The phenomenon of transient chaos has been extensively researched across diverse domains, including hydrodynamics, ecology, economics, coupled electronic circuits, electrophysiology, neural networks, and power grids. Additionally, the occurrence of extreme events (EEs) in these fields has demonstrated profound impacts, leading to a crucial question: Do extreme events emerge during a transient chaotic state in complex networks? This underscores the importance of investigating EEs phenomenon within transient chaotic systems. However, the precise identification and characterization of transient extreme events (TEEs) have not been addressed in the existing literature. In this manuscript, we identify extreme events that emerge during the transient state from both a dynamical system and network perspective. This study provides the first evidence of transient extreme events within network and mathematical modeling, offering valuable insights into real-world occurrences. Here, we investigate the emergence of typical extreme events and transient extreme events within a mutually coupled star network of the Brusselator model. The identification of TEEs was accomplished using the Finite-Time Lyapunov Exponent and statistical methods, which are employed to assess rare events. Additionally, we investigated the transitions of EEs by evaluating the average synchronization error and the system’s energy, which quantifies the degree of coherence among the oscillators in the network. To validate the robustness and generalizability of our results, we expanded our study to networks of varying sizes, consistently identifying similar patterns and phenomena. This detailed analysis not only deepens our comprehension of TEEs but also opens up possible potential applications across various scientific and engineering fields, including enhancing the reliability of mathematical models and forecasting extreme events.

1. Introduction

In recent years, there has been a noticeable increase in attention towards investigating extreme events, which are occurrences in dynamical systems that happen suddenly and exhibit unusual dynamical phenomena. Extreme events phenomena are extensively studied within various isolated dynamical systems [? ? ], providing critical insights into their underlying mechanisms. However, real-world scenarios often involve dynamical systems that are not isolated but are components of larger, interconnected networks. Studying extreme events in the context of networks is crucial for understanding the occurrence and impact of extreme events in both natural systems, such as ecological networks [? ], neuron networks [? ] and man-made systems [? ? ]. Hence, extreme events phenomenon is observed across various types of network models, including static networks [? ], random walks on networks [? ? ], scale-free networks [? ], multilayer, interdependent complex networks [? ], and time-varying networks [? ].

Therefore, a wide range of natural and man-made phenomena are studied in scale-free topology [? ? ], and the star network motif, representing scale-free networks, consists of a central hub node connected to several peripheral nodes, star network makes it an ideal model for understanding the dynamics of scale-free networks. Further, numerous intriguing dynamical behaviors like chimera states [? ], remote synchronization [? ], and explosive synchronization [? ] have been demonstrated regarding the star network motifs in phase oscillators [? ], the spread of an epidemic [? ] and this network found within neural networks [? ] that play vital roles in both cognitive processes [? ] and sensory functions [? ]. However, the phenomenon of extreme events has not yet been studied in this network.

Moreover, prior research has shown that extreme events emerge within chaotic and nonchaotic systems [? ] in isolated, coupled, or networked dynamical systems. The interesting question arises: is the behavior of the extreme events induced during the transient state, and if so, how does it manifest in the asymptotic state?, In the past, transient chaos has been extensively studied in various fields, such as hydrodynamics [? ], ecology [? ], economics [? ], coupled electronic circuits [? ], electrophysiology [? ], neural networks [? ], and power grids [? ]. Similarly, significant impacts of extreme events [? ] have been observed in these same fields. However, while these phenomena have been explored independently, this underscore the importance of investigating EEs phenomenon within transient chaotic systems. Additionally, transient chaos exhibits over very long iterations of the transient period [? ? ] and extreme transient behaviors, where the system’s transient states become extremely prolonged and occur spontaneously [? ]. This makes it particularly interesting to investigate the behavior of extreme events in the transient state of a system, which is responsible for the system’s asymptotic state. Hence, we classified the extreme events in the system’s transient state as a special class of extreme events, which we denote as transient extreme events (TEEs).

Besides, the first experimental demonstrations of chaotic behavior in chemical oscillators were observed in the Belousov-Zhabotinsky (BZ) reaction[? ]. Since then, the BZ reaction has been extensively studied in isolated, coupled, and network systems, showcasing a wide range of nonlinear phenomena, including Turing patterns, waves, clusters, and chaos[? ? ]. However, investigations into BZ reactions in star network configurations remain limited, and experimental studies in these networks[? ] have created significant interest in star network configurations. In the context of chemical oscillators, we recently investigated extreme events in the Brusselator model[? ] influenced by an external periodic force for the first time. Motivated by the previous research on star network configurations of the Belousov-Zhabotinsky reaction, this manuscript explores extreme events and transient extreme events (TEEs) using the theoretical Brusselator model in a star network configuration.

This paper is organized as follows: Section 2 describes the mathematical model. Section 3 provides the numerical analysis, covering transient life time , generalized extreme value distribution, average synchronization error, and the system’s energy function. Section 4 discusses the significance and potential applications of this study. Finally, in section 5 presents the conclusions.

2. Model Description

We consider a coupled Brusselator oscillators arranged in a star network topology as defined by the N ×N connectivity matrix A_nm,

and where each oscillator (n = 2,…,N) in the network is coupled to the central hub oscillator (n = 1) as displayed in Fig. 1 through the cross-feedback, which is symmetric owing to bidirectional couplings. The governing equations are provided by:

where n = 1,2,…,N. The system state variables are x and y, which represent concentrations of autocatalysts, while a and b are constant system parameters representing concentrations of reactants and ϵ represents the global chemical coupling strength. The d_n = (N -1,_{N-1 times}) are represents the in-degree of node n, which is used for normalising the input towards node. The equilibrium points for the system Eq. (1) is obtained by setting the derivatives of x and y with respect to time equal to zero. Thus, the system has only one equilibrium point in each oscillator, which is (x_n0, y_n0) = (a, b∕a). The characteristic eigenvalue equation is derived for the equilibrium point of an N oscillator are

Hence, the equilibrium points stability depends on the system parameters of a and b. If b > (a² +1), the equilibrium point is unstable. If b < (a² +1), it’s stable. When b = (a² +1), it can be a centre or an ellipticpoint, showing oscillatory behaviour.

3. Numerical Results

Our study of the proposed network system (1) began with an analysis of its dynamics using a bifurcation diagram and its corresponding Lyapunov spectrum of the central hub with varying coupling strength (ϵ) within the range of ϵ ∈(0, 1). Here, for convenience in the present study, we contemplate a system consisting of a total of N=6 nodes for an oscillatory network, with other system parameters being constant as a = 0.21 and b = 1.061. For all numerical analyses, the initial conditions for the oscillators in the network were set as (x_i(0),y_i(0)) = (0.01,0.01) if i is odd, (0.02,0.02) if i is even. The system demonstrates distinct complex dynamics under varying initial conditions, which will be discussed in a later section 3.2. A bifurcation diagram is computed for the transient and asymptotic states with a step size of 0.01. The transient state is depicted in Fig. 2(a) in blue color dots, calculated for the iterations of T = 2×10⁴ to 2×10⁵ time units. The asymptotic state is depicted in the Fig. 2(a) in red color dots, calculated for the iterations of T = 2×10⁵ to 7×10⁵ time units. Our analysis revealed that the network demonstrates periodic behavior in both transient and asymptotic states at weak coupling.

However, at coupling strength ϵ =0.16, a sudden expansion is occurs in the amplitude of the network, resulting in an extreme events within both system states. This phenomenon persists until the coupling strength reaches ϵ =0.605 in the asymptotic state, at this point, the system returns to a periodic state due to synchronization across the network induced by strong coupling. This transition from a chaotic state to a completely synchronized state takes a prolonged time to settle, inducing transient chaos with extreme events embedded within system. The system exhibits extreme events only in the transient state; this events region is depicted as a rectangular box in Fig. 2(a) from ϵ =0.483 to 0.605. We named this new phenomenon as Transient Extreme Events (TEEs), which is the system that exhibits the extreme events occurrence within a transient chaotic state. The following subsection explains this new phenomenon using relevant numerical and statistical measures.

Figure 2: (online color) The panel shows (a) the bifurcation diagram of the central hub , with the transient state depicted as blue dots and the asymptotic state as red dots. (b) displays the Lyapunov exponent of the system central hub.

To examine the typical extreme events behavior in our network, the numerically obtained time series is depicted in Fig. 3. Throughout the entire iteration, the system parameters are fixed as given above, and the global coupling strength is set to ϵ = 0.1657; hence, the system shows chaotic behavior, and its trajectory spontaneously exhibits extreme events (rare events). To distinguish extreme events from normal events, a threshold height (H_s) is determined using a statistical approach that identifies significant deviations from the mean value of the system’s state variable. Here, H_s is calculated as H_s = ⟨x_max⟩+mσ_xmax, where ⟨x_max⟩ represents the mean of the peak state variable, σ_xmax is the standard deviation of x_max, and m is an integer specific to the system.

Figure 3: (online color) Numerically computed (a) time evolution of a typical extreme events of the state variable with the threshold (H_s = 1.3152) marked as a horizontal dashed line for m = 8, and (b) the corresponding probability distribution function (PDF) with the threshold marked as a vertical dashed line.

To validate the existence of EEs, we plotted the critical threshold (H_s = 1.3152) in the time series (x_h(t)) as a dashed horizontal line for m =8, as illustrated in Fig. 3(a). This abrupt expansion of the network is classified as an extreme events. To confirm that the system is in an asymptotic state [? ], very long time iterations of T = 1×10⁸ were taken to allow the system to evolve. To confirm the presence of extreme events in the proposed model, we have plotted the probability distribution function for the state variable x_h(t) of the central hub. We considered a long time span, running the system for 2×10⁸ time units, and allowing it to evolve through transient states with constant parameters as used previously.

In figure 3(b) shows a continuously heavy-tail distribution, surpassing the threshold indicated by the vertical dotted black line. This indicates the occurrence of extreme events and confirms the low probability of large-amplitude events occurring beyond the H_s mark, which is calculated for m = 8 using the respective probability distribution function data of ϵ =0.1657 to confirm the presence of the typical extreme events in the network.

3.1. Transient Extreme Events

To study the behavior of transient extreme events in the proposed network Eq. (1) by presenting our primary findings from the numerically obtained time series for the central hub (x_h(t)) in Fig. 4. For the global coupling strength ϵ =0.6, the system exhibits transient chaos, displaying chaotic behaviour for long-time iterations of 1.4388×10⁵ before settling into a periodic state. This transient chaotic state exhibits extreme events in every oscillator of the network and this events are characterized by threshold height (H_s). We observed that chaotic state behavior remains confined to the transient state, with the EEs behavior showing a sudden expansion in phase space, represented in blue color in Fig. 4(a). In the asymptotic state, the system settles into a periodic state, represented in red color as shown in the subplot of Fig. 4(a). This transient chaotic behavior is confined to a specific temporal region, with EEs phenomenon characterized by rare and extreme amplitude bursts, depicted in blue color in Fig. 4(b). After a transient state, the system settles into a periodic state (period 10 limit cycle), represented in Fig. 4(b) with a red color. For the case of transient extreme events with ϵ =0.6, the probability distribution function is depicted in Fig. 4(c), which shows a continuously heavy-tailed distribution that exceeds the threshold value indicated by the vertical dotted black line. This demonstrates the occurrence of extreme events during the transient state, depicted in blue color, beyond the threshold, confirming the low probability of large amplitude events. Additionally, distribution indicates that the periodic asymptotic state is confined within a bounded region depicted at Fig. 4(c) in red color.

Figure 4: (online color) Panel shows (a) the phase portrait in the () plane, with the asymptotic periodic state depicted in the subplot. (b) the time evolution of the central hub variable , with transient states shown in blue color with a subplot labeled "chaos" and asymptotic behavior in red color with a subplot labeled "periodic". The TEEs threshold H_s is indicated by a dashed black line. (c) the Probability Distribution Function (PDF), with the asymptotic periodic distribution depicted in the subplot. and (d) the corresponding Finite-Time Lyapunov Exponent (FTLE) of the time series () for ϵ = 0.6.

To confirm the transition of the system from a chaotic state to a periodic state, we utilize the Finite Time Lyapunov Exponent (FTLE), which quantifies the amount of stretching or folding experienced by a trajectory along a specific direction over a finite time interval. This is illustrated in Fig. 4(d), which shows the maximum Lyapunov exponent for each oscillator in the system. During the extended transient state, the system demonstrates chaotic behavior, as indicated by the positive values of the Lyapunov exponent. These positive values confirm the chaotic behaviour in the transient state. Subsequently, as the system evolves, it transitions into a periodic state. This transition is validated by all the Lyapunov exponent becoming non-positive at t ~ 1.43×10⁵, signifying that the system has settled into a stable periodic behavior.

3.2. Transient life Time

To validate the transient life time and the escape rate from the proposed network, a finite time Lyapunov exponent analysis were performed to differentiate the transient and asymptotic behaviours of the system. Specifically, we considered finite time largest Lyapunov exponent to denote periodic behavior and other positive values of as indicating chaotic state. In the transient chaotic regime, the distribution of transient chaos is dependent on the initial conditions of the system. The system parameters are the same as the TEEs values and the hub oscillator initial conditions x_h(0) are varied in the ranges of 0 to 2.5 and we have plotted the phase diagram in the (-) plane, as illustrated in Fig. 5. For these initial conditions, the finite-time Lyapunov exponents were calculated using a total of iterations. In Fig. 5, the regions exhibiting chaotic behavior are shown in gray color region, while regions showing periodic behaviors are denoted by red color region. The lifetime of transient chaos is measured as the duration from the initial time to the point when the largest Lyapunov exponent becomes negative. This process is repeated for a range of initial conditions of the central hub between 0 and 2.5. The mean value of these measurements is determined to be , representing the average transient lifetime.

Figure 5: (online color) The phase diagram constructed with initial conditions versus transient time, using the Finite-Time Lyapunov Exponent. In the periodic regime (gray color), the Lyapunov exponents have negative values. In the chaotic region (red color), the Lyapunov exponents have positive values.

3.3. Generalized Extreme Value (GEV) distribution

To study the probability distribution of the network across the coupling (ϵ), we calculated the Generalized Extreme Value (GEV) distribution [? ], a primary statistical model in Extreme Value Theory[? ]. GEV allowed us to determine the shape of the tail, where extreme events occur. GEV distribution is widely used across different disciplines, such as engineering, climatology, finance, and insurance[? ? ]. This distribution offers a comprehensive framework to characterize these events, which is crucial for understanding of extreme phenomena. From the bifurcation diagram in Fig. 2, we deduced that the network exhibits periodic behavior at weak and strong coupling (ϵ), as well as periodic intermittency. Consequently, these periodic states follows a discrete distribution. Therefore, we excluded these regions and performed calculations only for the states that fit the Generalized Extreme Value distribution.

The cumulative distribution function of the generalized extreme value (GEV) distribution is calculated by,

where μ is the location parameter, ρ is the scale parameter, and ξ is the shape parameter. The shape of the distribution depends on the value of ξ , When ξ is negative, extreme events follow a Weibull distribution, and the tail of the distribution is bounded. Conversely, for positive ξ values, extreme events exhibit a Fr�chet distribution, displaying a heavy tail distribution. In cases where ξ equals to zero, the extreme value distribution mirrors a Gumbel distribution, showcasing an exponential decay in the tail of the distribution.

By examining the evolution of the sign of with coupling strength (), as shown in Fig. 6, In this present study, we observe that the network 1 follows a Fr�chet distribution (), indicating a heavy-tailed distribution. This implies that the probability of observing extremely large values decays slowly, meaning that large deviations from the mean are possible and occur with a non-negligible probability. Although in many instances, a high shape parameter () induces the distribution to be more skewed to the right (indicating high amplitudes), this does not surpass the threshold due to frequent large amplitudes, as depicted in Fig. 6 by blue dots and in the subplot representing the PDF of non-extreme events for . At certain coupling strengths, events that qualify with a minimum positive shape parameter () induce extreme events by exceeding the threshold value, depicted in Fig. 6 by red dots and in the subplot representing the PDF of extreme events for .

Figure 6: (online color) The generalized extreme value distribution shape parameter as a function of coupling strength ϵ is illustrated. Non-extreme events are shown in blue color, while extreme events are depicted in red color. The corresponding Probability Distribution Function (PDF) is displayed in the subplots.

The return period (or recurrence interval) for a Generalized Extreme Value (GEV) distribution is the average interval of time between occurrences of an events [? ], which is calculated by RP = , where p_(xpeak) is the empirical cumulative distribution function that represents the probability that a value is less than or equal to x_peak. In Fig. 7, the return period is plotted for the TEEs parameters of the central hub oscillator with threshold (H_s) as dashed horizontal line. From Fig. 7, it is evident that extreme events exceeding the threshold have a long return period ranging from 200 to 6250, with the very low probability () of occurrence of events between 0.005 and . This indicates that the rarity and infrequency of such occurrences, confirming the presence of extreme events in the network.

3.4. Average Synchronization Error

To study the synchronization of the network, the average synchronization error between the hub and the peripheral nodes of the network is measured, ⟨E⟩∈ [0,1] is defined by

where vector norm ||x|| = , a value of the average synchronization error () is equal to zero indicates no synchronization error, implying perfect or complete synchronization (CS) among the oscillators. Conversely, positive values of indicate that the central oscillator is out of synchronization with the rest of the network oscillators.

Additionally, focus on the scenario where the peripheral nodes synchronize into a single cluster while the hub remains separate, forming its own distinct cluster. This phenomenon, known as remote synchronization (RS) [? ], occurs when two or more nodes synchronize not through direct connections but via intermediate nodes. We introduce an additional measures to account for this situation. The average synchronization error for among peripheral ⟨E_p⟩ is measured by

Hence, for our system, the average synchronization error ⟨E⟩ is calculated for both the transient and asymptotic states, which are shown in blue and red colors respectively, as depicted in Fig. 8. The coherence among the peripheral nodes ⟨E_p⟩ is depicted in green color. The figure 8 illustrates the average synchronization error () across different coupling strengths (ϵ). These measurements have been computed from the numerical integration of system (1) using the parameter configuration for TEEs described in Sec. 3.1. From Fig. 8(a), it is clear that there is no notable difference in synchronization error between both the transient and asymptotic states of the network. The average synchronization error among the peripheral nodes suggests that the peripheral nodes are synchronized independently from the hub at weak coupling of ϵ ∈ [0,0.145], a phenomenon denoted as remote synchronization (RS).

At strong coupling of ϵ ∈ [0.6733,1], the network achieves complete synchronization (CS), as confirmed by the zero average synchronization error between the hub and peripherals depicted in blue color. In addition to understanding the synchronization dependence on initial conditions, we calculated the average synchronization error ⟨E⟩ for ten random sets of initial conditions, as shown in Fig. 8(b). The dark black color line indicates the calculated average of ⟨E⟩. It is evident that for any chosen initial conditions, the system exhibits remote synchronization at weak coupling and complete synchronization at strong coupling, as illustrated by the corresponding time series are shown as subplots in Fig. 8(b), with the hub depicted in blue color and the peripheral nodes in red color.

3.5. Energy Function Calculation

A mathematical framework to derive an energy function for chaotic systems, drawing upon the Hamiltonian formalism, is proposed by Sarasola et al.[? ]. In this formalism, the Hamiltonian function serves as an energy function for conservative systems, encapsulating the system’s dynamics through its variables in phase space. For dissipative chaotic systems, where energy dissipation occurs, the dynamics are expressed as a sum comprising a skew-symmetric Poisson bracket and a symmetric bracket, termed the generalised Hamiltonian formalism. In general theory, in a dissipative dynamical system, energy is not conserved over time. This means that the total amount of energy in the system decreases due to dissipation or energy transfer. Therefore, in this section, the energy of our proposed network is calculated with respect to coupling co-efficient and to explore how the energy transfer to one system to another systems as well as the cumulative energy flows of the network. The velocity vector field f(x) of an dissipative autonomous dynamical system can be separated into two distinct components:

where f_c is a divergence-free vector field responsible for the complete rotational tensor and f_d is a gradient vector field carrying the entirety of whole divergence. The change in the system’s energy along a trajectory is solely attributable to the contribution from f_d; the energy is dissipated, passively or actively, due to the divergent component of the velocity vector field, Thus, it obeys the relation:

For each dynamical system, there exists a partial differential equation from which the energy function H(x) can be determined by

A solution H which is satisfied Eq. (3) of the system Eq. (1) represented by the non-definite quadratic form:

in accordance with Eq. (2), the rate of change of Hamilton energy along a trajectory of the system (1) is,

This energy calculations are very useful for examining the changes in average Hamilton energy within a network. We evaluated the total mean Hamilton energy for a range of coupling strengths ϵ, from 0 to 1, while keeping other system parameters constant as used for TEEs. The calculated results are shown in Fig. 9, where the cumulative energy of all oscillators is represented by the green line. The red and blue lines represent in subsets showing how the energy of the central hub and one peripheral node change, respectively.

In figure 9, we can see that when the coupling strength (ϵ) is low, the cumulative energy (H) remains constant and evolves linearly due to remote synchronization. However, there is a sudden change in energy when the coupling strength ranges from ϵ ∈ (0.19,0.62) and the system transitions into a chaotic state. These abrupt changes in the transition of cumulative energy indicate that there is an exchange of energy between the hub and its peripherals. The network exhibits chaotic oscillations. During this range of coupling strength (ϵ), the system displays a variety of complex dynamics such as extreme events, generalized synchronization, transient chaos, and more. Furthermore, under strong coupling (ϵ ∈ (0.62,1.0)), the total energy remains stable, and the individual oscillator energies of the hub and peripherals are synchronized. This is illustrated in the subplot of Fig. 9 and indicates perfect or complete synchronization (CS) in the system.

By analyzing the energy diagram, we observed that at the coupling parameter value ϵ = 0.67 (marked by the vertical dotted line), the system undergoes a transition from a desynchronized chaotic state to a synchronized periodic state. During this transition period, the system exhibits typical transient chaos before establishing a strong coupling that leads to a synchronized periodic regime. This transient chaos exhibits extreme events within it and is characterized as transient extreme events (TEEs). Further to check the robustness of extreme events (EEs) and transient extreme events (TEEs) in networks of different sizes, We plotted the spatiotemporal pattern of a network with oscillators, which exhibits extreme events in all oscillators. This is depicted in Fig. 10, where the black gradient color indicates the large amplitude of extreme events (EEs) in the network, as qualified by the threshold from the respective temporal data. Additionally, the top panel of Fig. 10 shows the temporal plot for a peripheral node, chosen as node index as an example, with the threshold line indicated. Additionally, we plotted the two-parameter diagram in the parameter space of for ranging from 0 to 1 and network sizes chosen in the range of to oscillators, as shown in Fig. 11. This diagram distinguishes the periodic and chaotic states using the largest Lyapunov exponent, with periodic states depicted in gray color and chaotic states (non-extreme events) in green color and extreme events were classified based on a threshold value . Additionally, to differentiate between transient extreme events (TEEs) and typical extreme events (EEs), we performed calculations for both the transient and asymptotic states, with TEEs shown in blue color and EEs in red color. From Fig. 11, similar results are observed as depicted in Figs. 2 and 8, the system exhibits a periodic state in both weak and strong coupling regimes and displays transient extreme events (TEEs) before achieving complete synchronization.

Figure 10: (online color) The top panel displays the temporal plot of for node index in a network of oscillators, with the threshold indicated by a horizontal line, while the bottom panel depicts the spatiotemporal pattern of a network with oscillators. The dark black gradient color indicates the large amplitude of EEs in the network.

Figure 11: (online color) The two-parameter diagram in the parameter space of illustrates the dynamical behavior of the network across a range of oscillators, from N = 3 to 15. Blue points represent the parameter values where TEEs exist, red points indicate the existence of EEs, green color points denote Non-extreme events (chaotic states), and gray color points signify periodic states.

4. Possible Application

We have discussed some possible applications based on our results. In particular, our results on TEEs, supports some perspectives of applications for researchers and policymakers in a various field of domain. For instance, the integration of renewable energy sources into the power grid can also introduce transient power fluctuations, which may manifest as extreme events and transient [? ]. Understanding and predicting these TEEs can be crucial for developing strategies to mitigate their impact, thereby improving the overall reliability and robustness of power grids.

Understanding the mechanisms behind long transients and their implications for forecasting is now a key challenge in both theoretical and empirical ecology. The identification of mechanisms that drive these transients, along with an understanding of the scaling laws governing their lifetimes, could significantly enhance the accuracy of long-term forecasting and improve crisis anticipation. The study of Transient Extreme Events (TEEs) offers a valuable perspective for addressing these challenges. By examining the mechanisms underlying long transients within ecosystems [? ], researchers can develop a unifying theory that integrates both physical and ecological insights. The application of TEEs in ecological contexts holds the potential to bridge gaps in current research and pave the way for advancements in the quantitative analysis of ecological transients.

On the other hand, the network of Brusselator provides a valuable framework for studying pattern formation, particularly through the analysis of sudden expansions (extreme events) and transient extreme events (TEEs). These phenomena offer insights into how oscillatory behavior within the network can lead to the emergence of spatial patterns, such as stripes or spots. Beyond chemical reactions, the Brusselator model has been extended to study pattern formation in other systems, including neural networks and ecological systems, where similar mechanisms of diffusion and instability play crucial roles. The application of the theoretical Brusselator star-coupled network to transient extreme events could involve studying how these networks respond to sudden changes or perturbations, resulting in extreme behaviors. This approach could provide valuable insights into the mechanisms driving extreme events across various systems.

5. Conclusion

In this study, we observed the transition of a network from a chaotic (unsynchronized) state to a periodic (synchronized) state as the coupling strength was varied. During this transition, extreme events (EEs) emerged in a chemical model configured in a star network. Our analysis reveals that the proposed model exhibits complex dynamics, including the occurrence of extreme events. Specifically, we identified prolonged excursions of trajectories away from the bounded attractor during the transient state, with chaotic attractors displaying rare higher-amplitude events characterized as transient extreme events (TEEs). To confirm the presence of extreme events (EEs) and transient extreme events (TEEs), we determined a critical threshold through statistical methods. The dynamical transitions of the attractors and the occurrence of transient extreme events were analyzed using Finite-Time Lyapunov Exponents. The observed extreme events were further validated through statistical analyses, including the probability distribution function, Generalized Extreme Value distribution, and return period plots, to ensure the rarity of these events, and the transition to synchronization was confirmed by measuring the average synchronization error. Additionally, we calculated energy variations to verify the sudden expansions and transitions within the network. This research sheds light on the conditions under which extreme events can arise during prolonged transient states, significantly enhancing our understanding of such events in physical, engineering, and natural systems exhibiting transient dynamics.

Data available on request from the authors