1 5 (zi,1 (t) + 4), zi,2 (t) → zi,2 (t) − 6, yi (t) → yi (t) − 1, and obtain    z˙i,1 (t) = 1000 − yi2 (t) − 2yi (t) − zi,1 (t) ,        z˙i,2 (t) = 5 4yi (t) + 4.472 − zi,2 (t) , (5)   y˙ i (t) = 1000 − yi3 (t) + 3yi (t) − 8      + 5zi,1 (t) − zi,2 (t) + E + ui (t) ,

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(a)

Figure 2. Electronic circuit realization of the HR neuron.

(b)

Figure 3. Approximated practical synchronization region (shaded area) for two coupled HR circuits S∗ . The stars indicate the measured boundary points. where the membrane potential yi (t) ∈ R is the output of the ith neuron, zi (t) := col(zi,1 (t), zi,2 (t)) ∈ R2 are the internal states, ui (t) ∈ R is an external input channel, which can be used to communicate with other neurons, and E ∈ R is a constant parameter. We set the constant E = 3.3, for which the Hindmarsh-Rose circuit operates in a chaotic bursting mode. Because the change of variables is linear and invertible, it is possible to show that (5) is strictly C 1 -semipassive and its internal dynamics is exponentially convergent. Figure 2 shows the electronic circuit board that implements the system (5). A detailed description of the circuit diagram and the coupling interface can be found in Neefs et al. [2010]. 5.1 Practical synchronization of Hindmarsh-Rose circuits on Cartesian-product networks Because of the inherit imperfections of the experimental setup, we can not expect that differences between the outputs (and internal states) of the systems converge asymptotically to zero. It is necessary to allow for a mismatch between them, which, of course, needs to be small enough in order to consider that the systems are practically synchronized. Definition 2. (Practical synchronization). Consider k dynamical systems with outputs yi (t) ∈ Rm , i = 1, . . . , k, defined on an interval [t0 , t2 ). The k systems are said to be practically synchronized with bound , if there is t1 (), t0 ≤ t1 () < t2 , such that |yi (t) − yj (t)| <  for all i, j ∈ I and t ∈ [t1 , t2 ). For notational convenience, we select the value of  and refer to practical synchronization with bound  simply as practical synchronization. In the following experiments, we say that the circuits practically synchronize if the eventual difference between the outputs does not exceed  = 0.5[V]. 249

Figure 4. The scaled copies R2 and R6 of the region S∗ (light grey), their intersection R2 ∩ R6 (dark grey), and the (boundary of the) practical synchronization region for the Cartesian-product network (∗). (a) For α = 0.5. (b) For α = 1.0. Although this value of  looks rather large, one has to realize that a small mismatch in the timing of the spikes results in a relatively large synchronization error. Figure 3 depicts the practical synchronization region of two HR circuits coupled through the full delay coupling (3) with a12 = 1. We define the practical synchronization region as the set of all γ, τ ∈ R≥0 for which the coupled circuits practically synchronize with bound  = 0.5 [V]. This practical synchronization region is constructed by, for a fixed value of the time-delay τ , increasing the coupling strength γ from 0 [-] to 5 [-] and recording the values for which the two coupled systems begin to practically synchronize and for which practical synchronization is lost. The resulting boundary points are indicated by stars in Figure 3. The practical synchronization region, denoted by the grey area in Figure 3, is obtained through simple linear interpolation of the measured boundary points (the stars). We denote this region by S∗ . Consider the strongly connected graphs G1 and G2 depicted in Figure 1a witha1 = a2 = a3 = b = 1. We remark that, for G1 , maxi j∈Ni aij = 2, which violates  our assumption maxi j∈Ni aij = 1. This violation is done for practical purposes. Nevertheless, from a theoretical point of view, the factor 2 could be absorbed in the coupling strength γ (by  re-defining γ); hence, violating the assumption of maxi j∈Ni aij = 1 has no consequences for our theoretical results. Next, consider six HR circuits interconnected through the time-delay coupling (4) with γ2 = αγ1 and α = ( 12 , 1) on the Cartesian-product network G1 G2 depicted in Figure 1b. Using the practical synchronization region of two coupled circuits S∗ and Corollary 1, we can predict the values of γ1 and τ for which the six HR coupled circuits practically synchronize. Figure 4 depicts the intersection of the two scaled copies R2 and

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chronization diagram of two full delay coupled HR circuits. These experimental results indicate that the theoretical result presented in Corollary 1, which is derived for networks of noise-free identical systems, can be successfully applied to real-world applications. Our result gives a fairly accurate prediction of the practical synchronization region for full delay coupled HR circuits on Cartesian-product networks. REFERENCES (a)

(b)

Figure 5. Predicted synchronization region (the gray area). The (boundary of the) experimentally obtained practical synchronization region for the Cartesian-product network (∗). (a) For τ = 0.25[ms]. (b) For τ = 0.50[ms]. R6 of S∗ defined in Corollary 1, and the experimentally obtained boundary of the practical synchronization region for the Cartesian-product network (the stars). The region 2 R2 is obtained by scaling S∗ by a factor λ22 = min(3,2α) over the γ-axis, and R6 is a copy of S∗ scaled by the 2 factor λ26 = 2α+3 over the γ-axis. Figure 5 depicts projections of the synchronization region in the parameter space (γ1 , γ2 ) ⊂ R≥0 × R≥0 , for τ = 0.25[ms] and τ = 0.50[ms]. The gray region in Figure 5 is the predicted practical synchronization region obtained using Corollary 1, and the stars are the experimentally obtained points delimiting the boundary of the practical synchronization region. 6. CONCLUSION We have presented a theoretical result on synchronization in Cartesian-product networks of coupled semipassive systems that interact through diffusive time delay couplings of the form (4). The predictive value of this theoretical result in practical situations is tested using an experimental setup built around electronic circuit board realizations of the Hindmarsh-Rose model. To account for the inevitable dissimilarities of the electronic HR circuits, we have introduced the notion of practical synchronization, which states that the circuits may be called synchronized if, after some transient time, the differences between their outputs are sufficiently small on a long finite time interval. In a first experiment, we have determined the practical synchronization region of two diffusively time-delay coupled HR circuits. Next, we have successfully applied the theory presented in Corollary 1 to construct the practical synchronization region in a Cartesian-product network of six full delay coupled HR circuits from the practical syn250

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