Research paperSpiral wave chimeras in reaction-diffusion systems: Phenomenon, mechanism and transitions
Introduction
An intriguing phenomenon observed in systems of coupled identical oscillators is the coexistence of coherent and incoherent regions in the space, knowing as the chimera states [1], [2], [3], [4]. This counterintuitive dynamical behavior is discovered first by Kuramoto and Battogtokh [1], and is named later as “chimera state” for its analogy to the monster in Greek mythology which owns lion’s head, goat’s body, and serpent’s tail [2]. Since its discovery, chimera state has inspired extensive theoretical and experimental studies during the past two decades, with the systems investigated ranging from physical to chemical and to biological systems. In theoretical studies, chimera states have been observed in a variety of modeling systems, including networks of coupled identical oscillators (e.g., phase oscillators [1], [2], time-delayed oscillators [5], [6], chaotic maps [7], limit-cycle oscillators [8], excitable oscillators [9], [10] and bistable oscillators [11]), networks of coupled non-identical oscillators [12], [13], [14] and even quantum systems [15], [16]. In experimental studies, chimera states have been generated successfully in coupled systems such as optical oscillators [17], chemical oscillators [18], [19], mechanical oscillators [20], electronic oscillators [21], electrochemical systems [22] and lasers [23]. With these studies, the conditions for generating chimera states as adopted in the seminal works have been largely relaxed [24], [25], [26], [27], [28], [29], and the concept of chimera states has been largely broadened and generalized [30], [31], [32], [33], [34], [35], [36]. For instance, instead of nonlocal couplings (which has been regarded as a necessary condition for generating chimera states), recent studies show that chimera states can also be generated in systems with global [24], [25] or local couplings [27], [28], [32], [34], [37], [38], [39], [40]. Meanwhile, the concept of chimera states has been largely generalized and a variety of chimera-like states have been reported, e.g., clustered chimeras [5], [41], amplitude and amplitude mediated chimeras [34], [42], alternating chimeras [43], chimera death [31], [44], spiral wave chimeras [35], [36], switching chimeras [45] and traveling chimeras [37], [46]. In addition, besides regular networks, chimera-like states have been also reported and studied in networks of complex structures [47], [48], [49], [50], [51], [52], [53], [54]. In particular, chimera-like states have been observed in complex network of coupled neurons [51], [52], [53], and are regarded as having important implications to the neuronal functions, saying, for example, the unihemispheric slow-wave sleep (USWS) of some aquatic mammals (e.g. dolphins and whales) and birds [55], in which one half of the brain is in sleep while the other part of the brain remains awake.
Whereas chimera states are normally generated in one-dimensional systems, recent studies show that sophisticated chimera-like patterns can be also generated in higher dimensional systems [35], [36], [56], [57]. One example is the spiral wave chimera (SWC) [35], [36], which combines the features of spiral waves and chimera states, and is typically observed in two-dimensional systems of nonlocally coupled oscillators. Spiral waves are a special type of patterns observed in many realistic spatiotemporal systems [58], including Belousov–Zhabotinsky (BZ) reaction [59], the catalytic oxidation of CO on Pt [60], Rayleigh–Bénard systems [61], aggregating slime-mould cells [62] and neuron networks [63]. In particular, spiral waves have been observed in cardiac tissue [64], and are believed to be one of the key reasons causing cardiac arrhythmias. In the classical form, a spiral wave consists of a spiral tip (defined as the point of phase singularity, i.e., the topological defect) and spiral arms (waves) and, during the process of system evolution, the spiral arms are rotating around the tip and propagating outwards from the tip to the peripheral region. As the system parameter changes, complicated spiral waves can be generated, including meandering spiral waves and spiral turbulence [65], [66], [67], [68]. Different from the classical spiral waves whose dynamics is governed the by a singular point (the tip), in SWC the system dynamics is governed by a core constituted by a group of desynchronized oscillators and occupies a small, circular region in the space. Interestingly, it is shown that despite of the incoherent inner core, spiral arms can be propagating stably in the outer region. For the model of nonlocally coupled phase oscillators, an analytical description of SWC has been given in Ref. [36] and, by the perturbation theory, both the rotation speed of the spiral arms and the size of the asynchronous core can be predicted. Since the discovery of Shima and Kuramoto [35] in nonlocally coupled periodic oscillators [35], SWC has received growing interest in the field of nonlinear science in recent years, particularly for researchers working on pattern formations in reaction-diffusion (RD) systems [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81]. In theoretical studies, SWCs in nonlocally coupled chaotic oscillators have been reported in Ref. [70], in which it is found that SWCs are characterized by the presence of synchronization defect lines, along which the dynamics follows a periodic behavior different from that of the bulk; SWCs in locally coupled oscillators have been reported in Refs. [71], [74], [76]; the stability of SWCs has been analyzed in Ref. [77]. Experimental verification of SWCs has been given in Ref. [82], in which a large-size two-dimensional array of nonlocally coupled Belousov–Zhabotinsky (BZ) chemical oscillators are employed and some new dynamical features of SWCs are revealed, including the erratic motion of the asynchronous spiral core, the growth and splitting of the cores, and the transition from SWCs to incoherent states. Despite the progresses made, the mechanisms and properties of SWCs remain elusive and many questions remain not clear, e.g., the roles of the phase-lag parameter in generating SWCs [36], the transitions from SWCs to other states in the parameter space [82], and the observability of SWCs in systems of locally coupled oscillators [71], [74].
For experimental physicists and chemists, a question of particular interest is whether SWCs can be generated in the general RD systems in which the dynamical elements are locally coupled through diffusions. Whereas results based on numerical simulations indicate that SWCs could be generated in locally coupled systems [71], [76], the models employed in these studies seems somewhat artificial and are difficult to be realized in experiments. The experiment conducted by Totz et al. [82], whereas is able to generate SWCs successfully, relies on the non-physical, nonlocal couplings that are realized through computer interface. As such, from the point of view of experimental studies, an urgent question to be answered is whether SWCs can be observed in the general RD systems possessing local, diffusive couplings. Should the answer be positive, the following-up questions are: (1) What are the properties of the SWCs? (2) How the SWCs are destabilized and transited to other states as the system parameters vary? and (3) Can the theoretical models be realized in experiments? In the present work, we attempt to address these questions by investigating the dynamics of an experimentally feasible RD system of local couplings. We are able to demonstrate that, even though the system elements are coupled locally, stable SWCs can still be generated in a wide region in the parameter space. We conduct a detailed numerical analysis on the properties of SWCs, and also the transitions of SWCs to other states in the parameter space. It is found that, while the SWCs share the properties of the conventional SWCs as observed in nonlocally coupled systems, they do possess some unique features, including the presence of SWCs in partial variables, the destabilization scenarios of SWCs, and the newly revealed phenomenon of shadowed spirals. In particular, in shadowed spirals, regular spirals are emerged on top of the desynchronization background, which manifests from a new viewpoint the coexistence of coherence and incoherence in spatiotemporal systems, generalizing thus the concept of chimera states. Furthermore, treating the system as an ensemble of oscillators coupled through a common medium, we conduct a phenomenological analysis on the formation of SWCs, which provides insights on the mechanism of SWCs.
The rest of the paper is organized as follows. In Section 2, we will present the model of a three-component FitzHugh–Nagumo-type RD system, describe the numerical methods used in simulations, and give an analysis on the bifurcation diagram of the local dynamics. In Section 3, we will demonstrate the typical SWC states observed in simulations and, by the conventional approaches, characterize the properties of SWCs. In Section 4, we will propose the phenomenological theory, based on which the underlying mechanism of SWCs will be explored. In Section 5, we will study the transition behaviors of SWCs in the parameter space, in which the two destabilization scenarios, namely core breakup and core expansion, will be discussed and the new phenomenon of shadowed spirals will be presented. Candidate experiments for verifying the theoretical findings will be given in Section 6, together with discussions and conclusion.
Section snippets
Model and numerical methods
Our model of locally coupled RD system reads [74], [83], [84],which describe the dynamics of the concentrations of three chemical reactants, and . This three-component RD system consists of a FitzHugh–Nagumo (FHN) kernel (consisting of and ) coupled to the third component and has been used in literature to investigate pattern formations in BZ systems dispersed in a water-in-oil Aerosol OT (AOT) microemulsion (BZ-AOT
Spiral wave chimeras and properties
Setting and we plot in Fig. 2(a) a snapshot of the component taken around . We see that the whole space is occupied by a single spiral wave centered at (0,0). As time increases, the spiral wave is rotating inwardly (antispiral), i.e., the spiral arms are moving towards the center, with the angular speed . A zoom-in plot of the core region is plotted in Fig. 2(b). We see that, encircled by the spiral arms, a circular region consisting of a group of disordered,
Mechanism analysis
The fact that SWCs can be generated in locally coupled RD systems seems contradictory to the existing studies on spiral waves, as it is well known that the presence of diffusion in RD systems will lead to a smooth distribution of the reactants in space (except the point at the spiral tip) [58]. The key to generating SWCs in our model of locally coupled RD systems lies in the special scheme of single-component diffusion, i.e., diffusion exists only for the component while are absent for
Transitions from spiral wave chimeras to other states
By varying the system parameters, the system may transit from SWC to other states. In the current study, we focus on the transitions of SWCs to other states with respect to the variations of and . As discussed in Section 2, plays as the bifurcation parameter of FHN oscillator and characterizes the reaction rate of the diffusive component . We thus expect that by varying and rich dynamics could be observed. In what follows, we will present two typical scenarios governing the
Discussions and conclusion
Whereas chimera-like patterns have been reported in a variety of systems in literature, most of the studies rely on the adoption of nonlocal couplings. As nonlocal couplings are absent in typical RD systems in which elements are interacted through local diffusions, it is commonly believed that chimera-like pattens such as SWC can not be observed in typical RD systems. This believing is validated further by experiments in Ref. [82]. There, to generate SWC in chemical BZ oscillators, nonlocal
CRediT authorship contribution statement
Bing-Wei Li: Conceptualization, Data curation, Visualization, Funding acquisition, Investigation, Methodology, Writing - original draft, Writing - review & editing. Yuan He: Investigation, Data curation, Visualization, Formal analysis, Validation. Ling-Dong Li: Investigation, Data curation, Formal analysis, Software. Lei Yang: Investigation, Data curation, Formal analysis, Validation. Xingang Wang: Conceptualization, Funding acquisition, Data curation, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by Natural Science Foundation of Zhejiang Province under grant no. LY16A050003. XGW was supported by the National Natural Science Foundation of China under grant no. 11875182.
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