Coupled Hypergraph Maps and Chaotic Cluster Synchronization
aa r X i v : . [ n li n . C D ] F e b Coupled Hypergraph Maps and Chaotic Cluster Synchronization
Tobias B¨ohle a , Christian Kuehn a,b , Raffaella Mulas c,d,e , and J¨urgen Jost c,f a Faculty of Mathematics, Technical University of Munich, Boltzmannstr. 3, 85748 Garching b. M¨unchen, Germany b Complexity Science Hub Vienna, Josefst¨adter Str. 39, 1080 Vienna, Austria c Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany d The Alan Turing Institute, The British Library, London NW1 2DB, UK e University of Southampton, University Rd, Southampton SO17 1BJ, UK f Santa Fe Institute for the Sciences of Complexity, 1399 Hyde Park Road Santa Fe, New Mexico 87501, USA
Coupled map lattices (CMLs) are prototypical dynamical systems on networks/graphs. They exhibitcomplex patterns generated via the interplay of diffusive/Laplacian coupling and nonlinear reactionsmodelled by a single iterated map at each node; the maps are often taken as unimodal, e.g., logisticor tent maps. In this letter, we propose a class of higher-order coupled dynamical systems involvingthe hypergraph Laplacian, which we call hypergraph coupled maps. By combining linearized (in-)stability analysis of synchronized states, hypergraph spectral theory, and numerical methods, wedetect robust regions of chaotic cluster synchronization occurring in parameter space upon varyingcoupling strength and the main bifurcation parameter of the unimodal map. Furthermore, wefind key differences between Laplacian and hypergraph Laplacian coupling and detect various otherclasses of periodic and quasi-periodic patterns. The results show the high complexitiy of coupledgraph maps and indicate that they might be an excellent universal model class to understand thesimilarities and differences between dynamics on classical graphs and dynamics on hypergraphs.
To study dynamical systems induced by iterating cou-pled maps on networks is an established paradigm innonlinear dynamics originally termed coupled map lat-tices (CMLs) [1]. Each node/vertex i ∈ { , , . . . , d } of aconnected network/graph G evolves according to a time-discrete map f . Typical examples are the logistic map f ( x ) = µx (1 − x ) or the tent map f ( x ) = µ min { x, − x } each with parameter µ ∈ [0 , x n ( i ) ∈ R at node i at time n ∈ N is defined via x n +1 ( i ) = f ( x n ( i )) − ǫ (∆ G f )( x n ( i )) , (1)where ǫ ∈ R is a parameter controlling the diffusive cou-pling, and the normalized Laplacian ∆ G is defined as(∆ G u )( x ( i )) := u ( x ( i )) − i X j ∼ i u ( x ( j )) , (2)where i ∼ j when i, j are connected by a link/edge, andwe call them neighbors in that case, and deg i is thenumber of neighbors of i . Classically, one has consid-ered lattices G = L ( d , d ) ( d , ∈ N ) with d d nodes,or complete graphs G = K d ( d ∈ N ) on d nodes; bothclasses already display very surprising phenomena. Sub-sequently, triggered by the rise of network science, it wasdiscovered that new effects may arise in CMLs when thegraph is not complete or a lattice [2–4]. A commonlyencountered theme in all classes of CMLs is synchroniza-tion [5]. Dynamics is called synchronized if x n ( i ) = x n ( j )for all i, j and all times n ≥ n for some n ∈ N . Impor-tantly, synchronized dynamics need not be constant in n , but could, for instance, show itself chaotic behavior.In such a case, one speaks of the (complete) synchro-nization of chaos [6]; we refer to [7] for the discovery ofthe general effect of chaotic synchronization in coupled oscillators. An important object is the (complete) syn-chronization manifold M := { x (1) = x (2) = · · · = x ( d ) } ,and one is interested in the transverse stability of M .Consider an orbit ¯ γ = { ¯ x n } ∞ n =1 of the given map f . Ifthe CML is uncoupled ( ǫ = 0), then the homogeneoussolution γ = { x n ( j ) = ¯ x n } for all j is synchronized andremains in M . Then one may linearize around γ , de-rive a variational equation, and link stability for ǫ > f and the eigenvalues of∆ G . As shown in [2], (complete) synchronization on M is transversally linearly stable for (1) if | e µ (1 − ǫλ k ) | < ∀ k ∈ { , . . . , d } (3)where µ = lim N →∞ N N − X n =0 log | f ′ (¯ x n ) | is the Lyapunov exponent of f and λ k are the nonzeroeigenvalues of the Laplacian (2), which can be ordered as0 = λ < λ ≤ · · · ≤ λ d . The eigenvalue 0 is simple because we assumed that G is connected. Obviously, it suffices to check (3) for theeigenvalues λ and λ d . Thus, in favorable cases, thereis a certain range of values of the coupling parameter ǫ for which (3) is satisfied. Note that the eigenfunctionsfor the simple eigenvalue λ = 0 are the constants corre-sponding to the tangential direction of the synchroniza-tion manifold. If we assume temporal instability with µ >
0, then (3) is not satisfied for λ = 0. Thus, when(3) is satisfied for all other eigenvalues, the constantsare the only unstable directions at a synchronized state,and this precisely means that M is linearly transversallystable. The concept of chaotic synchronization can becombined with the idea of cluster synchronization , i.e.,different subsets/clusters of nodes synchronizing amongthemselves, but not across clusters [8, 9].Yet, graph coupling is often insufficient in many ap-plications, which require higher-order interactions. Forexample, higher-order coupling is of crucial importancein ecology [10, 11], neuroscience [12], cell biology [13],epidemiology [14], or opinion formation [15]; see also [16]for a recent survey of the area, which displays a high levelof recent activity [17–21] in the context of dynamics. Inthis work, we are interested in the effect higher-order cou-pling between nodes can have on CMLs with a focus onchaotic dynamics. So far, CMLs have been considered ongraphs and we propose a very general extension to chem-ical hypergraphs, which includes classical hypergraphs asa special case.A hypergraph consists of vertices connected by hyper-edges, which can couple more than two vertices. Forchemical hypergraphs [22], we also additionally label ver-tices of each hyperedge as inputs and outputs; note thatthese classes need not be disjoint. The (chemical) hyper-graph Laplace operator is then given by∆ H u ( x ( i )) := P h in: i input (cid:18) P i ′ input of h in u ( x ( i ′ )) − P j ′ output of h in u ( x ( j ′ )) (cid:19) hypdeg i + − P h out: i output (cid:18) P ˆ i input of h out u ( x (ˆ i )) − P ˆ j output of h out u ( x (ˆ j )) (cid:19) hypdeg i with hypdeg i := X h,i ∈ h ( | h | − | h | is the number of vertices contained in the hyper-edge h . The definition ∆ H is modified from [22], in orderto ensure an appropriate normalization for our dynamics.The hypergraph Laplacian ∆ H again has real spectrum.Analogously to (1), we want to couple the dynamics on ahypergraph via ∆ H for a given map f : [0 , → [0 ,
1] ateach node. The hypergraph Laplacian may fail to satisfythe maximum principle; this is the case when it has non-constant eigenfunctions for the eigenvalue 0. Therefore,we define the function σ ( x ) := ( x − n if x ∈ [2 n, n + 1]2( n + 1) − x if x ∈ [2 n + 1 , n + 2]for n ∈ Z and put x n +1 ( i ) = σ ( f ( x n ( i )) − ǫ (∆ H f )( x n ( i ))) , (4)which makes the unit interval invariant under the dynam-ics. The spectral properties of ∆ H are richer than thoseof ∆ G [22]. In particular, ∆ H can possess the eigenvalue 0 with multiplicity >
1, and none of the eigenfunctionsneed to be constants. An example is the hyperflower H c,t,ℓ defined via three parameters c , t and ℓ [23]. It isa generalization of the star graph. There is a set of c central vertices and ℓ sets each consisting of t peripheralvertices . Each set containing peripheral vertices is calleda leaf . Central vertices are contained in all hyperedges,but each hyperedge additionally includes only peripheralnodes from one leaf, so in total there are ℓ hyperedges.By convention we classify central vertices as inputs andperipheral nodes as outputs. An example for the dy-namics of (4) in Figure 1 shows complex chaotic clustersynchronization. x n ( i ) center 1st leaf 2nd leaf 3rd leafNodes ( i )FIG. 1. Numerical Integration of (4) for f ( x ) = µx (1 − x ) ona hyperflower with c = t = 7 and ℓ = 3 for µ = 1 . ǫ = 8.Plotted iterations are 5000 < n ≤ x n ( i )are alternately plotted in red, cyan, green and purple uponincreasing n . To understand synchronization patterns, the resultsfor CMLs on graphs motivate us to consider the eigen-value/eigenfunction structure of hyperflowers H c,t,ℓ . Thefunction which equals − c + t ) / ( c + t − − t/ ( c + t − ℓ − /c on centralnodes and 1 /t on peripheral nodes. Furthermore, everyfunction that is +1 on one node, − c − ℓ ( t −
1) such linearlyindependent functions. Altogether, we have generated c + tℓ linearly independent eigenfunctions, which is therequired number.We start by analyzing linear stability of the synchro-nized solution on H c,t,ℓ , which follows a similar patternas for graphs as we just have to replace ∆ G by ∆ H . Firstnote, that for a synchronized solution to exist, we needto require c = t . Then, a necessary condition to retain atleast partial synchronization ( x ( i ) = x ( j ) for some i = j )is stability in the direction of eigenfunctions, which are+1 on one vertex − µ < . (5)This is in clear contrast to the assumption µ > µ >
0, was necessary to have non-stationarydynamics of a synchronized solution. Given the condition µ < G , the hypergraph Laplace on the hy-perflower has further eigenfunctions, which are constanton certain components of the hyperflower. By requir-ing instability of the synchronized solution with respectto perturbations in direction of these eigenfunctions, wemay still hope to retain non-stationary dynamics of par-tially synchronized solutions. In other words, the eigen-functions that are constant on each of the componentsand thus corresponding to positive eigenvalues are takingover the job of the constant eigenfunction correspondingto the eigenvalue 0 on graphs. Instability in direction ofthe positive eigenvalue ˜ λ = ( c + t ) / ( c + t − λ = t/ ( c + t − | e µ (1 − ǫ ˜ λ ) | > , (6) | e µ (1 − ǫ ˆ λ ) | > . (7)Even though one actually needs to find additional stabil-ity conditions around a partially synchronized solution,our numerical simulations reveal that the instability con-ditions around the completely synchronized solution doalready provide great insight about the existence of non-stationary partially synchronized solutions. Especially, if f is given by the tent map f ( x ) = µ (cid:18) − (cid:12)(cid:12)(cid:12)(cid:12) x − (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) (8)this makes sense, as f is piecewise linear and thus stabil-ity conditions derived from a linearization of f ( x ) are tosome extent independent of the particular state x . Forthe tent, the Lyapunov coefficient can explicitly be givenby µ = ln( µ/ µ, ǫ ) fulfill the stability conditions (5), (6)and (7). In particular, we marked areas in which theconditions are fulfilled by diagonal lines seen in Figure 2.Further, a numerical integration of the system (4), start-ing from a slight perturbation of a completely synchro-nized state, yields areas in which one has non-stationary µǫ Stability Region for the Tent MapFIG. 2. The diagonal lines represent areas in which (5), (6)and (7) are satisfied for the tent map. The green region de-picts ( µ, ǫ ) values for which numerical simulations revealednon-stationary partial synchronization with different dynam-ics on each components. x n ( i ) center 1st leaf 2nd leaf 3rd leafNodes ( i )FIG. 3. Numerical Integration of (4) for f ( x ) given by thetent map (8) on a hyperflower with c = t = 7 and ℓ = 3 for µ = 1 . ǫ = 3. Plotted iterations are 5000 < n ≤ x n ( i ) are alternately plotted in red, cyan, greenand purple upon increasing n . partial synchronization with different dynamics in each ofthe components of the underlying hyperflower (see greenregions in Figure 2. As seen in Figure 1, a closer lookat the dynamics for parameter values in the green re-gion shows chaotic dynamics on each component of thehyperflower. We observe several interesting phenomena.First, the results suggest that (5) is sufficient for par-tial synchronization. Second, cluster synchronization ofchaos only appears when the conditions (5), (6) and (7)are satisfied and third, chaotic dynamics can appear forvalues of µ < µǫ FIG. 4. Numerical simulations of (4) for f ( x ) = µx (1 − x )on a hyperflower with c = 10, ℓ = 5 and t = 3 revealedcluster synchronization of chaos in yellow regions. Doublysynchronized chaos occurs for all parameter values ( µ, ǫ ) inthe yellow regions. By neglecting the requirement of stability condition(7), i.e. allowing perturbations that are − µ, ǫ ) = (1 . ,
3) satisfies (5) and (6) butnot (7). The resulting dynamics can be seen in Figure 3.Even though our analytical derivations of stability con-ditions require assumptions about the hyperflower, nu-merical simulations can of course be performed for thecases not covered by our analytical derivations. Specif-ically, we consider simulations on a hyperflower with c = 10, ℓ = 5 and t = 3. For a given parameter pair( µ, ǫ ), we numerically infer synchronization of the centralnodes if the standard deviation over i = 1 , . . . , c of x n ( i )drops below a certain threshold ( ≈ − ) as n → ∞ .Similarly, we infer chaos in the center of the hyperflowerif the leading Lyapunov coefficient is positive on the cen-tral nodes. In the same way we deduce synchronizationand chaotic behavior of nodes in the first leaf of the hy-perflower. Based on those four criteria this allows usto classify the dynamical behavior for given parametervalues and initial conditions. In particular, we say thatthe dynamics shows doubly synchronized chaos if both ofthe leading Lyapunov coefficients for the two clusters arepositive and the values of x n synchronize within the twoclusters (but not necessarily across the clusters). Now,we conduct numerical simulations for different parame-ter values of µ and ǫ but with the same initial conditionfor each simulation and investigate for each parameterpair ( µ, ǫ ) the occurrence of doubly synchronized chaos.The yellow regions in Figure 4 depict such areas, whereasthere is no doubly synchronized chaos in the blue region. x n ( i ) Nodes ( i )FIG. 5. Numerical Integration of (4) for f ( x ) = µx (1 − x )with µ = 2 .
868 and ǫ = 6 .
04 on a cyclic hypergrpah with e = 10, ℓ = 6, m = 1 and s = 2. Plotted iterations are5000 < n ≤ x n ( i ) are alternately plottedin red, cyan, green and purple upon increasing n . On hyperflowers, we have detected a variety of otherpatterns, including steady and periodic synchronizationpatters, as well as chaotic cluster patterns, where a sin-gle cluster chaotically forces clusters. Yet, above we haveonly shown the most complex interaction, where clustersare chaotically synchronized yet not correlated. Further-more, we have considered less symmetric hypergraphs,e.g., the cyclic hypergraphs Z e,ℓ,m,s , which is a class de-fined by four parameters e, ℓ, m, s . One can view Z e,ℓ,m,s as a set of es nodes, which are arranged in a circle. Thereare e edges each encompassing ℓ neighbors. These edgesare distributed uniformly around the circle such that ifone edge starts at a node i on the circle, the next edgestarts at node that is s nodes away from i . If one goesaround the circle, the first m nodes of each edge are spec-ified as input nodes, whereas the remaining ones are out-put nodes. While for some parameters, this class of hy-pergraphs has symmetries under permutation of nodes,it does not for others. If we consider, for example, thecyclic hypergraph with e = 10 edges, ℓ = 6, m = 1and s = 2, there is no symmetric subgroup that leavesthe hypergraph Laplace operator ∆ H invariant. Permut-ing two nodes would either cause edges to be spannedover non-neighboring nodes or edges not to start withnodes specified as input, both contradicting with a pos-sible invariance of the hypergraph Laplacian. However, anumerical simulation starting from a completely synchro-nized initial condition with small perturbation, see Fig-ure 5, shows that both even and odd nodes form a clus-ter within which the dynamics synchronizes and showschaotic behavior but there is no synchronization acrossthe two clusters. ACKNOWLEDGMENTS
TB thanks the TUM Institute for Advanced Study(TUM-IAS) for support through a Hans Fischer Fellow-ship awarded to Chris Bick. TB also acknowledges sup-port of the TUM TopMath elite study program. CKwas supported a Lichtenberg Professorship of the Volk-swagenStiftung. RM was supported by The Alan TuringInstitute under the EPSRC grant EP/N510129/1. [1] K. Kaneko.
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