Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model

Zimmermann, Martín; Firle, Sascha O.; Natiello, Mario A.; Hildebrand, Michael E.; Eiswirth, M.; Bär, Markus; Bangia, Anil K.; Kevrekidis, Ioannis G.

doi:10.1016/s0167-2789(97)00112-7

Cited by 73 publications

(47 citation statements)

References 34 publications

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“…Continuous spectrum crossing has been associated with the onset of ''chemical turbulence'' in models of CO oxidation [28]; this motivates future careful investigation of the ''detachment dynamics'' of the pulse beyond the turning point, as well as of the dynamics in the neighborhood of the unstable pulses. Notice also on the unstable branch the (visual) distance between the steadily dragged pulse and the (moving) temperature spot.…”

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confidence: 98%

Gentle Dragging of Reaction Waves

et al. 2003

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Using a recently realized ''addressable catalyst surface'' [Science 294, 134 (2001)] we study the interaction of chemical reaction waves with prescribed spatiotemporal fields. In particular, we study how a traveling chemical pulse is ''dragged'' by a localized, moving temperature heterogeneity as a function of its intensity and speed. The acceleration and eventual ''detachment'' of the wave from the heterogeneity is also explored through simulation and stability analysis. DOI: 10.1103/PhysRevLett.90.018302 PACS numbers: 82.40.Bj, 05.45.-a, 82.40.Np, 82.45.Jn As the elements of spontaneous pattern formation are progressively understood through theory, experimentation, and scientific computation [1], ways to purposefully interact with the coherent structures (pulses and fronts) that constitute the building blocks of spatiotemporal patterns are becoming the focus of extensive research [2,3]. In particular, the stabilization and control of various patterns [3-9] through local, nonlocal, or global feedback, and pattern formation in media with designed heterogeneities [10 -12] are the source of novel insights for spatiotemporal dynamics.To explore such phenomena, we have recently constructed an ''addressable catalyst'': A focused laser beam, manipulated through computer-controlled mirrors and capable of ''writing'' spatiotemporal temperature heterogeneity patterns on a metal single crystal catalyst. The loop between this actuation and sensing (both resolved in space and time) through nonintrusive microscopies is then closed through the computer or the experimentalist herself/himself in real time. Our model system is the low-pressure catalytic oxidation of CO on Pt(110), a reaction exhibiting well-documented spatiotemporal patterns [13][14][15][16], and whose macroscopic modeling has reached an advanced level [17][18][19][20].In this Letter we study the interaction of reactive pulses with a single, spatially coherent but temporally mobile heterogeneity. In particular, we use a temperature heterogeneity, localized in space and steadily moving in time, to ''drag'' spontaneously isothermally forming reactive pulses and fronts with speeds differing from their natural speed. We examine the shapes acquired by these dragged waves and their limits of stability (that is, the range of dragging speeds for which they can exist). Through computer-aided analysis we examine the nature of the detachment instability, marking the loss of the ability of the heterogeneity to drag a pulse in 1D or 2D. Two possible paths to instability (depending on the linearized spectrum crossing) are detected. We explore the postdetachment transient dynamics and the interactions of a pulse with successive elements of a 1D periodic, constant speed array of identical heterogeneities. The theme of pulse dragging is currently also being explored in a variety of physically relevant Hamiltonian systems, or weakly perturbed dissipative variations thereof (targeted transfer of pulses in optical media or the motion/displacement of harmonic traps or optical l...

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Gentle Dragging of Reaction Waves

et al. 2003

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“…Among the available scenarios, few of them consider the framework of stochastic partial differential equations (SPDE) [5][6][7] to describe STC. A successful example is, however, the mapping of the Kuramoto-Shivashinsky equation (describing a STC regime named phase turbulence) to the stochastic model of surface growth known as KardarParisi-Zhang equation [8].A particular instance of STC is a regime called spatiotemporal intermittency (STI) which is present in a large variety of systems [9][10][11][12]. Generally speaking, this regime is a chaotic spatiotemporal evolution (the turbulent phase) irregularly and continuously interrupted by the spontaneous formation of domains with a wide range of sizes and lifetimes, where the behavior is ordered (laminar).…”

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Stochastic Spatiotemporal Intermittency and Noise-Induced Transition to an Absorbing Phase

et al. 2000

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We introduce a stochastic partial differential equation capable of reproducing the main features of spatiotemporal intermittency (STI). Additionally the model displays a noise induced transition from laminarity to the STI regime. We show by numerical simulations and a mean-field analysis that for high noise intensities the system globally evolves to a uniform absorbing phase, while for noise intensities below a critical value spatiotemporal intermittence dominates. A quantitative computation of the loci of this transition in the relevant parameter space is presented.Spatiotemporal chaos (STC) is a complex behavior, common to many spatially extended nonlinear dynamical systems [1][2][3][4]. This behavior is characterized by a combination of chaotic time evolution and spatial incoherence made evident by correlations decaying both in space and time. In spite of considerable theoretical and experimental effort devoted to give a precise definition of STC and its different regimes, the present status is still unsatisfactory. A possible strategy to make progress in the understanding of STC is to investigate scenarios based on simple models, with few controlled ingredients, that reproduce the spatio-temporal structures under study. Such models are instrumental in searching for generic mechanisms leading to such complex behavior. Among the available scenarios, few of them consider the framework of stochastic partial differential equations (SPDE) [5][6][7] to describe STC. A successful example is, however, the mapping of the Kuramoto-Shivashinsky equation (describing a STC regime named phase turbulence) to the stochastic model of surface growth known as KardarParisi-Zhang equation [8].A particular instance of STC is a regime called spatiotemporal intermittency (STI) which is present in a large variety of systems [9][10][11][12]. Generally speaking, this regime is a chaotic spatiotemporal evolution (the turbulent phase) irregularly and continuously interrupted by the spontaneous formation of domains with a wide range of sizes and lifetimes, where the behavior is ordered (laminar). The borders of the laminar regions propagate as fronts and eventually cause the collapse of the corresponding region into the turbulent background. There are strong indications that the STI regime has many features in common with phenomena of probabilistic nature. For example, it appears in some systems that critical exponents at the onset of STI are in the universality class of directed percolation [13]. STI has also been related to nucleation [11], another process associated with stochastic fluctuations. However, no description of STI in terms of SPDEs has been put forward so far.The purpose of this Letter is to introduce a model, entirely based on a simple SPDE, that describes the main features of STI, and report the existence of a noise induced transition from laminarity to STI. The laminar phase is associated to an equilibrium state called absorbing in the SPDE parlance. The role of the turbulent phase is played by a strongly fluctuating sta...

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“…The essentials of this paradigm were first proposed by Turing in 1952 in a seminal article that set up the chemical basis of morphogenesis (Turing, 1952). Decades later, experimental work in chemical systems such as the Belousov-Zhabotinskii or catalytic driven surface reactions confirmed this hypothesis by being able to predict a rich variety of stationary as well as oscillatory spatial patterns as discussed in, for instance, Zimmermann et al (1997), Beaumont et al (1998), Lebiedz and Brandt-Pollmann (2003) or Lebiedz and Maurer (2004).…”

Section: Introductionmentioning

confidence: 99%

“…This paradigm has also been extensively employed to understand the routes that drive these systems to instability. In this way, dynamic analysis of diffusion-reaction systems and in particular the FHN system, has been the subject of intensive research, especially in what refers to the detection of instability conditions and bifurcation analysis leading to moving fronts, spiral waves and pattern formation (Zimmermann et al, 1997;Beaumont et al, 1998;Gear et al, 2002;Sweers and Troy, 2003;Bouzat and Wio, 2003).…”

Section: Introductionmentioning

confidence: 99%

Stabilization of inhomogeneous patterns in a diffusion–reaction system under structural and parametric uncertainties

Vilas¹,

García²,

Banga³

et al. 2006

Journal of Theoretical Biology

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Many phenomena such as neuron firing in the brain, the travelling waves which produce the heartbeat, arrythmia and fibrillation in the heart, catalytic reactions or cellular organization activities, among others, can be described by a unifying paradigm based on a class of nonlinear reaction-diffusion mechanisms. The FitzHughNagumo (FHN) model is a simplified version of such class which is known to capture most of the qualitative dynamic features found in the spatiotemporal signals. In this paper, we take advantage of the dissipative nature of diffusion-reaction systems and results in finite dimensional nonlinear control theory to develop a class of nonlinear feed-back controllers which is able to ensure stabilization of moving fronts for the FHN system, despite structural or parametric uncertainty. In the context of heart or neuron activity, this class of control laws is expected to prevent cardiac or neurological disorders connected with spatiotemporal wave disruptions. In the same way, biochemical or cellular organization related with certain functional aspects of life could also be influenced or controlled by the same feed-back logic. The stability and robustness properties of the controller will be proved theoretically and illustrated on simulation experiments.

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Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model

Cited by 73 publications

References 34 publications

Gentle Dragging of Reaction Waves

Gentle Dragging of Reaction Waves

Stochastic Spatiotemporal Intermittency and Noise-Induced Transition to an Absorbing Phase

Stabilization of inhomogeneous patterns in a diffusion–reaction system under structural and parametric uncertainties

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