Pinning control of chimera states in systems with higher-order interactions
Riccardo Muolo Lucia Valentina Gambuzza Hiroya Nakao Mattia Frasca





corresponding author: muolo.r.aa@m.titech.ac.jp
Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan


 
Department of Electrical, Electronics and Computer Science Engineering, University of Catania, 95125 Catania, Italy


Department of Systems and Control Engineering, Tokyo Institute of Technology, Tokyo 152-8552, Japan
International Research Frontiers Initiative, Tokyo Institute of Technology, Kanagawa 226-8501, Japan


Department of Electrical, Electronics and Computer Science Engineering, University of Catania, 95125 Catania, Italy
Istituto di Analisi dei Sistemi ed Informatica “A. Ruberti”, IASI-CNR, 00185 Roma, Italy





§ ABSTRACT 

Understanding and controlling the mechanisms behind synchronization phenomena is of paramount importance in nonlinear science. In particular, chimera states, patterns in which order and disorder coexist simultaneously, continue to puzzle scholars, due to their elusive nature. Recently, it has been shown that higher-order interactions greatly promote the onset of chimera states, which are easier to found and more resilient when the system units interact in group. In this work, we show that the higher-order framework is fertile not only for the emergence of chimera states, but also for their control. Via pinning control, a technique consisting in applying a forcing to a subset of the nodes, we are able to trigger the emergence of chimera states with only a small fraction of controlled nodes, at striking contrast with the case without higher-order interactions. We show that our setting is robust for different higher-order topologies and types of pinning control and, finally, we give a heuristic interpretation of the results via phase reduction theory. Our numerical and theoretical results provide further understanding on how higher-order interactions shape nonlinear dynamics.  

Keywords Chimera states, pinning control, higher-order interactions, synchronization, hypergraphs




Pinning control of chimera states in systems with higher-order interactions
    Mattia Frasca 
    September 9, 2024
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§ INTRODUCTION


Understanding the mechanisms underlying self-organization phenomena on networks is a paramount task in the study of complex systems, which is complemented by the development of efficient methods to control such dynamics <cit.>. This is particularly relevant in the framework of synchronization dynamics, where, depending on the applications, it is fundamental to achieve a synchronized state, e.g., in power grids <cit.>, or to break it into an asynchronous one, e.g., in neuroscience <cit.>, where synchronization is often associated to pathological states. The network framework remains still relevant in the modeling of complex systems, nonetheless, over the past years scholars have started considering more complex structures such as hypergraphs and simplicial complexes <cit.>. This is because networks do not capture interactions beyond the pairwise setting, i.e., two-by-two, while many systems have shown evidence of higher-order, i.e., group, interactions <cit.>. Examples come from, but are not limited to, neuroscience <cit.>, ecology <cit.> and social behaviors <cit.>. Higher-order interactions have been proven to greatly affect the collective behavior, for instance, in random walks <cit.>, synchronization dynamics <cit.>, contagion <cit.> and pattern formation <cit.>, to name just a few. Given the ubiquity of group interactions <cit.>, it is important to understand how to control the dynamics in such systems. While significant progress has been made in the control of networks <cit.>, the investigation into the control of systems with higher-order interactions has only recently begun <cit.>. 

The focus of this work is an intriguing type of synchronization pattern called chimera state, which consists of the coexistence of coherent and incoherent domains of oscillations. Coexistence of coherence and incoherence was first observed by Kaneko for globally coupled chaotic maps <cit.> and was then found in several numerical settings with global <cit.> and nonlocal <cit.> coupling schemes. Despite all the previous research on the subject, the work that historically is considered to be the first to characterize the emergence of chimera states is the well-known paper by Kuramoto and Battogtokh <cit.>, made popular by a successive work of Abrams and Strogatz, who, with a creative intuition, compared the coexistence of different dynamical state to the chimera, a mythological creature in which parts of different animals coexisted <cit.>. Besides the pure theoretical relevance of such an astonishing phenomenon, a great part of the interest has been generated by the existence of analogous patterns in real systems: for instance, in Josephson junctions <cit.> and electronic circuits <cit.>, laser <cit.>, mechanical <cit.> and nano-electromechanical systems <cit.>, to name a few. Particular attention has been devoted to neuroscience <cit.>, specifically to unihemispheric sleeping patterns in animals <cit.>. Except for some particular configurations in which robust chimera patterns are induced by the network structure <cit.>, in both numerical and experimental settings chimera states are often elusive and characterized by a rather short lifetime. Hence, there is a vast literature on networked systems, consisting in looking for different settings (e.g., parameter ranges, network topologies, coupling configurations, etc.) making such patterns easier to find and with a longer lifetime. Moreover, after the first definition by Kuramoto and Battogtokh <cit.>, several kinds of chimera states have been defined, e.g., amplitude chimeras <cit.> or phase chimeras <cit.>. We will not thoroughly discuss such studies, inviting the interested reader to consult a book <cit.> and a review <cit.> on the subject. In the context of higher-order interactions, chimera states have been proven to be enhanced in some pioneering works considering both pairwise and higher-order interactions <cit.>. This claim was further corroborated in <cit.> for systems with pure higher-order interactions, where the emergence of chimera states on higher-order topologies is compared with the absence of such patterns when the interactions are pairwise. 

In this work, we consider the setting studied in <cit.> and implement a control to further trigger the emergence of chimera states. Our control approach will rely on the so called pinning control, a technique used to drive networks onto a desired dynamical state by using a control input applied to a small subset of nodes <cit.>. Such technique has been successfully used in the framework of opinion dynamics <cit.>, epidemics <cit.>, pattern formation <cit.> and synchronization dynamics <cit.>, to name a few. The latter includes the control of chimera states with pairwise interactions, which we hereby extend to the higher-order framework. Indeed, in <cit.> it was shown that it is possible to trigger the emergence of chimera states via pinning. Nonetheless, to achieve such task, at least half of the network nodes need to be controlled. In what follows, after introducing the setting in Sec. <ref>, we show that higher-order interactions considerably facilitate the work of the controllers and chimera states can be obtained by controlling only a small fraction of the nodes. Such results are shown in Sec. <ref> for two different kinds of pinning approaches, that we named additive pinning and parametric pinning. Moreover, we show that, rather than the number of nodes, what matters is the size of the hyperedges, i.e., the group of nodes interacting with each other. Then, before the discussion of some potential future directions, we give a heuristic interpretation of the results based on the phase reduction theory <cit.> in Sec. <ref>.





§ THE MODEL AND THE SETTING


In this Section, we introduce the system exhibiting chimera states, which is analogous to that studied in <cit.>. We consider coupled Stuart-Landau oscillators, a paradigmatic model for the study of synchronization dynamics, given that it is the normal form of the supercritical Hopf-Andronov bifurcation <cit.>. The coupling takes place through pure higher-order interactions, namely, by mean of a higher-order topology called nonlocal hyperring, which is a generalization of the nonlocal pairwise coupling <cit.>. The type of chimera state that we will hereby consider is that of phase chimeras, states have been first observed by Zajdela and Abrams <cit.>, which consist in oscillation patterns where the amplitude and the frequency of each oscillator are the same, but the phases exhibit a chimera behavior, i.e., a part of the oscillators have the same phase, while the other phases are distributed along the unit circle. The peculiarity of such pattern is that, once obtained, it does not vary, because the frequency is the same for all the oscillators. Hence, we would observe the same exact pattern after each period. For this reason, we find the description given by Zajdela and Abrams, "frozen patterns of disorder", perfectly fitting. The reader can find a thorough characterization of these patterns in the Refs. <cit.>. On a side, let us note that multitailed phase chimeras have only been found in the pairwise setting <cit.>, but not yet in the higher-order one. In what follows, every chimera state discussed and shown in the figures will be a phase chimera. For sake of simplicity, we will refrain from using the word "phase" and will call them simply "chimeras".




 §.§ Stuart-Landau oscillators coupled via nonlocal hyperrings


We consider a system made of N interacting Stuart-Landau units. In the absence of any interaction, each unit i of the system is described by the following equations

    ẋ_i = α  x_i - ω y_i  -(x_i^2  + y_i^2 )x_i=f(x_i,y_i), 
    
    ẏ_i = ω x_i + α y_i - (x_i^2  + y_i^2 )y_i=g(x_i,y_i)  ,

where α is a bifurcation parameter and ω is the frequency of the oscillators. Let us stress that the units are homogeneous, meaning that the parameters α and ω are the same for each and every system. Each isolated system exhibits a stable limit cycle for α>0, which is the case we will consider throughout this study. 



To model the higher-order interactions we use a generalization of the links (or edges) called hyperedges, whose structure can be encoded by using adjacency tensors, a generalization of the adjacency matrices for networks <cit.>. From the literature dealing with simplicial complexes <cit.>, we adopt the convention that a hyperedge of (d+1) nodes (i.e, encoding a (d+1)-body interaction) is called a d-hyperedge. As an example, let us consider the 3-rd order[Let us note that in tensor algebra the order (or rank) of a tensor is given by its indices, e.g., a scalar is a 0-rank tensor, a vector is a 1-rank tensor, a matrix is a 2-rank tensor, etc. Hence, A^(3)={A_ijkl^(3)} would be a 4-rank tensor and the adjacency matrix, given that it is a matrix, a 2-rank tensor. However, here we follow that notation that is most commonly used in the literature on higher-order interactions.] adjacency tensor (i.e., encoding 4-body interactions) A^(3)={A_ijkl^(3)}. We have that A_ijkl^(3)=1 if nodes i,j,k,l are part of the same hyperedge and 0 otherwise. This is the analogous of the adjacency matrix for networks, which is, indeed, a 1-st order adjacency tensor. The chosen higher-order topology is that of nonlocal d-hyperrings, a pure higher-order structure introduced in <cit.> and consisting of hyperedges of size d+1 (d is the order of the interaction) attached through one node and set in a circular structure. In Fig. <ref>, we depict a 3-hyperring, in panel a), together with its pairwise counterpart, namely, the clique-projected network (cpn) obtained by projecting each hyperedge into a clique having the same size, in panel b). Observe that a hyperring is a uniform hypergraph, the hyperedges all having the same size. 

Following the analyses carried out in previous works <cit.>, we assume the coupling to involve only the first state variable of each oscillator, i.e., x. Let us stress that other coupling configurations can be considered, as discussed in Appendix <ref>. With the above assumption, the equations for systems (<ref>) coupled via a generic d-hyperring read


    ẋ_i =f(x_i,y_i) + ε∑_j_1,...,j_d=1^N A_i,j_1,...,j_d^(d)(h^(d)(x_j_1,...,x_j_d) - h^(d)(x_i,...,x_i) ),
    
    ẏ_i =g(x_i,y_i)+ ε∑_j_1,...,j_d=1^N A_i,j_1,...,j_d^(d)(h^(d)(x_j_1,...,x_j_d)- h^(d)(x_i,...,x_i) ),

where ε>0 is the coupling strength[Note that such configuration involves only interactions of order d, i.e., (d+1)-body, hence it is not necessary to denote the coupling strength with ε_d.]. Because we consider identical oscillators, together with the fact that the coupling is diffusive like, i.e., it vanishes when all oscillators are in the same state, system (<ref>) admits a fully synchronous solution. Such a coupling is a special type of non-invasive interaction <cit.>. Moreover, we will consider coupling functions such that the higher-order interaction cannot be decomposed into pairwise ones[It was shown by Neuhäuser et al. that the higher-order coupling functions need to be nonlinear, otherwise the many-body interaction can be decomposed into pairwise ones <cit.>. Successively, it was further shown that such assumption may not be enough and, if the nonlinear functions h are the sum of nonlinear terms which separately account for each unit, e.g., h^(d)(x_j_1,...,x_j_d)=h(x_j_1)+...+h(x_j_d), they can still be decomposed into pairwise ones <cit.>.]. Eq. (<ref>) can be rewritten in compact form for each unit i as 
    Ẋ⃗̇_i=F⃗(X⃗_i)+𝔻∑_j_1,...,j_d=1^N A_i,j_1,...,j_d^(d)(H⃗^(d)(X⃗_j_1,...,X⃗_j_d)-H⃗^(d)(X⃗_i,...,X⃗_i)),
 where X⃗_i=(x_i,y_i)^⊤, F⃗=(f,g)^⊤, H⃗^(d)=(h^(d),h^(d))^⊤ and 𝔻=ε D=ε[ 1 0; 1 0 ]. As stated previously, in Appendix <ref> we report additional results for different coupling matrices D.

To highlight the effects of higher-order interactions, we will perform a comparison between the dynamics on the hyperring and on its respective clique-projected network (cpn), as in <cit.>. The equations for the dynamics with pairwise interactions are 

    ẋ_i =f(x_i,y_i) + ε∑_j=1^N A_i,j^(1)(h^cpn(x_j) - h^cpn(x_i) ),
    
    ẏ_i =g(x_i,y_i)+ ε∑_j=1^N A_i,j^(1)(h^cpn(x_j)- h^cpn(x_i) ),
 where the coupling functions for the dynamics on the clique-projected network h^cpn is determined from its corresponding h^(d) (see below). 

We will perform pinning control on hyperrings of 4 different orders, namely, 3-,4-,5- and 6-hyperrings, involving 4-,5-,6- and 7-body interactions, respectively. In line with other previous works, e.g., <cit.>, we will consider odd-degree polynomials as coupling functions. The coupling functions for each hyperring and its clique-projected network are the following



    h^(3)(x_j_1,x_j_2,x_j_3)=x_j_1x_j_2x_j_3                                   ,   h^cpn(x_j)=x_j^3 , 
    
    
         h^(4)(x_j_1,x_j_2,x_j_3,x_j_4)=x_j_1^2x_j_2x_j_3x_j_4                        ,   h^cpn(x_j)=x_j^5 , 
    
    
         h^(5)(x_j_1,x_j_2,x_j_3,x_j_4,x_j_5)=x_j_1x_j_2x_j_3x_j_4x_j_5              ,   h^cpn(x_j)=x_j^5 , 
    
    
         h^(6)(x_j_1,x_j_2,x_j_3,x_j_4,x_j_5,x_j_6)=x_j_1^2x_j_2x_j_3x_j_4x_j_5x_j_6   ,   h^cpn(x_j)=x_j^7 ,

where the adjacency tensor accounts for all the permutations of the indexes. As an example, let us explicit the equations for the 4-body case (3-hyperring) and its corresponding clique-projected network. The equations for a 3-hyperring are the following 


    ẋ_i =α  x_i - ω y_i  -(x_i^2  + y_i^2 )x_i + ε∑_j_1,j_2,j_3^N A_i,j_1,j_2,j_3^(3)(x_j_1x_j_2x_j_3 - x_i^3),
    
    ẏ_i =ω x_i + α y_i - (x_i^2  + y_i^2 )y_i+ ε∑_j_1,j_2,j_3^N A_i,j_1,j_2,j_3^(3)(x_j_1x_j_2x_j_3 - x_i^3 ),


while those for the clique-projected network are


    ẋ_i =α  x_i - ω y_i  -(x_i^2  + y_i^2 )x_i + ε∑_j^N A_i,j(x_j^3 - x_i^3 ),
    
    ẏ_i =ω x_i + α y_i - (x_i^2  + y_i^2 )y_i+ ε∑_j^N A_i,j(x_j^3 - x_i^3 ).


The equations for interactions of different orders can be constructed analogously by means of the coupling functions (<ref>) (see also the SM of <cit.>).



 §.§ Hyperedge-based local order parameter




To quantify the synchronization of an ensemble of oscillators it is common to use the Kuramoto order parameter <cit.>, which gives a global measure of how much the oscillators are synchronized. However, chimera states involve the coexistence of coherent (i.e., synchronized) and incoherent (i.e., desynchronized) oscillators and the respective regions are localized. Hence, a global measure of the synchronization does not provide useful information on the chimera state. For this reason, scholars have proposed a local Kuramoto order parameter, which measures the synchronization in a given region of the network, by quantifying the differences between neighboring oscillators, as was done, for instance, in <cit.>. In the case of hyperrings, a natural extension of the local order parameter would need to account for the synchronization inside each hyperedge rather than some arbitrary neighborhood. Partially inspired by a work by Shanahan <cit.>, where the order parameter is defined with respect to the communities of a network, we hereby define a hyperedge-based local order parameter as follows: 
    ℛ_i^ℰ(t)=|1/ℰ_i∑_j∈ℰ_i e^ϑ_j(t)| ,

where  is the imaginary unit, ϑ_j is the (time evolving) phase of the j-th oscillator and ℰ_i is the hyperedge(s) node i is part of. By taking as example a 3-hyperring, Fig. <ref>a), we can see that, if node i is a junction node, then it is part of 2 different hyperedges and will have 6 neighboring nodes, while non-junction nodes will have only 3 neighbors. From that, we can observe that non-junction nodes will have the same hyperedge-based local order parameter ℛ_i^ℰ. The number of nodes with the same ℛ_i^ℰ in each hyperedge increases with the order of the hyperring: for instance, in 3-hyperrings they will be 2, while in 6-hyperrings they will be 5. 
 In analogy with the hyperedge-based local order parameter, for the clique-projected network we define a clique-based local order parameter as follows 
    ℛ_i^𝒞(t)=|1/𝒞_i∑_j∈𝒞_i e^ϑ_j(t)| ,
 where 𝒞_i is the clique(s) node i is part of.



 §.§ Pinning control


Chimera states are often elusive patterns, emerging only for limited ranges of the parameters and specific initial conditions <cit.>. Higher-order interactions greatly enhance the possibility of observing such a behavior <cit.>. Nonetheless, ad hoc initial conditions remain a fundamental prerequisite for the chimera to emerge. Our goal is, hence, to induce chimera states in settings where they would not spontaneously appear. For this, we put to use a popular technique in control theory, called pinning control, which consists in externally acting on a subset of the nodes to drive the dynamics of the whole ensemble of nodes towards a desired state <cit.>, and has been successfully applied in the context of chimera states on networks <cit.>.  In our setting, the pinning will act as a perturbation on a subset of the nodes, with the goal of developing a region of incoherence, while leaving the unperturbed nodes in their synchronous state. 




In order to implement the control, we need to determine which nodes to pin and for how much time. Hence, let us define the pinning time t_p, i.e., the time during which the control will be active, and the number of pinned nodes, N_p< N. We will denote all the nodes with the index i=1,...,N and the subset of pinned nodes with i_p=1,...,N_p. Let us observe that, in the context of pairwise interactions, a chimera state is obtained when about half of the nodes are controlled <cit.>. In Sec. <ref>, where we show the numerical results, the reader will appreciate the great advantage offered by the presence of higher-order interactions. The pinning control setting is schematically depicted in Fig. <ref>.



  §.§.§ Control protocol I: additive pinning control


Let us now proceed with setting up the control protocol. The first type of pinning control is called additive, and it was successfully implemented in <cit.> to control chimera states with pairwise interactions. In such a setting, the pinned nodes receive an input in an additive fashion. The equations read


    Ẋ⃗̇_i=F⃗(X⃗_i)+𝔻∑_j_1,...,j_d=1^N A_i,j_1,...,j_d^(d)(H⃗(X⃗_j_1,...,X⃗_j_d)-H⃗^(d)(X⃗_i,...,X⃗_i)) +U⃗_i,
 where U⃗_i=0⃗ for non controlled nodes and, for i=i_p, U⃗_i_p=(u_i_p(t),u_i_p(t))^⊤, with 
    u_i_p(t)= λ_i_p[Θ(t)-Θ(t-t_p)],
 where λ_i_p are parameters drawn from a uniform distribution of a given interval and Θ is the Heaviside step function, whose value is 1 when the argument is positive and 0 when it is null or negative. This way, the control is active as long as t≤ t_p.



  §.§.§ Control protocol II: parametric pinning control


The second type of pinning control we are going to implement consists in acting on the controlled nodes by modifying the parameters of the dynamical system. This protocol, which we will call parametric pinning control, is given by the following equations


    Ẋ⃗̇_i=F⃗_i(X⃗_i)+𝔻∑_j_1,...,j_d=1^N A_i,j_1,...,j_d^(d)(H⃗(X⃗_j_1,...,X⃗_j_d)-H⃗^(d)(X⃗_i,...,X⃗_i)),

where F⃗_i= F⃗ for the nodes that are not pinned, while, for i≡ i_p=1,...,N_p, it is given by F⃗_i= (f_i_p,g_i_p)^⊤, where


    f_i_p( x_i_p, y_i_p) = α  x_i_p(t) - Ω_i_p(t) y_i_p(t)  -(x_i_p^2(t)  + y_i_p^2(t) )x_i_p(t), 
    
    
     g_i_p( x_i_p, y_i_p) = Ω_i_p(t) x_i_p(t) + α y_i_p(t) - (x_i_p^2(t)  + y_i_p^2(t) )y_i_p(t) .

The frequency of the controlled nodes, Ω_i_p(t), is given by 
    Ω_i_p(t)=ωΘ(t-t_p)+ω_i_pΘ(t_p-t),
 
where ω_i_p is new frequency induced by the pinning and Θ is the Heaviside step function, whose value is 1 when the argument is positive and 0 when it is null or negative. This ensures that Ω_i_p=ω_i_p for t≤ t_p and it switches to ω as soon as the control is switched off. In our numerical implementation, the new frequencies ω_i_p will be drawn from a uniform distribution of a given positive interval. As a last remark, let us note that the control acts only on the frequency and not on the amplitude, because we have numerically verified that acting on the amplitude has no effect whatsoever.




§ NUMERICAL RESULTS


In this section we show the numerical results of our pinning approaches. We start by comparing the case of a 3-hyperring, where chimera states occur with both pinning approaches, with its corresponding clique-projected network, where chimera states do not emerge. Then, we will exploit the hyperring structure and develop a pinning strategy that maximizes the width of the incoherence region while the number of controlled nodes remains low. For the latter, we will show the results only for additive pinning, leaving the discussion of the analogous results obtained through parametric pinning in Appendix <ref>.





 §.§ Comparison between higher-order and pairwise interactions


Let us proceed in testing our pinning approaches to control the emergence of chimera states on a 3-hyperring and compare it with the clique-project network case. In Figs. <ref> and <ref>, we show the results for the additive pinning on a hyperring and a clique-projected network, respectively, while, in Figs. <ref> and <ref>, we show the results for the parametric pinning on a hyperring and a clique-projected network, respectively. For both pinning approaches, we see that the control induces a chimera state when the topology is higher-order (Figs. <ref> and <ref>). Moreover, such state persists for long integration times[In the figures, we show the temporal evolution until 1000 time steps, while the chimera persists until about 4000 time steps. After such integration time, the chimera state turns into an incoherent state where variation between adjacent phases is smooth, but ℛ_i^ℰ is low. We have found this state to persist with constant ℛ_i^ℰ until 20000 time steps (the maximum integration time tested).]. These features can be visualized through the hyperedge-induced local order parameter ℛ_i^ℰ, which shows that the nodes which have been controlled are not oscillating coherently with respect to their neighbors sharing the same hyperedge(s) and that such incoherence persists. On the other hand, the pairwise case does not yield chimeras (Figs. <ref> and <ref>). In fact, the initial decoherence induced by the control is quickly reabsorbed by the system and, although a clear trace of the pinning remains, the difference between the phases of adjacent oscillators is small and the variation smooth. This can be visualized through the clique-based order parameter ℛ_i^𝒞. Note that in <cit.> the latter state was distinguished from a chimera one through the total phase variation. Our approach based on a local order parameter is complementary.





Additionally, let us stress that all results hereby shown, regardless of the order of the hyperring, coupling and pinning approach, are due to the higher-order topology and no chimera states are found when performing the simulations with the same setting but on the corresponding clique-projected network, exactly as in the figures shown in this section. Note, though, that there is one particular exception discussed in Appendix <ref>, where the observed pattern is not due to the higher-order topology but due to the coupling configuration, and, in fact, it is found also in the corresponding pairwise system. Such results provide further evidence that higher-order interactions promote the presence of chimera states and are consistent with the existing literature <cit.>.







Let us conclude by pointing out that in previous works chimera states were obtained for specific values of the initial conditions, while random or uniform initial conditions did not yield the same result. In our numerical study, the initial conditions do not matter as long as the system starts on the synchronous solution, i.e., on the limit cycle of the Stuart-Landau oscillator (<ref>). Hence, we will choose the initial conditions to be uniform and without noise, i.e., (x_j^0=1,y_j^0=0) for every oscillator.





 §.§ Scaling of the pinned subset with respect to the hyperring size





It is already remarkable that the higher-order framework favors the onset of chimera states with respect the pairwise one, but here we also show that the percentage of nodes required to obtain a chimera state, by exploiting the interactions of the hyperring, is small. Hence, we set up a pinning protocol in which we try to maximize the number of nodes affected by one single controller. Due to the hyperring structure, a way can be to pin every other junction node, so that each controlled node can, in principle, affect two hyperedges, as schematically shown in Fig. <ref>. To observe how the number of pinned nodes scales with the size of the hyperring, we keep constant the number of hyperedges, so that the total number of nodes increases, but the backbone of the structure remains unchanged. Given that each pinned node is part of 2 hyperedges, in principle, we are able to control all the nodes in these hyperedges. E.g., in a 3-hyperring, with each pinned nodes we would control 7 nodes, in a 4-hyperring 9 nodes, in a 5-hyperring 11 nodes and in a 6-hyperring 13 nodes. For brevity, we present here the results for the additive pinning. The results for the parametric pinning are similar and can be found in Appendix <ref>. 




In Fig. <ref>, we show the results obtained with such control scheme on d-hyperrings with d=3,4,5,6, i.e., 4-, 5-, 6- and 7-body interactions, where we have fixed the number of hyperedges. Indeed, we can observe that we obtain a chimera state by inducing a large region of incoherence (more than half of the nodes) with a control that involves only a small fraction of the nodes. Moreover, the pinned nodes N_p are kept constant for every structure, meaning that N_p does not scale with the number of nodes, but rather with the number of hyperedges, which allows to control large structures with only a handful of nodes. The pinned nodes are ≃ 8.8% of the total nodes for the 3-hyperring (panels a) and b)), ≃ 6.6% of the total nodes for the 4-hyperring (panels c) and d)), ≃ 5.3% of the total nodes in the 5-hyperring (panels e) and f)) and ≃ 4.4% of the total nodes in the 6-hyperring (panels g) and h)). In the latter case, our pinning scheme does not work as well as for the other structures and the coupling strength need to be significantly increased in order to observe a chimera state. However, we can observe from Fig. <ref>h) that the chimera is not anymore stable and the front of incoherence enlarges. Let us stress that stable chimera states can be easily obtained also in 6-hyperrings by reducing the distance between the controlled nodes, as in the previous section, instead of the pinning scheme of Fig. <ref>.

Let us note that the parameters λ_i_p need to be all positive or all negative in order for this pinning scheme to yield persistent chimera states. When such random inputs are drawn in a symmetric interval with respect to the 0, e.g., [-2,2] as in Fig. <ref>, a chimera states forms but vanishes at about 400 time units. On the other hand, when the control inputs have the same sign, e.g., [0,2] as in Fig. <ref>, the chimera is persistent until about 4000 time units, i.e., 10 times longer.

In Appendix <ref>, we show analogous results for the parametric pinning, except for the case 6-hyperring, where chimera states do not emerge by pinning every other junction node, even with stronger couplings. The fact that our results are robust with respect to different control approaches is a good indication of the pervasiveness of the phenomenon. 

Let us conclude the Results Section by pointing out that the chimera behavior can be further enhanced by increasing the number of pinned nodes and/or reducing the distance between them. Moreover, by increasing the magnitude of the parameters λ_i_p, we can also obtain chimera states through controlling even less nodes than in the simulations hereby shown. However, we have presented a setting in which the parameters λ_i_p have a magnitude comparable with the involved parameters, in order to make it suitable for applications.   






§ HEURISTIC INTERPRETATION THROUGH PHASE REDUCTION THEORY


In this section we will give a heuristic interpretation of the results based on the phase reduction approach <cit.>, which consists of reducing a given oscillatory system to a phase, i.e., Kuramoto-type, model <cit.>. In a nutshell, starting from a system of N highly dimensional units in a limit cycle regime, e.g., 
    Ẋ⃗̇_i=F⃗_i(X⃗_i)+ε∑_j=1^N A_ijG⃗_ij(X⃗_j,X⃗_i),
 under given assumptions (see <cit.> for details), we can reduce to a system of N interacting phase oscillators of the form 
    ϑ̇_i=ω_i+ε∑_j=1^N A_i,jZ⃗(ϑ_i)·G⃗_ij(ϑ_j,ϑ_i),
 where ω_i is the frequency of the i-th oscillator and Z⃗ is the phase sensitivity function. The key in the reduction process is to find an expression for Z⃗, which has an analytical expression only for some specific cases, among which the Stuart-Landau model <cit.> and weakly nonlinear oscillators <cit.>, while it needs to be obtained numerically for general oscillators. 


The phase reduction theory has been applied also to systems with higher-order interactions <cit.>, obtaining higher-order versions of the Kuramoto model, which exhibit much richer behaviors than the pairwise one. Indeed, the first evidence of higher-order-induced exotic behaviors, which triggered the interest of the community towards this new framework, has come from higher-order phase models (although not derived through phase reduction) <cit.>. In what follows, we will apply the phase reduction to our model on a 3-hyperring and on the corresponding clique-projected network, Eqs. (<ref>) and (<ref>) respectively, to give an intuition of why it is easier to induce chimera behavior via pinning when the topology is higher-order. 

For the Stuart-Landau model, the phase sensitivity function is Z⃗(ϑ)=(-sin(ϑ),cos(ϑ))^⊤ <cit.>. Let us consider system (<ref>) in polar coordinates, i.e., X⃗_i=(cos(ϑ_i),sin(ϑ_i)), and proceed with the reduction by computing the following 
    Z⃗(ϑ_i)·Ẋ⃗̇_i    =   sin^2(ϑ_i)ϑ̇_̇i̇+cos^2(ϑ_i)ϑ̇_i= 
          -αcos(ϑ_i)sin(ϑ_i)+ωsin^2(ϑ_i)+cos(ϑ_i)sin(ϑ_i)+ωcos^2(ϑ_i)+αcos(ϑ_i)sin(ϑ_i) -cos(ϑ_i)sin(ϑ_i)+
          ε∑_j_1,j_2,j_3^N A_i,j_1,j_2,j_3^(3)( cos^3(ϑ_i)sin(ϑ_i)-cos(ϑ_j_1)cos(ϑ_j_2)cos(ϑ_j_3)sin(ϑ_i)+cos(ϑ_j_1)cos(ϑ_j_2)cos(ϑ_j_3)cos(ϑ_i)-cos^4(ϑ_i) ),
 which gives 
    ϑ̇_i=ω+ε∑_j_1,j_2,j_3^N A_i,j_1,j_2,j_3^(3)Φ( ϑ_i, ϑ_j_1, ϑ_j_2, ϑ_j_3 ),
 where 
    Φ( ϑ_i, ϑ_j_1, ϑ_j_2, ϑ_j_3 )=  cos^3(ϑ_i)sin(ϑ_i)-cos(ϑ_j_1)cos(ϑ_j_2)cos(ϑ_j_3)sin(ϑ_i)+cos(ϑ_j_1)cos(ϑ_j_2)cos(ϑ_j_3)cos(ϑ_i)-cos^4(ϑ_i).

Observe that ω is the same for all the oscillators because we started from identical Stuart-Landau systems. If we apply the same procedure to the system on the clique-projected network, i.e., Eq. (<ref>), we obtain 


    ϑ̇_i=ω+ε∑_j^N A_i,jΨ( ϑ_i, ϑ_j ),

where 
    Ψ( ϑ_i, ϑ_j ) =(cos(ϑ_i)-sin(ϑ_i))(cos^3(ϑ_j)-cos^3(ϑ_i)).
 

Through the averaging method <cit.>, the 4-body coupling (<ref>) can be approximated as 
    Φ( ϑ_i, ϑ_j_1, ϑ_j_2, ϑ_j_3 ) ≃3/8( sin(ϑ_j_1+ϑ_j_2+ϑ_j_3-3ϑ_i) + cos(ϑ_j_1+ϑ_j_2+ϑ_j_3-3ϑ_i)  -1),
 while the pairwise coupling as (<ref>) 

    Ψ( ϑ_i, ϑ_j )  ≃3/8( sin(ϑ_j-ϑ_i)+cos(ϑ_j-ϑ_i)-1 )= √(2)3/8sin(ϑ_j-ϑ_i+π/4)-3/8.

 
 The coupling given by Eq. (<ref>) steers the system towards synchronization, because Ψ( ϑ_i, ϑ_j ) has form of the well-known Kuramoto-Sakaguchi coupling, i.e., sin(ϑ_j-ϑ_i+α), which is known to be attractive for |α|<π/2 <cit.>. On the other hand, Eq. (<ref>) allows for a much richer dynamics, given that there are many more combinations of the phases and the coefficients such that the coupling term vanishes, as it is the case of the higher-order Kuramoto model <cit.>. This fact, gives an intuition not only of the much richer dynamics observed when higher-order interactions are present <cit.>, but also of why higher-order systems favor the presence of chimera states and it is much easier to induce such behavior via pinning, compared to the pairwise case. Moreover, given the form of the higher-order coupling terms, once such state is achieved, it is more difficult for the higher-order system to steer towards synchronization, which provides a qualitative explanation of why the chimera states are also persistent. Let us remark that the first intuition of this behavior was given by Zhang et al. in <cit.>, where it was shown, for the higher-order Kuramoto model, that, when the systems leaves the attraction basin of the synchronous state, it is more difficult to synchronize again because higher-order interactions cause a shrinking of such attraction basin, which becomes smaller. In our case, the control pushes the system away from the synchronous solution creating a chimera state and the higher-order interactions favor the persistence of this state. 

 




As a further corroboration of the results shown in this work and of the correctness of our phase reduction approach, let us conclude this section by applying our pinning protocols to the reduced phase models and show that the outcome confirms our claims. We will show the results of additive pinning (<ref>), leaving those for the parametric approach (<ref>) in Appendix <ref>.
In Figs. <ref> and <ref>, we show the result of the additive pinning procedure on a 3-hyperring and its corresponding clique-projected network, respectively. The setting is analogous to that of Figs. <ref> and <ref>, and so are the results: while the control induces a chimera state in the higher-order phase reduced model, no chimera emerges in the pairwise setting. The same results are obtained also when performing parametric pinning control (see Appendix <ref>).





§ DISCUSSION


In this work we have shown how pinning control can be applied to higher-order system to trigger the emergence of chimera states and how higher-order interactions are a key feature for the chimera state to develop and persist. It was already known from previous works that higher-order interactions enhanced chimera states, however, the set of parameters, initial conditions and couplings allowing for such behavior remained limited. Thanks to our pinning schemes, that we called additive pinning and parametric pinning, we were able to overcome such limitation and observe chimera patterns for a wide range of settings. Moreover, and this is the most remarkable result, the higher-order framework makes possible to control the presence of chimeras by only acting on a small fraction of the nodes, at striking contrast with the network case, where half of the nodes are to be controlled to achieve this objective. Lastly, our heuristic interpretation of the results goes beyond this work and provides a possible explanation of other previous results regarding synchronization patterns observed in higher-order systems, in particular the claim made by Zhang et al. that higher-order interactions shrink the attraction basin of the synchronized state <cit.>.

Our results clearly show that it is easier and more efficient to trigger the emergence of chimera states when higher-order interactions are present. A further study could be to determine how much energy is needed to control the chimera state in comparison with the pairwise setting, by relying on energy aware controllability measures <cit.>. Another efficient control strategy could be to apply an intermittent pinning, analogously to the occasional coupling setting developed in the framework of amplitude death <cit.>, where techniques from piecewise-smooth systems could be used <cit.>. Another interesting direction would be to apply pinning control to directed higher-order structures, such as directed <cit.> and m-directed hypergraphs <cit.>, which have been proven to greatly affect nonlinear dynamics in the context of synchronization <cit.> and Turing pattern formation <cit.>, respectively. Pinning approaches in systems with directed higher-order interactions have been developed in some pioneering works <cit.>, but not yet in the context of chimera states. Lastly, the recent implementation of higher-order interactions in electric circuits <cit.> opens the way to further applications in this direction. 

In conclusion, this work is one of the first in which tools from control theory are applied to systems with higher-order interactions and it shows the numerous possibilities offered by this novel framework. We believe that there is plenty of exciting research to be done in this direction and that the ground we built with this work is the basis for further studies shedding more light on the interplay between dynamics with higher-order interactions and its control.





  §.§.§ Acknowledgements
 R.M. and H.N. acknowledge JSPS, Japan KAKENHI JP22K11919, JP22H00516, and JST, Japan CREST JP-MJCR1913 for financial support. The contribution of L.V.G. and M.F. to this work is framed into the activities of the project "CoCoS: Control of Complex Systems", funded by the University of Catania, under the PIA.CE.RI. 2024-26 initiative. We are grateful to Timoteo Carletti for useful discussions, including pointing us to Ref. <cit.>. R.M. is particularly grateful to Giacomo Baggio and Francesco Lo Iudice for the discussions on pinning control and to Iván León and Yuzuru Kato for the discussions on the phase reduction.











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§ NUMERICAL RESULTS FOR DIFFERENT COUPLING SCHEMES






In the Main Text, we have considered systems of the form of Eq. (<ref>), where the coupling matrix is D=[ 1 0; 1 0 ]. Let us observe that such setting is the one in which it is easier to observe chimera states induced by the initial conditions. However, through our pinning approach, it is possible to observe chimera states also for different configurations of the coupling.  In what follows, we give a brief survey of which configurations yield chimera states, obtained by performing parametric pinning control on a 3-hyperring of 204 nodes, where the parameters are α=1 and ω=1, the coupling strength is ε=0.01, and by pinning N_p=18 nodes spaced every 3 for t_p=100 time units and with the parameters ω_i_p drawn from a uniform distribution of the interval [0.5,2.5]. We have performed 10 simulations for each identical setting except for the parameters ω_i_p, which changed at every iteration. 

The following coupling configurations always lead to chimera states for the examined range of parameters, couplings and pinning features: D = [ 0 0; 0 1 ]  ,   [ 0 1; 0 1 ]  ,   [ 0 1; 1 0 ]  ,   [ 1 1; 0 0 ]  ,   [ 0 0; 1 1 ]  ,   [ 1 1; 0 1 ]. 

The following configurations lead 40%-60% of times to chimeras, depending on the parameters ω_i_p: 
 D = [ 1 0; 0 0 ]  ,   [ 1 0; 0 1 ]  ,   [ 1 1; 1 0 ]  ,   [ 0 1; 1 1 ]. Despite not being as easy as in the former case, chimera states in the latter can be achieved by increasing the coupling strength, the intensity of the parameters ω_i_p, the pinned nodes and the pervasiveness of the pinning (e.g., pinning every node instead of 1 every 3). 

The following coupling configurations never lead to chimera states for the examined range of parameters, couplings and pinning features: D = [ 1 0; 1 1 ]  ,   [ 1 1; 1 1 ]. 

Particularly interesting are the following coupling configurations: D = [ 0 1; 0 0 ]  ,   [ 0 0; 1 0 ], where a chimera state emerges, but it is unstable and the incoherence region grows until the whole system develops a fully incoherent state. Moreover, such behavior is independent on the number of pinned nodes and occurs also when only one node is perturbed, which makes this setting interesting for applications in which incoherence needs to be achieved. Let us note that, in this case, higher-order interactions do not play a role, but the key feature is the coupling configuration. In fact, we obtain the same result in the pairwise setting. 

Lastly, let us point out that also different hyperring topologies and the additive pinning configuration lead to analogous behaviors and that no chimera nor incoherent states are observed when the higher-order interactions are "flattened" onto the corresponding clique-projected networks with none of the two pinning approaches. Again, let us stress that all the above does not apply to the coupling configurations leading to full incoherence, namely, [ 0 1; 0 0 ]  ,   [ 0 0; 1 0 ], whose behavior is determined by the coupling and not by the presence of higher-order interactions. In fact, the same behavior is observed also for pairwise interactions. 






§ SCALING WITH RESPECT TO THE NUMBER OF NODES FOR THE PARAMETRIC PINNING CONTROL APPROACH






In this Appendix, we complement the Main Text by showing the results obtained for the case of parametric pinning, which are qualitatively analogous to those of Sec. <ref> obtained through the additive pinning approach, i.e., those on the scaling of the fraction of pinned nodes with respect to the hyperring size. In this setting, we are able to keep the number of pinned nodes constant as we increase the size of the structure. Again, let us stress that the total number of nodes in the hyperring increases, but the number of hyperedges is kept constant. 

In Fig. <ref>, we show the results obtained with such control scheme on d-hyperrings with d=3,4,5,6, i.e., 4-, 5-, 6- and 7-body interactions, where we have fixed the number of hyperedges. Indeed, we can observe that we obtain a chimera state by inducing a large region of incoherence (more than half of the nodes) with a control that involves only a small fraction of the nodes. Moreover, the pinned nodes N_p are kept constant for every structure, meaning that N_p does not scale with the number of nodes, but rather with the number of hyperedges, which allows to control large structures with only a handful of nodes. The parameters ω_i_p are the same for all the simulations. The pinned nodes are ≃ 8.8% of the total nodes for the 3-hyperring (panels a) and b)), ≃ 6.6% of the total nodes for the 4-hyperring (panels c) and d)), ≃ 5.3% of the total nodes in the 5-hyperring (panels e) and f)) and ≃ 4.4% of the total nodes in the 6-hyperring (panels g) and h)). In the latter case, the pinning scheme consisting in controlling one every 2 junction nodes does not yield a chimera state, not even with stronger coupling, as shown in Fig. <ref>g-h), where, moreover, we see that the region of incoherence enlarges. This does not change if we increase the number of pinned nodes, but only if we reduce the gap between them.








§ PARAMETRIC PINNING CONTROL OF THE REDUCED PHASE MODEL






In this Appendix, we complement the numerical results of Sec. <ref> of the Main Text on additive pinning control for the phase reduced model by showing analogous results for the parametric pinning approach. In Figs. <ref> and <ref>, we show the results of the parametric pinning procedure on a 3-hyperring and its corresponding clique-projected network, respectively. The setting is analogous to that of Figs. <ref> and <ref> of the Main Text. We see that, while the control induces a chimera state in the higher-order phase reduced model, only a slight incoherence emerges in the pairwise setting, which is not a chimera, as can be seen by looking at the clique-based local order parameter.








