Spatio-temporal patterns of non-autonomous systems on hypergraphs: Turing and Benjamin–Feir mechanisms

This paper examines the Turing patterns and the spatio-temporal chaos of non-autonomous systems defined on hypergraphs. The analytical conditions for Turing instability and Benjamin–Feir instability are obtained by linear stability analysis using new comparison principles. The comparison with pairwise interactions is presented to reveal the effect of higher-order interactions on pattern formation. In addition, numerical simulations due to different non-autonomous mechanisms, such as time-varying diffusion coefficients, time-varying reaction kinetics and time-varying diffusion coupling are provided respectively, which verifies the efficiency of theoretical results.


Introduction
Over the years, network science has established as a backbone to investigate complex dynamical systems in diverse areas including physics [1], sociology [2], biology [3], the neurosciences [4] and the climate sciences [5]. The natural and artificial systems can be described in terms of networks, with nodes and links representing individual elements and interactions between elements respectively. Such a high-level abstraction provides a compact view of the overall topology and further helps to understanding the behaviors of complex systems [6][7][8]. And the methodologies of network science are spread across many disciplines, including graph theory [9], statistical physics [10] and statistics [11], which amplifies the potential and applicability of network theory.
In the majority of related studies, mathematical representations of networks capture only pairwise interactions. But growing evidence indicates that the function of real-world systems is not limited to dyadic interactions, but rather involves the actions of larger groups [12,13]. Examples include the contagion networks [14], the network of neurons [15], the social networks [16] and the protein networks [17]. Therefore, higher-order representations such as hypergraphs and simplicial complexes have renewed interest and helped to uncover a wealth of new physical phenomena. For example, researchers have discovered the intriguing chimera states that only emerge in higher-order topologies [18]. And a few works have observed the promotion of cluster synchronization and explosive transitions in higher-order generalizations [19][20][21][22][23][24]. As exploration deepens, we can not only gain insight into the emergent phenomena arising from higher-order interactions, but also make it possible to further modulate them. Hypergraphs allow an arbitrary number of agents to interact and represent them in the form of hyperedges to encode the higher-order interactions. As for simplicial complexes, although very powerful, this approach requires mutual inclusion for the interactions, which is restrictive for general purposes [25]. By contrast, the hypergraph is a more versatile tool for addressing multi-body interactions.
With the improvement of descriptive ability, analysis complexity and difficulty increase significantly, especially for dynamic processes on the networks. One of the key challenges is to understand the heterogeneous pattern dynamics that generate regular spatial patterns, spatio-temporal chaos, etc. There are several basic mechanisms to enable such dynamics for network-coupled systems, and here, we focus on the Turing instability (TI) and Benjamin-Feir instability (BFI). TI, first introduced by Turing in 1952, has become a popular mechanism for spatial patterns [26]. Under certain conditions, the presence of diffusion can seed the instability by perturbing the homogeneous state via the reaction-diffusion network, and then leading to the spatially heterogeneous density distribution [27]. Similarly, BFI, usually used to understand modulational instability, is also a type of diffusion-driven instability. But it perturbs the homogeneous but time-period state to form the spatio-temporal patterns such as chaos. As for hypergraphs, researches on heterogeneous dynamics have made preliminary progress. For example, Carletti et al considered the dynamical systems defined on hypergraphs and analyzed Turing patterns and synchronisation phenomena by introducing a new combinatorial Laplace operator [28]. Anwar and Ghosh investigated the intralayer and interlayer synchronization in multiplex networks with higher-order interactions, and illustrated the effect of higher-order architecture on synchronisation over multiplex networks [29].
However, there are very few reports focusing on the non-autonomous systems anchored on hypergraphs. The time-dependent properties common in realistic scenarios prompt the research, but the development has been delayed compared to other counterparts due to the analysis difficulty [30][31][32][33]. Studying heterogeneous dynamics of such paradigm is not an easy task since the standard spectra analysis that relies on temporal eigenvalues can break down. Recent enlightening results to face this problem are provided by Van Gorder in the [34]. The author proposed a theory to analyze the onset of the diffusive instability for non-autonomous systems on continuum domains using comparison principles. Further, this theory has been extended to solve similar problems on the temporal network of pairwise interactions [35]. Inspired by the above, we study the pattern formations of non-autonomous systems defined on the hypergraphs in this paper. The main motivation of this work can be summarized as follows: • We explore the combined effect of higher-order interactions and time-dependent properties, and analyze the conditions for two basic diffusive instability mechanisms, i.e. TI and BFI. • Emergent patterns such as Turing patterns and spatio-temporal chaos from non-autonomous reactiondiffusion systems are uncovered as well as their characteristics. • Self-organization processes on the hypergraph and corresponding projected network are compared to provide some ideas on the control of pattern formations.
The remainder of this paper proceeds along the following lines. In section 2, we review the formalism of hypergraphs and introduce non-autonomous systems defined on it. Section 3 presents the linear stability analysis of the equilibrium of the system, and conditions for the occurrence of TI and BFI are discussed. Then, for three different time-varying terms, numerical simulations are provided in section 4 to illustrate the Turing patterns and spatio-temporal chaos that occur on hypergraphs. Finally, we summarize the results and conclude in section 5.

Basic mathematical structure
In this section, we provide a brief review of the hypergraph and the non-autonomous system anchored on it to establish the studied network-coupled model.

Hypergraphs
An undirected hypergraph is generally defined by the pair consists of a set of unordered nodes and can describe the interactions of arbitrary order. Note that if all hyperedges have the size |E α | = 2, we recover a normal network, whereas we have a simplicial complex if the subsets of each hypergraph that |E α | > 2 are contained in E.
We then give some definitions about hypergraphs to describe the higher-order structures. The incidence matrix e, describing the adjacency of vertices on hyperedges, can be defined as Based on this matrix, we can construct the n × n adjacency matrix A = ee T , whose element A ij describes the number of hyperedges that connect the vertices i and j. And the main diagonal elements are set to zero. In addition, the m × m hyperedge matrix C can be obtained by C = e T e, whose entry C αβ describes the number of nodes shared by hyperedges E α and E β . Observe that C αα indicates the size of the hyperedge E α . Based on the adjacency matrix A, corresponding Laplace matrix L can be obtained with entry [36,37]. Here, k i = ∑ j A ij stands for the number of hyperedges passing through the node i and δ ij represents the Kronecker delta. This is a simple but inflexible generalization from the binary networks to the hypergraphs since the higher-order couplings cannot be fully extracted. Especially, the effect of the size of hyperedges on the transition among nodes is not taken into account. To fill this gap, authors of [38]. proposed a biased random walk process according to the fact that the spreading among nodes tends to occur in those belonging to the same hyperedges. The resulting Laplace matrix integrates the number and size of hyperedges, and assigns weight values to reflect the closeness of different nodes. Specifically, the entries of the new adjacency matrix A H are Here,Ĉ is a diagonal matrix with diagonal elements equal to those of the matrix C. Then the new Laplace matrix L H arrives at which is employed in the following investigations for instability mechanisms. D H is a diagonal matrix with the nonzero elements D H ii = ∑ j A H ij . And the matrix A H considers the size and the number of incident hyperedges of different nodes to encode the higher-order interactions, which can be regarded as the adjacency matrix of the equivalent weighted network, i.e. a weighted adjacency matrix. If the size of all hyperedges is equal to 2, the random walk Laplace matrix L H degenerates into the standard Laplace matrix L for pairwise interactions.
It is worth emphasizing that the dynamical behavior evolving over such weighted network is equivalent to that over the corresponding hypergraphs [39], which allows us not only to use the analysis methods of binary network, but also to avoid the difficulties when dealing with the tensors. In addition, the projected network of the hypergraph is also a useful tool to illustrate the higher-order characteristics. Projected network is constructed by mapping the hyperedges into a complete clique of suitable size. If the hypergraph only consists of simple hyperedges, this operation is reversible and the result of the reverse operation will be unique. The schematic diagram of the hypergraph and the projected network is shown in figure 1. We then compare the dynamics evolving on the hypergraph and its projected network in the following contents.

Non-autonomous systems
For a complex network system, there are two basic factors at play behind the emerging behavior: the intrinsic dynamics of each node and the coupling of different nodes. Next, we turn our attention to the dynamical process on hypergraphs. We mainly concentrate on the non-autonomous systems including time-dependent diffusion coefficients, non-autonomous reaction kinetics and time-dependent diffusion coupling. Generic mathematical structures are provided in this section which consider all these three terms, and detailed analysis of each term will be carried out later.
According to the above-mentioned random walk process set-up on hypergraphs, we arrive at the following generic system: The system consists of n identical units linked through a hypergraph and each unit is hosted on a node with the m-dimensional variable u i (t) denoting the activity at time t. The first term on the right-hand side of the equation indicates the local dynamics of each single node: F provides the non-autonomous kinetics. The second term describes the interaction between different nodes: the propensity is reflected by the weights relying on the hyperedge degree C αα , while the diffusion coupling is defined by the continuous function G; D(t) stands for the global diffusive coefficient. It is clear that system (1)  Dynamical behavior of such system is not a simple superposition of each single units, but instead tends to display collective behaviors. Hence we explore two primary mechanisms which generate transitions from the homogeneous states to the hetergeneous spatial or spatio-temporal states, i.e. TI and BFI. More precisely, assume that the interconnected systems always admit a homogenerous state, the (in)stability of this equilibrium is proved by exploring the related eigenvalues of the linearized system, determining the outcome of the perturbations growing with time. However, due to the existence of non-autonomous features, conventional temporal eigenvalues can no longer convey the information on the stability of the system [40][41][42], and more general 'eigenvalue functions' and more valid stability criteria should be introduced.
In order to solve this problem, we employ the theory proposed by Van Gorder in [34,35,43]. Van Gorder introduced a method to deal with the dynamical processes governed by non-autonomous systems on temporal networks, which can not only yields concise and effective stability criteria, but also avoids limitations of strong assumptions. However, despite of its comprehensiveness, follow-up researches only address the standard binary network and lack the extensions to higher-order structures, which is the focus in this paper. We mainly study the loss of stability of the equilibrium due to diffusion. The equilibrium here refers to two types: the uniform constant base state and the simple limit cycle, which leads to Turing patterns and spatio-temporal chaos respectively. And the resulting patterns are compared with those of the corresponding projected networks to dig up more characteristics of higher-order interactions.

Linear stability analysis
Using definitions of the weighted adjacency matrix A H , we rewrite system (1) as follows: The assumptions on function G ensure that the uniform equilibrium u * governed by du i /dt = F(u i , t) also satisfies equation (2). Here, the equilibrium can be either fixed points or limit cycles. If the equilibrium proves to be unstable, small perturbations δu i near it will develop into spatio-temporal patterns. Introducing small perturbations u i = u * + δu i and substituting it into equations (2), we obtain the linearized regime where ∂ u F(u * ) and ∂ u G(0, t) are matrices of derivatives. Let us observe that the uncoupled equilibrium u * is also a solution of system (3) since the Laplace matrix L H always admits a zero eigenvalue associated to the eigenvector (1, 1, . . . , 1). Therefore, we can study its instability mechanisms to illustrate the formation of spatial patterns of system (2). Standard analysis relies on examining the real parts of the eigenvalues to see if there exist some spatial modes leading to the growing perturbations. However, this approach will fail in the non-autonomous case, when the eigenvalues tend to change over time. Then a targeted method will be applied to carry out the theoretical analysis upon two cases, TI and BFI [34,35,43]. As for TI, we consider a reaction-diffusion framework defined on the hypergraph including the time-dependent global diffusion parameters and the time-varying reaction kinetics. While for BFI, we focus on the complex Ginzburg-Landau equation (CGLE) describing the coupled oscillators on the hypergraph, with the time-varying diffusion coupling.

Turing instability analysis
TI occurs due to the coupling of local kinetics and diffusion, and hence we consider a generic two-variables reaction-diffusion process on the hypergraph. Assuming that nonlinear functions f 1 and f 2 define the non-autonomous reaction kinetics for u and v, the reaction part of the system is (4) Replicating this system to each node i (1 ⩽ i ⩽ n) of the hypergraph, we arrive at the studied mathematical framework for u i and v i . Unlike standard reaction-diffusion networks, diffusion between nodes on the hypergraph possesses a weight depending on the number and size of common hyperedges. To describe this feature, the change of the concentrations (in terms of u i ) due to diffusion is expressed bẏ According to the definitions of the adjacent matrix A H and the associated Laplace matrix L H , the above expressions can be simplified tȯ In addition, global diffusion parameters are set to be D u (t) and D v (t) for variables u i and v i , which determines the time-varying nature of diffusion regime. Therefore, the non-autonomous reaction-diffusion hypergraph can be written as follows Since the underlying hypergraph is undirected, the Laplace matrix L H is symmetric, non-negative with the smallest eigenvalue zero. More precisely, denoting the set of the eigenvalues of L H by (ρ H l ) 1⩽l⩽n with the associated eigenvectors (ϕ H l ) 1⩽l⩽n forming the orthonormal basis, ϕ H l · ϕ H m = δ lm , they satisfy L H ϕ H l = ρ l ϕ H l . Notably, although this representation is similar to the case of standard reaction-diffusion networks, but higher-order interactions are actually embedded in the matrix L H , which helps to circumvent the complexity caused by simplicial complexes.
The studied equilibrium of system (5) refers to the time-dependent solutions (u * (t), v * (t)) to the equation (4) which remain uniform in space. We then derive the necessary conditions for TI through linear stability analysis. First, introducing small perturbations about the base state (u * (t), v * (t)): and substituting it in the system (5), we obtain the compact form of the linearized equations with δu = (δu 1 , δu 2 , . . . , δu n ) T and δv = (δv 1 , δv 2 , . . . , δv n ) T . We expand the first-order perturbations on the eigenbasis ϕ H l of L H and take the form of Substituting the above expressions into the equation (6), we arrive at which determines the development of the lth perturbations. Here, J ij (i, j = 1, 2) is the partial derivative of the reaction kinetics with respect to u and v. For the autonomous context, there exist p l (t) ∼ exp(λ l t) and q l (t) ∼ exp(λ l t) with λ l denoting the eigenvalues of the system, which are not appropriate ansatz here because they are not only difficult to compute, but also cannot provide the information of stability. To solve this problem, we then employ the method introduced in [34,35,43], readers can refer to these literatures for more details. This method builds on an important conclusion that: for a non-autonomous second-order ODE of the form d 2 Y/dt 2 + M 1 (t)dY/dt + M 2 (t)Y = 0, there will be at least one fundamental solution with the growth rate at least exp(ηt) if M 2 (t) ⩽ −η 2 − ηM 1 (t) [34,35,43]. Let the limit η → 0 + , the strict inequality M 2 (t) < 0 permits a fundamental solution having the exponential growth [34,35,43]. For application, we transform the ODE (7) into a second-order form by eliminating p l (t) and q l (t) respectively, and the results are Then, determining the exact expressions of M 2 (t) and applying the above stability criterion, the function p l (t) admits a solution with exponential growth when ) .
Similarly, the condition that q l (t) has the same results is ) .
According to the requirements of TI, only one of the p l and q l needs to grow in time exponentially. Therefore, Turing conditions for system (5) can be written as Let us observe that the TI condition (8) may hold over a time interval, i.e. it is generally transient. At such a time interval, Turing patterns can emerge but will not exhibit a regular spatial structure, which is distinguished from the case of continuous media. In the numerical simulation section, we give some examples to prove the validity of theoretical results.

Benjamin-Feir instability analysis
BFI is another basic mechanism that can trigger heterogeneous spatial dynamics, but by destabilising the uniformly synchronized states [44]. For analysis of this mechanism, amplitude equations are usually derived from original systems, upon which the conditions of BFI near a supercritical Hopf bifurcation will be obtained. In this paper, to sometimes reveal some universal properties, we focus on the normal form for a supercritical Hopf bifurcation, i.e. the CGLE on hypergraphs, and illustrate the evolution of spatio-temporal dynamics. First, we consider a limit-cycle oscillator obeying to the CGLE. Let w i denote the complex amplitude of the oscillator on node i, local dynamics can be written as Here, a is a complex parameter and the CGLE admits a base state solution w i = W(t) = e −iat . Then, single oscillator is assumed to be coupled via the hypergraph, which can be expressed as b is second complex parameter. In contrast to the previous section, diffusion coupling here is of a more general type, defined by the time-dependent function G, and the non-autonomous nature reflected by it is the focus of our attention. Next, we consider the loss of stability of the limit cycle solution due to BFI. For the non-autonomous system (9), the form of spatial perturbations are employed as W(t)(1 + ϵδw i ). Therefore, compact form of the linear equations can be written as Here, δw denotes (δw 1 , δw 2 , . . . , δw n ) T and G w (0, t) is the partial derivative of G with respect to w. Expanding the perturbations upon the eigenvectors of the Laplace matrix L H , i.e. δw l = (p l (t) + q * l (t))ϕ H l , we obtain the ODE for complex-values functions p l (t) and q l (t): Transforming the above equations into new variables, x = p l + q l and y = p l − q l , we arrive at Let us observe that at least one of |x| and |y| grows when and only when at least one of |p l | and |q l | grows. The analysis of BFI can therefore be converted into the analysis of system (10). Eliminating either x or y gives the following second-order differential equations According to the comparison results in [34,35,43], we obtain the BFI condition as follows

Numerical simulations
In this section, we provide some examples to demonstrate the validity of the above theoretical results. Specifically, we adopt the Brusselator scheme to model the reaction kinetics for TI analysis and use the CGLE for BFI analysis. The interactions between different dynamical agents are inscribed with hypergraphs. The method for building a hypergraph is the same as in [28]. Specifically, based on a Barabási-Albert network, three new nodes are attached to the existing network every time according to a preferential attachment scheme, until there are 20 nodes in the hypergraph. The focus of this algorithm is not the scale free nature of the network but to make the hierarchical structures connected [28]. Finding all the cliques in this network and transforming them into hyperedges of appropriate size, then we can obtain the corresponding hypergraph. We present the spatial dynamics on the hypergraph and its projected network, in order to highlight the effects produced by higher-order connections. The simulation results and analysis are presented below in three parts, where the first two parts address the Turing patterns due to time-varying diffusion parameter and time-dependent reaction kinetics, respectively, and the last part examines the spatio-temporal chaos under the time-varying diffusion coupling.

Case I: time-varying diffusion parameter
First, we consider the effect of time-varying diffusion parameter on the Turing patterns. In order to highlight this non-autonomous term, hypergraph structure and reaction kinetics, dominated by the Brusselator model Based on the results of equation (8), conditions for the occurrence of Turing patterns on the hypergraph and its projected network are derived separately for the above configuration. In figure 2, we plot the variation of the functions on the left-hand side of inequality sign over time, and Turing patterns will appear when some of the curves locate below the axis. We observe that all curves are above the axis until t = 200, after which some of them stabilize below the axis. This is because the value of time-varying diffusion parameter D u switches around t = 200 with the change rate 0.1. We also find that the variation of curves corresponding to hypergraph is greater than that of the projected network, which is caused by a wider range of the eigenvalues of the hypergraph. This feature is consistent with the results mentioned in [28]. That is to say, time-varying properties do not affect this characteristic which can be used in the setting and control of Turing patterns in different network topologies.
We then plot the time series of u i on hypergraph and its projected network in figure 3. It is clear that dynamical behavior evolves into a heterogenerous state from the initial homogeneous steady state, i.e. Turing patterns emerge, which is consistent with theoretical results. Since the instability takes time to develop before the concentrations of u i begin to diverge, the point at which Turing patterns form lags behind the point at which the diffusion parameter switches. Furthermore, figure 3 shows at t = 500 the concentrations of u i start to differentiate, while at about t = 600 the concentrations stabilize. The transient process takes up little time, so that the system quickly locks onto the stable Turing patterns and remains in this state for an arbitrarily long time. The reason for this may be the rapid transition of the diffusion parameter, which then causes the rapid change of unstable modes. The emergence of Turing patterns on the projected network shows same features, but the divergence begins at an earlier point, i.e. the time required for the instability growth is shorter.

Case II: non-autonomous reaction kinetics
Next, we examine the case where the reaction kinetics defined on the hypergraph are non-autonomous. Specific setting is still based on the Brusselator model, which is multiplied by a time-period function to The diffusion part is determined by the above-mentioned hypergraph. The period of the switch of reaction kinetics is t = 1000 and (u * , v * ) = (1, b/c) is also the equilibrium in this case. In figure 4(a), we plot a diagram of the TI conditions, i.e. the left-hand side of the equation (8). It can be seen that both the time periods in which the TI conditions are satisfied or not are present and they alternate. Then, taking the same initial conditions as Case I, we plot the time series of u i on the hypergraph in figure 4(b). Notably, the period of the change of u i is also t = 1000, with each cycle first experiencing a uniform steady state and then forming Turing patterns. This process of pattern variations is consistent with the results predicted by the Turing conditions. Since all u i have moved away from the equilibrium u * = 1 and undergone a differentiation, the simulation results validate the validity of Turing conditions. Compared to Case I, Turing patterns here are more intricate and not simply regular distributions. It indicates the intense competition between the unstable modes, leading to a longer transient stage and a stronger non-monotone evolution. In addition, figure 4 also shows a time lag when the system returns to the spatially homogeneous steady state.

Case III: time-varying diffusion coupling
In this case, we provide examples to verify the validity of the BFI analysis. In contrast to the above two cases, the simulated non-autonomous system consists of a time-varying diffusion coupling term, while the rest of the system is time-invariant. Specific mathematical structure is expressed byẇ 1(t − 150))). Such a setting makes k = 0.85 within the time interval (50, 150),  According to the theoretical results of BFI, we plot the left-hand side of the equation (11) in figure 5(a). Clearly, there exist some modes making the function always greater than zero, while others make the function value less than zero in the time interval (50, 150). Hence, BFI occurs during this time interval. The original uniform synchronous state will be broken, changing to a spatially heterogeneous pattern which is presented as chaos here. In figure 5(b), we plot the evolution of the modulus of w i at each node to show the spatio-temporal dynamics defined over the hypergraph. Initial conditions are set to be Re(W) = Im(W) = √ 2/2 plus the uniform random variables on (−0.01, 0.01). We find that during the time interval (50, 150), the modulus leaves the equilibrium state |W| = 1 and develops into a chaotic state of clutter. But this chaotic state does not last for long, and then returns to the equilibrium state. It should be noted that the coupling strength required to activate and suppress the chaos is different. With a weaker coupling strength, spatio-temporal chaos are easier to occur, but suppressing this phenomenon needs to increase the coupling strength significantly. Simulation results are consistent with the above BFI conditions, demonstrating the validity of theoretical results.

Conclusion
In this work, we have studied the spatio-temporal patterns of the non-autonomous system defined on hypergraphs. To avoid the inherent limitations of dealing with binary exchanges, hypergraphs endowed with many-body interactions are employed to express the coupling among dynamical units. In addition, non-autonomous systems consisting of non-autonomous reaction kinetics, time-varying global diffusion parameters and time-dependent diffusion coupling are considered to investigate their roles in spatio-temporal patterns.
Due to the presence of non-autonomous mechanisms, standard approaches that explore the temporal eigenvalues will fail. Therefore, new criteria for the transition from homogeneous states to heterogeneous spatial or spatio-temporal states are introduced to overcome this problem [34,35,43]. On this basis, conditions for the onset of TI and BFI are analyzed, which generates Turing patterns and the spatio-temporal chaos on hypergraphs. Numerical simulations are carried out not only to verify the efficiency of theoretical results but also to compare the pattern formation on the hypergraph with its pairwise projected network, which illustrates the special features of higher-order structures. Specifically, the eigenvalues of the hypergraph cover a much wider range compared to the projected network, which indicates that Turing patterns are more likely to be triggered on the hypergraph, as also verified in the [28]. This feature is not affected by the time-dependent nature of the reaction-diffusion system. But self-organization process on the hypergraph takes more time to lock onto the emergent patterns, and the time when the homogeneous states begin to diverge is relatively late. The reason may be the more intense competition between the unstable modes on the hypergraph than that on the projected pairwise network. Figure 3 shows the characteristics of the emergent Turing patterns on hypergraph and projected network are almost the same and both exhibit regularity and simple symmetry. On the other hand, for emergent chaos due to the BFI, we also observe some interesting phenomena. The most notable one is that it is possible to generate and suppress the spatio-temporal chaos on the hypergraph, but the required diffusion coupling strengths of the oscillators are different. Specifically, weak coupling is suitable to trigger the desired chaos, but relatively stronger coupling is needed to suppress them after excitation. All of the above phenomena are found in the non-autonomous reaction-diffusion systems, revealing the self-organization process on the hypergraph due to the TI and BFI mechanisms and the important characteristics of the emergent patterns.
Using theoretical results in this paper, we can not only understand the self-organized patterns on the hypergraph, but also gain insight into the solution of problems in some applications. The control of spatio-temporal patterns is one of the attractive topics. Physical mechanisms for this issue include thermal forcing [45] and illumination [46] by nonlinear kinetics of reactions. According to our analysis in Case II, corresponding researches can be extended to the hypergraph structure and the control time can be set in a finite interval to reduce the cost, while time-dependent diffusion parameters in Case I can provide global control to tune the emergent patterns. In addition to the non-autonomous reaction-diffusion framework, time-dependent properties may also appear in the base states such as limit cycle. Our analysis about the BFI of CGLE defined on the hypergraph provides general results for the reference when carrying out related studies. On the other hand, this study can also help to solve some problems in the field of ecology. For example, higher-order models have been emphasized in addressing epidemic spreading [47,48], metapopulation [49] and so on. For the reaction-diffusion process, conclusions here can provide theoretical framework for dealing with the case where coefficients are time-dependent.
Understanding the emergent self-organized patterns on the hypergraph is the basis of many topics. For example, corresponding results can be applied in turn to infer the underlying hypergraph topology by observing pattern formations. Besides, theoretical results can also provide ideas for the study of the spatial patterns in the practical model constructed by the real data. Finally, the interesting phenomena observed in this paper motivate us to reconsider the previous studies from the perspective of non-autonomous systems on hypergraphs, which may lead to some new conclusions. This work takes a step forward in exploring the emergent patterns of more complex network-coupled dynamical systems that encode higher-order relations.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).