| The chemical reaction systems with positive and negative feedbacks, which arefar away from the thermodynamic equilibrium state, can generate chemicaloscillations. Various self-organizing spatiotemporal patterns can emerge in thesenonlinear chemical systems by the coupling with diffusion, convection, transfer ofmass and heat, ion migration in electromagnetic field or other types of transport. Dueto the existence of more than two feedback loops coupled reciprocally, thereaction-diffusion system with multi-feedbacks can show more complex structuresand be closer to the real chemical reaction system. In present paper, a three-variablemodel with multi-feedback is adopted for the numerical simulation ofreaction-diffusion system. A series of spatiotemporal patterns, which are particular tothe multi-feedback systems, are observed. And multiple analyzes methods are appliedto systematically investigated these complex patterns for their generation andevolution. At last, some universal laws of the spatiotemporal patterns inmulti-feedback systems are obtained.The three-variabledynamical model applied in this paper contains two positivefeedbacks and one negative feedback, which can constitute two feedback loops. Theintersections of these two coupled feedback loops form the attractors of thehomogenous oscillations. That is why the HP model can show plentiful homogenousoscillatory activates by different dynamical bifurcations, such as the mixed-modeoscillations (11,12,…1n) via Fold-Hopf bifurcations and the period-adding oscillations(period-2, period-4,…) via period-doubling bifurcations.Both one-dimensional and two-dimensional spatial systems are adopted for thereaction-diffusion numerical simulations of the multi-feedback models. A series ofcomplexly but regularly spatiotemporal self-organizing structures can be observed,including the stable multi-armed patterns, the line-defect spiral waves and the mostinterestingly amplitude-modulated spirals with superstructures which are generated inthe mixed-mode oscillatory media. These modulations of wave amplitude are causedby the damping of diffusion which throws off the local oscillations of patterns fromthe homogenous trajectories and results in the transition zones between1n-1and1nmixed-mode oscillations in reaction-diffusion media. The trajectories of localoscillations in these transition zones periodically alternate between1n-1and1nhomogenous dynamical trajectories, which form quasiperiodic oscillatory activity.This new oscillatory activity of local kinetics can lead to the amplitude modulation of patterns in mixed-mode oscillatory media and constitute the superstructures. Itdemonstrates the mixed-mode oscillations of multi-feedback systems and thediffusion-induced instability are two requirements of the modulated patterns withsuperstructures. And a geometrical model contained two Archimedes spirals isdesigned to reproduce the periodical modulations of wave amplitude. Thisgeometrical model illustrates the generation mechanism of the superstructures, suchas overtarget and superspiral on the basic waves.For special sets of diffusion for different components, the reaction-diffusionsystem with multi-feedback in mixed-mode oscillatory media also can generate theChimera spirals with coexistence of incoherent spiral cores and coherent spiral arms.Different with the Chimera patterns in nonlocally coupling system, the incoherentregions of Chimera spirals in reaction-diffusion system are steady discontinuousstructures which are constituted by the spatial fixed point of the reaction-diffusionmodel. The oscillation quenching of the fast limit cycle oscillatory activities inmixed-mode oscillations lead to these new fixed points, which are spatiotemporaldefects in space. Both the coupling between different feedback loops in system andthe unbalanced diffusions of three variables are requirements of the Chimera patternsin multi-feedback systems. On the other hand, the transition between1n-1and1nmixed-mode oscillations in local kinetics had been discovered to visibly influence thestability of the steady incoherent regions in Chimera patterns. |
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