Explosive death in direct and indirectly coupled oscillators: Review

The transition in the dynamical behavior in the coupled system has several applications in science. The phase transitions of synchronization and oscillation suppression have both been thoroughly researched for a very long time. The second-order transition, which is continuous and reversible, is demonstrated by the standard results in the vast majority of cases in the coupled system. Recently, the first-order transitions reported in oscillation suppression have been reported in the complex network of the coupled oscillators through direct and indirect interaction. Explosive death is a transition that is not only abrupt but also irreversible in its parameters. We currently have a very good grasp of first order transition in oscillation death in networked systems and a variety of significant contributions and advancements have substantially improved it. Here, we aim to provide a review on the explosive death in various direct and indirect coupled oscillator scenarios while reviewing the previous findings.


Introduction
In nature, individual systems do not exist in isolation.Communications are complex beyond belief due to the many connections, interactions, and communications between the oscillators [1].One of the key phenomena that oscillators exhibit is synchronization, which involves the coordination of their movements or oscillations [2][3].Synchronization can be broadly defined as the process by which two or more systems adjust their behavior in a way that allows them to operate in a coordinated and consistent manner.A variety of scenarios are studied in science when it comes to synchronization like neurons, cardiac pacemaker etc. [3].Typically, synchronization phenomena are characterized by second-order transitions, wherein the order parameter changes smoothly as the coupling strength is modified.However, a novel form of non-trivial synchronization, known as explosive synchronization, has been identified more recently in a network comprising interconnected oscillators [4].The nature of phase transitions is determined by a combination of an oscillator's characteristic dynamics and the topology of the network to which it belongs.In other words, the way in which an oscillator behaves and the way it is connected to other oscillators in a network will play a decisive role in the types of phase transitions that occur.[5].When undergoing an explosive transition, the order parameter experiences a sudden change at the point of transition.Moreover, it exhibits hysteresis, which entails different transitions when moving forwards versus backwards through the transition.In other words, the behavior of the order parameter is dependent on the direction in which the transition is taking place.[6][7][8].A small change in coupling strength in the network can lead to significant changes in the performance of the oscillators, ultimately resulting in synchronization [3,8].
In addition to synchronization, coupled oscillators can exhibit another form of collective behavior known as oscillation suppression.This phenomenon involves the cessation of oscillatory behavior, leading to the attainment of a fixed point state.In other words, the oscillators cease their motion and settle into a static, non-oscillatory state.The suppression of oscillations can be classified into two distinct categories: amplitude death (AD) [9,10] and oscillation death (OD) [11,12].The cessation of oscillatory behavior can take one of two forms, each with distinct characteristics and behaviors.Amplitude death occurs when the oscillation of the coupled oscillators gets stabilized and attains the same fixed point state which was unstable [13,14].Oscillation death is a phenomenon in which coupled oscillators stop oscillating and reach different steady states due to the nature of their coupling [11].This leads to the production of an inhomogeneous steady-state (IHSS) and a coupling-dependent homogeneous steady-state (HSS) [11,12].Amplitude death (AD) has been extensively studied because it is useful for suppressing unwanted oscillations in various systems such as lasers and mechanical engineering systems [15,16].Oscillation death, in contrast to AD, is a phenomenon that is especially relevant in biological systems, including those involved in cell differentiation [17,18].In many cases, the shift from an oscillating state to a death state in a system happens through a second order transition, but recent research indicates that it may also occur via a first-order transition known as explosive death [19,20].During this type of transition, referred to as coexistence, two distinct states are present and stable over a range of coupling strengths.This phenomenon is observed in many models of coupled limit cycles, chaotic systems, and physical systems.In other words, these systems may exhibit both oscillatory and non-oscillatory behavior depending on the coupling strength between the elements [19][20][21][22][23][24][25][26][27].Furthermore, the coupling scheme also plays an essential role in interacting systems.In the last few years, several types of coupling schemes are studied for coupled systems in literature, such as linear or nonlinear coupling, instantaneous or dynamical coupling, time delay coupling, and conjugate coupling, where systems interact through different variables [3,10,12].Besides direct coupling, the dynamical systems may interact indirectly via an intermediate system or a dynamical environment [28][29][30].This intermediate system or dynamical environment works as a medium or channel to transfer information between the indirectly coupled systems.Various examples of systems that exhibit indirect coupling have been studied in the last decade.For example, population of cells that utilize small molecules to communicate, which diffuse freely into a shared medium [31][32][33], same wooden beam hold multiple pendulum clocks [34], Chemical oscillators that interact via common solution [35] , Circadian oscillators that exhibit global neurotransmitter oscillators In population [36], Cold atoms that interact with an electromagnetic field as an ensemble [37], Laser that exhibit longitudinal modes and are connected through the saturation of a common amplifying medium [38], and nonlinear system that are indirectly coupled and show periodic or chaotic behavior [28][29][30]29].
The present letter provides an overview of explosive death in coupled oscillators, which occurs through both direct and indirect interactions.In section 2, we present and analyze the findings related to explosive death in models of directly coupled systems.Section 3 focuses on the results of explosive death in various scenarios involving indirect interactions.Finally, in Section 4, we have summarized our findings and drawn our conclusions regarding this topic.

Explosive death through direct interaction
Our current topic of discussion pertains to the phenomenon of explosive death that can occur in a network of directly coupled oscillators.
The behavior can be analyzed by examining a collection of Van der Pol oscillators that are coupled together through mean-field interaction.The equations below describe the dynamics of the system [20], In the given equation, i=1, 2, …, N (=100) denotes the oscillator index.The parameter b represents both the damping and nonlinear strength of the oscillator.denotes the mean value of the state variable x, while Q (0<=Q<=1) is the control parameter that regulates the mean-field intensity in diffusing coupling.At a given k, the order parameter A(k) for the coupled oscillators is calculated by a(k)/a(0) of a single, isolated oscillator.In the case of an oscillatory a(k) = (∑ i N <x i,max > t -<x i,min > t )/N state, a(k) takes a value greater than 0, while for a steady state, a(k) equals 0.  1).The dynamical regimes described above are observed within a specific range of parameters in both (d) the (b -k) phase plane, with Q fixed at 0.5, and (e) the (Q -k) phase plane, with b fixed at 3. These results were obtained for a network of 100 oscillators [20].
The variation of A(k) is displayed with k at fixed Q=0.5 shown in Fig. 1.When b is less than 1, the forward continuation of A(k) with coupling strength undergoes to the amplitude death state through the second order transition.While in backward continuation, A(k) shows the first order phase transition from amplitude death to rhythmic behavior and makes the semi-hysteresis area (SHA) in Fig. 1(a).The forward and backward transition points between the oscillatory state and the death state coincide, and correspond to a second-order phase transition occurring at a specific k at b=1 in Fig. 1(b).On the contrary, for values of b larger than 1, as depicted in Figure 1(c), a remarkable discontinuous transition in the order parameters A(k) is observed, where they drop sharply from a high value to zero.The forward and backward transition points exhibit dissimilarities and give rise to the hysteresis region (HA).According to the findings of this study, if the system undergoes a first-order transition, it will stabilize at the coupling-dependent HSS, while a second-order transition results in 2.1 Mean-field coupled oscillators stabilization at the origin.Additionally, the parameter 'b' is found to have a positive impact on the HA within the parameter space, meaning that increasing 'b' leads to higher values of HA.
In the parameter space, Figure 1(d-e) depicts the various dynamical states using k in forward and backward direction.The diagram includes the following representations: oscillation state (OS), amplitude death (AD), homogeneous steady state (HSS), and hysteresis area (HA), Semi hysteresis area (SHA).The transition of the partial second order takes place when b is less than by making a semi hysteresis area, and second order transition at b = 1.When the value of b exceeds 1, the transition to HSS of the first order takes place.In first-order transition, a coupled oscillator stabilizes at the HSS, which depends on the coupling, while when there is a second-order transition, oscillators stabilize at the origin.Moreover, as b increases, the hysteresis area (HA) expands in the parameter space [20].

Explosive death through direct interaction
There are numerous indirect interactions between physical, chemical, and biological systems due to common environments.Chemical signal molecules are used, for instance, by bacteria to communicate with one another.In order to communicate chemically, bacteria must produce, release, detect, and react to tiny hormone-like molecules known as "autoinducers."Bacteria may keep track of other bacteria in the environment through a technique known as quorum sensing [23].In this part, we looked at phase transitions in the context of the suppression of oscillation in an indirectly coupled oscillator

Explosive death through an environment
A collection of N identical Stuart-Landau oscillators, which interact via a shared environment, govern the behavior of the i th oscillator and its surrounding indirectly connected system [21].
in which i = 1, 2, …, N. ω represents the intrinsic frequency of the oscillator.The common environment's dynamics are shown by the variable s, which is represented by decay rate γ>0 when there is no coupling.The N = 100 used across this entire set of findings.
At the fixed values of parameters α = 0.5, A(ε) as a function of ε (for forward and backward direction) for different values is γ=0.3 and 0.7, illustrated in Fig. 2(a).In the forward continuation, A(ε) steadily drops with ε for γ=0.3.When ε= 2.7, the order parameter A(ε) exhibits an abrupt transition from a finite to a zero which indicates the coupled oscillator undergoes a first order transition from oscillatory state to a steady state.In the backward continuation scenario, we compute the A(ε) starting at steady state at ε = 4, and decreasing ε in steps of δε = 0.02.At ε = 1.8, a coupled oscillator exhibits a sharp transition in A(ε) from zero to finite value, indicating the resumption of oscillation.These sharp transition points, which occur at different values of ε, show that a hysteresis behavior exists.Similarly for γ=0.7, but the transition point is different and the hysteresis area is slightly shrunk.Fig. 2(b) is variation of A(k) with ε for different values of α at γ=0.5.This is a unique observation in environmentally coupled limit cycle oscillators.The results of the investigation on explosive death (ED) in (ε − γ) and (ε − α) parameter space are presented in Fig. 2(c-d).The different dynamical regime of the system is labeled as OS, HA, and HSS.The region denoted as OS contains only oscillatory solutions, while the region labeled as HSS consists of only homogeneous steady state solutions.In the HA zone, both the oscillation state and homogeneous steady state coexist.The collective behavior of a system is determined by α and γ, which are crucial factors.It's worth noting that as the feedback factor α decreases, the hysteresis area increases and eventually disappears when α reaches a certain low value [21].

Explosive death in complex network
For this instance, identical coupled Rössler oscillators exhibit chaotic behavior is taken into consideration.These oscillators interact with their two nearest neighbors (2P) as well as a shared environment.The equation below can be used to describe the behavior of the system [23], where i=1, 2, ..., N. x i , y i and z i are the state variables.The intrinsic parameters of the Rössler oscillators in the system are set to r = 0.1, b = 0.1, and c = 18.The state variable of the shared environment is denoted by s and decays at a rate of γ in the absence of coupling.The strength of the coupling between each oscillator and the environment is represented by ε. λ is the strength of coupling that determines the interaction between oscillators.P represents the count of neighboring oscillators in each direction that are coupled with a given oscillator in a ring network illustrated in Figure 3(a).The oscillators in the system are connected either globally or locally, depending on the value of P. When P=N/2, which corresponds to R =1/2, oscillators are globally connected (all-to-all coupling).Conversely, when P = 1, which corresponds to R=1/N, are locally connected.For value R between (1/N) and (1/2), the oscillators have a nonlocal coupling.In coupled oscillators, the mean amplitude of all oscillator is quantified by the order parameter A(λ) and to distinguishes the various death state with two additional order parameters mean =<|<x i > t |> i and variance ρ = 2 (<x i > t ).All the oscillator settle at either homogeneous steady state (HSS) ( >0 and ρ=0), or inhomogeneous steady state (IHSS) ( , ρ>0) for oscillation death.The numerical path of oscillation suppression is first evaluated by setting the network size to N = 100 and calculate A(λ) in forward and backward continuance.During forward calculation, the order parameter exhibited a sudden and discontinuous decrease, indicating a shift from oscillatory to non-oscillatory behavior.As the coupling strength continued to decrease during backward continuation, A(λ) jumped abruptly from zero to a finite value, signaling the resumption of oscillations from a non-oscillatory state in the coupled system.Notably, the forward and backward transition points (shown as λ f and λ b , respectively) occur at dissimilar values of λ, indicating the presence of hysteresis in the continuation diagram shown in Fig. 3(b).To explore the impact of the network on transition, we analyzed the parameter space (λ -R) when ε is set to 1.5, utilizing forward and backward calculation of λ.This analysis is depicted in Fig. 3(c).It illustrates that when the coupling strength R is set to 0.01 (i.e., P = 1), the systems consistently exhibit oscillatory behavior (OS).However, when R exceeds 0.08 and nonlocal coupling is introduced, all oscillators converge to a homogeneous steady state (HSS).The range of hysteresis observed in the parameter space is heavily influenced by the value of R. The area of the hysteresis region decreases as the forward transition point λ f is shifted, and when the coupled systems possess all-to-all coupling (i.e., at R=0.5), it nearly vanishes.To study how environmental coupling impacts this model, we analyzed a parameter plane (λ-ε) for R = 0.2 by employing forward and backward progression as a function of λ in Fig. 3(d).Observations indicate that the decisive value of environmental coupling makes a significant role in the first-order transition from the oscillating to dying state in coupled oscillators.This transition occurs after the critical value of ε.While, backward critical point shifts with ε, and forward critical point remains constant.Additionally, as the strength of coupling increases, the hysteresis region in parameter space expands.

Conclusion
We provide evidence that explosive deaths are possible in a range of dynamical systems, including networks of directly and indirectly connected oscillators.Such a shift from oscillation state to homogeneous or inhomogeneous steady-state is abrupt and irreversible.The underlying mechanism of this explosive death, which is essentially the same for coupled chaotic oscillators and coupled limit-cycle oscillators, is also explained.A hysteresis region with coexisting oscillation and death state is created by two transition points that occur at separate values and is evocative of a first order transition.Several biological and chemical systems exhibit coexisting oscillatory and steady states.In addition to explosive death, the observable concept of extinction and triggering in biological networks can be explained by the bistability of fluctuation and abeyance in these systems.Consequently, our study of explosive death is expected to not only provide further insight into various dynamical systems, but also a more profound comprehension of the mechanisms that give rise to explosive transitions in complex networks.

Fig. 1 :
Fig.1: (Color online) The A(k) as a function of k for forward (circle solid line) and backward (circle dash line) continuations for (a) b = 0.8, (b) b = 1, and (c) b = 2 for coupled VdP oscillators.The parameter planes display several dynamical areas of the system as given by Eq. (1).The dynamical regimes described above are observed within a specific range of parameters in both (d) the (b -k) phase plane, with Q fixed at 0.5, and (e) the (Q -k) phase plane, with b fixed at 3. These results were obtained for a network of 100 oscillators[20].

Fig. 3 :
Fig. 3: (a) A visual representation of a ring network made up of connected Rössler oscillators in the context of a shared environment.The oscillators are represented by black spheres, and the environment is represented by an orange sphere.(b) The graph shows forward (blue) and backward (red) iterations of A(λ) at P = 8 as the strength λ is varied.The different dynamical regimes are shown in the phase space (c) λ − R at ε = 1.5 and (d) λ − ε at P = 20, where regions labeled as OS, HA, and HSS [23].