Showing papers in "Chaos in 2002"
TL;DR: Cascading failure in a simplified transmission system model as load power demand is increased is examined and it is found that operation near critical points can produce power law tails in the blackout size probability distribution similar to those observed.
Abstract: From the analysis of a 15-year time series of North American electric power transmission system blackouts, we have found that the frequency distribution of the blackout sizes does not decrease exponentially with the size of the blackout, but rather has a power law tail. The existence of a power tail suggests that the North American power system has been operated near a critical point. To see if this is possible, here we explore the critical points of a simple blackout model that incorporates circuit equations and a process through which outages of lines may happen. In spite of the simplifications, this is a complex problem. Understanding the different transition points and the characteristic properties of the distribution function of the blackouts near these points offers a first step in devising a dynamical model for the power transmission systems.
554 citations
TL;DR: A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue as discussed by the authors.
Abstract: It has become widely accepted that the most dangerous cardiac arrhythmias are due to reentrant waves, i.e., electrical wave(s) that recirculate repeatedly throughout the tissue at a higher frequency than the waves produced by the heart's natural pacemaker (sinoatrial node). However, the complicated structure of cardiac tissue, as well as the complex ionic currents in the cell, have made it extremely difficult to pinpoint the detailed dynamics of these life-threatening reentrant arrhythmias. A simplified ionic model of the cardiac action potential (AP), which can be fitted to a wide variety of experimentally and numerically obtained mesoscopic characteristics of cardiac tissue such as AP shape and restitution of AP duration and conduction velocity, is used to explain many different mechanisms of spiral wave breakup which in principle can occur in cardiac tissue. Some, but not all, of these mechanisms have been observed before using other models; therefore, the purpose of this paper is to demonstrate them using just one framework model and to explain the different parameter regimes or physiological properties necessary for each mechanism (such as high or low excitability, corresponding to normal or ischemic tissue, spiral tip trajectory types, and tissue structures such as rotational anisotropy and periodic boundary conditions). Each mechanism is compared with data from other ionic models or experiments to illustrate that they are not model-specific phenomena. Movies showing all the breakup mechanisms are available at http://arrhythmia.hofstra.edu/breakup and at ftp://ftp.aip.org/epaps/chaos/E-CHAOEH-12-039203/ INDEX.html. The fact that many different breakup mechanisms exist has important implications for antiarrhythmic drug design and for comparisons of fibrillation experiments using different species, electromechanical uncoupling drugs, and initiation protocols. (c) 2002 American Institute of Physics.
512 citations
TL;DR: The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates and the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinate is introduced.
Abstract: The necessary and sufficient conditions for Lagrangian hyperbolicity recently derived in the literature are reviewed in the light of older concepts of effective local rotation in strain coordinates. In particular, we introduce the simple interpretation of the necessary condition as a constraint on the local angular displacement in strain coordinates. These mathematically rigorous conditions are applied to the winter stratospheric circulation of the southern hemisphere, using analyzed wind data from the European Center for Medium-Range Weather Forecasts. Our results demonstrate that the sufficient condition is too strong and the necessary condition is too weak, so that both conditions fail to identify hyperbolic lines in the stratosphere. However a phenomenological, nonrigorous, criterion based on the necessary condition reveals the hyperbolic structure of the flow. Another (still nonrigorous) alternative is the finite-size Lyapunov exponent (FSLE) which is shown to produce good candidates for hyperbolic lines. In addition, we also tested the sufficient condition for Lagrangian ellipticity and found that it is too weak to detect elliptic coherent structures (ECS) in the stratosphere, of which the polar vortex is an obvious candidate. Yet, the FSLE method reveals a clear ECS-like barrier to mixing along the polar vortex edge. Further theoretical advancement is needed to explain the apparent success of nonrigorous methods, such as the FSLE approach, so as to achieve a sound kinematic understanding of chaotic mixing in the winter stratosphere and other geophysical flows.
135 citations
TL;DR: A new chaos generator, a switching piecewise-linear controller, which can create chaos from a three-dimensional linear system within a wide range of parameter values is introduced.
Abstract: This paper introduces a new chaos generator, a switching piecewise-linear controller, which can create chaos from a three-dimensional linear system within a wide range of parameter values. Basic dynamical behaviors of the chaotic controlled system are investigated in some detail.
126 citations
TL;DR: It is found that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum.
Abstract: We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system’s variables are each car’s velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle–Takens–Newhouse scenario (limit cycles–two-tori–three-tori–chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum.
125 citations
TL;DR: A simplified model of human atria is developed that uses an ionic-based membrane model and whose geometry is derived from a segmented magnetic resonance imaging data set, but it does not take into account anisotropy.
Abstract: The maintenance of multiple wavelets appears to be a consistent feature of atrial fibrillation (AF). In this paper, we investigate possible mechanisms of initiation and perpetuation of multiple wavelets in a computer model of AF. We developed a simplified model of human atria that uses an ionic-based membrane model and whose geometry is derived from a segmented magnetic resonance imaging data set. The three-dimensional surface has a realistic size and includes obstacles corresponding to the location of major vessels and valves, but it does not take into account anisotropy. The main advantage of this approach is its ability to simulate long duration arrhythmias (up to 40 s). Clinically relevant initiation protocols, such as single-site burst pacing, were used. The dynamics of simulated AF were investigated in models with different action potential durations and restitution properties, controlled by the conductance of the slow inward current in a modified Luo–Rudy model. The simulation studies show that (1) single-site burst pacing protocol can be used to induce wave breaks even in tissue with uniform membrane properties, (2) the restitution-based wave breaks in an atrial model with realistic size and conduction velocities are transient, and (3) a significant reduction in action potential duration (even with apparently flat restitution) increases the duration of AF.
125 citations
TL;DR: The horizontal stirring properties of the flow in a region of the East Australian Current are calculated and recent theoretical results on the multifractal spectra of advected reactive tracers are applied to an analysis of a sea-surface temperature image of the study region.
Abstract: The horizontal stirring properties of the flow in a region of the East Australian Current are calculated. A surface velocity field derived from remotely sensed data, using the maximum cross correlation method, is integrated to derive the distribution of the finite-time Lyapunov exponents. For the region studied (between latitudes 36°S and 41°S and longitudes 150°E and 156°E) the mean Lyapunov exponent during 1997 is estimated to be λ∞=4×10−7 s−1. This is in close agreement with the few other measurements of stirring rates in the surface ocean which are available. Recent theoretical results on the multifractal spectra of advected reactive tracers are applied to an analysis of a sea-surface temperature image of the study region. The spatial pattern seen in the image compares well with the pattern seen in an advected tracer with a first-order response to changes in surface forcing. The response timescale is estimated to be 20 days.
123 citations
TL;DR: This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows and the spatial distribution of finite time LyAPunov exponents is discussed in relation to stirring properties.
Abstract: This paper discusses the application of Lyapunov theory in chaotic systems to the dynamics of tracer gradients in two-dimensional flows. The Lyapunov theory indicates that more attention should be given to the Lyapunov vector orientation. Moreover, the properties of Lyapunov vectors and exponents are explained in light of recent results on tracer gradients dynamics. Differences between the different Lyapunov vectors can be interpreted in terms of competition between the effects of effective rotation and strain. Also, the differences between backward and forward vectors give information on the local reversibility of the tracer gradient dynamics. A numerical simulation of two-dimensional turbulence serves to highlight these points and the spatial distribution of finite time Lyapunov exponents is also discussed in relation to stirring properties.
116 citations
TL;DR: This work investigates theoretically whether control can suppress alternans both temporally and spatially in homogeneous tissue paced at a single site and reveals that control failure above a critical cable length is caused by the formation of standing wave patterns of alternans that are remarkably analogous to sound harmonics in an open pipe.
Abstract: Electrical alternans are believed to be linked to the onset of life-threatening ventricular arrhythmias and sudden cardiac death. Recent studies have shown that alternans can be suppressed temporally by dynamic feedback control of the pacing interval. Here we investigate theoretically whether control can suppress alternans both temporally and spatially in homogeneous tissue paced at a single site. We first carry out ionic model simulations in a one-dimensional cable geometry which show that control is only effective up to a maximum cable length that decreases sharply away from the alternans bifurcation point. We then explain this finding by a linear stability analysis of an amplitude equation that describes the spatiotemporal evolution of alternans. This analysis reveals that control failure above a critical cable length is caused by the formation of standing wave patterns of alternans that are eigenfunctions of a forced Helmholtz equation, and therefore remarkably analogous to sound harmonics in an open ...
100 citations
TL;DR: A theoretical analysis of nanoscale pore formation during anodization reveals its fundamental instability mechanism to be a field focusing phenomenon when perturbations on the minima of the two oxide interfaces are in phase.
Abstract: A theoretical analysis of nanoscale pore formation during anodization reveals its fundamental instability mechanism to be a field focusing phenomenon when perturbations on the minima of the two oxide interfaces are in phase. Lateral leakage of the layer potential at high wave number introduces a layer tension effect that balances the previous destabilizing effect to produce a long-wave instability and a selected pore separation that scales linearly with respect to voltage. At pH higher than 1.77, pores do not form due to a very thick barrier layer. A weakly nonlinear theory based on long-wave expansion of double free surface problem yields two coupled interface evolution equations that can be reduced to one without altering the dispersion relationship by assuming an equal and in-phase amplitude for the two interfaces. This interface evolution equation faithfully reproduces the initial pore ordering and their dynamics. A hodograph transformation technique is then used to determine the interior dimension of the well-developed pores in two dimensions. The ratio of pore diameter to pore separation is found to be a factor independent of voltage but varies with the pH of the electrolyte. Both the predicted pH range where pores are formed and the predicted pore dimensions are favorably compared to experimental data. (c) 2002 American Institute of Physics.
94 citations
TL;DR: It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system and the choice of the observable.
Abstract: When a dynamical system is investigated from a time series, one of the most challenging problems is to obtain a model that reproduces the underlying dynamics. Many papers have been devoted to this problem but very few have considered the influence of symmetries in the original system and the choice of the observable. Indeed, it is well known that there are usually some variables that provide a better representation of the underlying dynamics and, consequently, a global model can be obtained with less difficulties starting from such variables. This is connected to the problem of observing the dynamical system from a single time series. The roots of the nonequivalence between the dynamical variables will be investigated in a more systematic way using previously defined observability indices. It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system. As will be mentioned, symmetries and the choice of observables also has important consequences in other problems such as synchronization of nonlinear oscillators.
TL;DR: It is shown that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a "sech"-profile.
Abstract: We study the extended Korteweg–de Vries equation, that is, the usual Korteweg–de Vries equation but with the inclusion of an extra cubic nonlinear term, for the case when the coefficient of the cubic nonlinear term has an opposite polarity to that of the coefficient of the linear dispersive term. As this equation is integrable, the number and type of solitons formed can be determined from an appropriate spectral problem. For initial disturbances of small amplitude, the number and type of solitons generated is similar to the well-known situation for the Korteweg–de Vries equation. However, our interest here is in initial disturbances of larger amplitude, for which there is the possibility of the generation of large-amplitude “table-top” solitons as well as small-amplitude solitons similar to the solitons of the Korteweg–de Vries equation. For this case, and in contrast to some earlier results which assumed that an initial disturbance in the shape of a rectangular box would be typical, we show that the number and type of solitons formed depend crucially on the disturbance shape, and change drastically when the initial disturbance is changed from a rectangular box to a “sech”-profile.
TL;DR: The full answer to Kelvin's question is that the regular vortex n-gon is stable at n/=8 it is unstable, and the general theory of stationary motions of a dynamical system with symmetry group is presented.
Abstract: This paper is devoted to the Lord Kelvin's (1878) problem on stability of the stationary rotation of the system of n equal vortices located in the vertices of a regular n-gon. During the last decades this problem again became actual in connection with the investigation of point vortices in liquid helium and electron columns in plasma physics. This regime is described by the explicit solution of the Kirchhoff equations. The corresponding eigenvalue problem for the linearization matrix can be also decided explicitly. This was used in the works of Thomson (1883) and Havelock (1931) to obtain exhaustive results on the linear stability. Kurakin (1994) proved that for n /=8 it is unstable. We also present the general theory of stationary motions of a dynamical system with symmetry group. The definitions of stability and instability are necessary to modify in the specific case of stationary regimes. We do not assume that the system is conservative. Thus, the results can be applied not only to various stationary regimes of an ideal fluid flows but, for instance, also to motions of viscous fluids. (c) 2002 American Institute of Physics.
TL;DR: It is found that the restitution hypothesis, which depends only on action potential duration restitution, can be extended to a more general condition that includes dependence on conduction velocity restitution as well as two other parameters.
Abstract: As a simple model of reentry, we use a general FitzHugh–Nagumo model on a ring (in the singular limit) to build an understanding of the scope of the restitution hypothesis. It has already been shown that for a traveling pulse solution with a phase wave back, the restitution hypothesis gives the correct stability condition. We generalize this analysis to include the possibility of a pulse with a triggered wave back. Calculating the linear stability condition for such a system, we find that the restitution hypothesis, which depends only on action potential duration restitution, can be extended to a more general condition that includes dependence on conduction velocity restitution as well as two other parameters. This extension amounts to unfolding the original bifurcation described in the phase wave back case which was originally understood to be a degenerate bifurcation. In addition, we demonstrate that dependence of stability on the slope of the restitution curve can be significantly modified by the sensitivity to other parameters (including conduction velocity restitution). We provide an example in which the traveling pulse is stable despite a steep restitution curve.
TL;DR: It is demonstrated, as a sequel to the earlier paper of Deng, that at the singular limit zeta=0, a singular Shilnikov's saddle-focus homoclinic orbit can exist as the reproduction rate ratio epsilon of the top-predator over the predator is greater than a modest value epsilus(0).
Abstract: Assume that the reproduction rate ratio ζ of the predator over the prey is sufficiently small in a basic tri-trophic food chain model. This assumption translates the model into a singularly perturbed system of two time scales. It is demonstrated, as a sequel to the earlier paper of Deng [Chaos 11, 514–525 (2001)], that at the singular limit ζ=0, a singular Shilnikov’s saddle-focus homoclinic orbit can exist as the reproduction rate ratio e of the top-predator over the predator is greater than a modest value e0. The additional conditions under which such a singular orbit may occur are also explicitly given.
TL;DR: By using the LAEP theorem, a new application of the GLM, glm, and alpha-model results for Lagrangian averaged ideal magnetohydrodynamics is highlighted.
Abstract: This paper employs the Lagrangian-averaged Euler- Poincaretheorem to provide a bridge between the gener- alized Lagrangian mean NGLMO equations and the Euler- alpha closure equations. The former NGLMO equations result from an exact theory of nonlinear waves on a La- grangian mean flow and are not closed, while the latter NEuler-alphaO equations are closed via linearization that introduces a length scale a, which Taylor's hypothesis renders constant if it is initially so. The Euler-alpha clo- sure equations with an additional Navier-Stokes viscous dissipation are of interest from the point of view of large eddy simulation and the object of the present work is to show how they relate to a more general theory. To clarify this, we derive them from a small amplitude approxima- tion to the GLM equations, the glm equations, by apply- ing Taylor's hypothesis of frozen-in fluctuations. This derivation from first principles is general enough to in- clude more physics, as we shall illustrate by deriving an alpha model for Lagrangian-averaged ideal magnetohy- drodynamics.
TL;DR: Basic concepts of cardiac electrophysiology are introduced and the main clinical methods being used to treat arrhythmia are described and a brief summary of the main themes contained in the articles in this Focus Issue is provided.
Abstract: This paper serves as an introduction to the Focus Issue on mapping and control of complex cardiac arrhythmias. We first introduce basic concepts of cardiac electrophysiology and describe the main clinical methods being used to treat arrhythmia. We then provide a brief summary of the main themes contained in the articles in this Focus Issue. In recent years there have been important advances in the ability to map the spread of excitation in intact hearts and in laboratory settings. This work has been combined with simulations that use increasingly realistic geometry and physiology. Waves of excitation and contraction in the heart do not always propagate with constant velocity but are often subject to instabilities that may lead to fluctuations in velocity and cycle time. Such instabilities are often treated best in the context of simple one- or two-dimensional geometries. An understanding of the mechanisms of propagation and wave stability is leading to the implementation of different stimulation protocols in an effort to modify or eliminate abnormal rhythms.
TL;DR: The derivation of a generally applicable reaction-migration equation, which describes the coupling via an integral over the whole electrode area, is presented and the types of patterns occurring under different geometries can be rationalized.
Abstract: The spatial coupling in electrochemical systems is mediated by ion migration under the influence of the electric field. Since field effects spread very rapidly, every point of an electrode can communicate with every other one practically instantaneously through migration coupling. Based on mathematical potential theory we present the derivation of a generally applicable reaction–migration equation, which describes the coupling via an integral over the whole electrode area. The corresponding coupling function depends only on the geometry of the electrode setup and has been computed for commonly used electrode shapes (such as ring, disk, ribbon or rectangle). The pattern formation observed in electrochemical systems in the bistable, excitable and oscillatory regime can be reproduced in computer simulations, and the types of patterns occurring under different geometries can be rationalized.
TL;DR: A dynamical model of the chaotic hunting behavior of Clione is formulated based on physiological in vivo and in vitro experiments and hypothesize that the chaotic activity of the receptor neurons can drive the complex behavior ofclione observed during hunting.
Abstract: In the presence of prey, the marine mollusk Clione limacina exhibits search behavior, i.e., circular motions whose plane and radius change in a chaotic-like manner. We have formulated a dynamical model of the chaotic hunting behavior of Clione based on physiological in vivo and in vitro experiments. The model includes a description of the action of the cerebral hunting interneuron on the receptor neurons of the gravity sensory organ, the statocyst. A network of six receptor model neurons with Lotka-Volterra-type dynamics and nonsymmetric inhibitory interactions has no simple static attractors that correspond to winner take all phenomena. Instead, the winnerless competition induced by the hunting neuron displays hyperchaos with two positive Lyapunov exponents. The origin of the chaos is related to the interaction of two clusters of receptor neurons that are described with two heteroclinic loops in phase space. We hypothesize that the chaotic activity of the receptor neurons can drive the complex behavior of Clione observed during hunting. (c) 2002 American Institute of Physics.
TL;DR: By means of numerical simulations and theoretical arguments, the behavior of front speed is described as a function of the stirring intensity, U, and for slow reaction, the front propagates with a speed proportional to U(1/4), conversely for fast reaction the frontspeed is proportional toU(3/4).
Abstract: The problem of front propagation in a stirred medium is addressed in the case of cellular flows in three different regimes: slow reaction, fast reaction and geometrical optics limit. It is well known that a consequence of stirring is the enhancement of front speed with respect to the nonstirred case. By means of numerical simulations and theoretical arguments we describe the behavior of front speed as a function of the stirring intensity, U. For slow reaction, the front propagates with a speed proportional to U1/4, conversely for fast reaction the front speed is proportional to U3/4. In the geometrical optics limit, the front speed asymptotically behaves as U/ln U.
TL;DR: Low-amplitude, high-frequency overdrive pacing, in combination with calcium channel inhibitors (e.g., class II or class IV antiarrhythmic drugs), may be useful for eliminating fibrillation.
Abstract: Recent findings indicate that ventricular fibrillation might arise from spiral wave chaos. Our objective in this computational study was to investigate wave interactions in excitable media and to explore the feasibility of using overdrive pacing to suppress spiral wave chaos. This work is based on the finding that in excitable media, propagating waves with the highest excitation frequency eventually overtake all other waves. We analyzed the effects of low-amplitude, high-frequency pacing in one-dimensional and two-dimensional networks of coupled, excitable cells governed by the Luo–Rudy model. In the one-dimensional cardiac model, we found narrow high-frequency regions of 1:1 synchronization between the input stimulus and the system’s response. The frequencies in this region were higher than the intrinsic spiral wave frequency of cardiac tissue. When we paced the two-dimensional cardiac model with frequencies from this region, we found that spiral wave chaos could, in some cases, be suppressed. When we coupled the overdrive pacing with calcium channel blockers, we found that spiral wave chaos could be suppressed in all cases. These findings suggest that low-amplitude, high-frequency overdrive pacing, in combination with calcium channel inhibitors (e.g., class II or class IV antiarrhythmic drugs), may be useful for eliminating fibrillation.
TL;DR: The model has three main components: the velocity of a single bubble in this unstable flow regime, an envelope velocity, which describes collective excitations in the mixing region, and a merger process, which drives an inverse cascade, with a steady increase of bubble size.
Abstract: In this paper we formulate a model for the merger of bubbles at the edge of an unstable acceleration driven (Rayleigh-Taylor) mixing layer. Steady acceleration defines a self-similar mixing process, with a time-dependent inverse cascade of structures of increasing size. The time evolution is itself a renormalization group (RNG) evolution, and so the large time asymptotics define a RNG fixed point. We solve the model introduced here at this fixed point. The model predicts the growth rate of a Rayleigh-Taylor chaotic fluid mixing layer. The model has three main components: the velocity of a single bubble in this unstable flow regime, an envelope velocity, which describes collective excitations in the mixing region, and a merger process, which drives an inverse cascade, with a steady increase of bubble size. The present model differs from an earlier two-dimensional (2-D) merger model in several important ways. Beyond the extension of the model to three dimensions, the present model contains one phenomenological parameter, the variance of the bubble radii at fixed time. The model also predicts several experimental numbers: the bubble mixing rate, alpha(b)=h(b)/Agt(2) approximately 0.05-0.06, the mean bubble radius, and the bubble height separation at the time of merger. From these we also obtain the bubble height to the radius aspect ratio. Using the experimental results of Smeeton and Youngs (AWE Report No. O 35/87, 1987) to fix a value for the radius variance, we determine alpha(b) within the range of experimental uncertainty. We also obtain the experimental values for the bubble height to width aspect ratio in agreement with experimental values. (c) 2002 American Institute of Physics.
TL;DR: The complex geometry of the heart may be an important determinant of VF activation patterns and uneven surface curvature and nonstationary reentrant propagation worked in concert to produce wave front fragmentation and complex activation patterns.
Abstract: The role of the heart’s complex shape in causing the fragmentation of activation wave fronts characteristic of ventricular fibrillation (VF) has not been well studied. We used a finite element model of cardiac propagation capable of simulating functional reentry on curved two-dimensional surfaces to test the hypothesis that uneven surface curvature can cause local propagation block leading to proliferation of reentrant wave fronts. We found that when reentry was induced on a flat sheet, it rotated in a repeatable meander pattern without breaking up. However, when a model of the rabbit ventricles was formed from the same medium, reentrant wave fronts followed complex, nonrepeating trajectories. Local propagation block often occurred when wave fronts propagated across regions where the Gaussian curvature of the surface changed rapidly. This type of block did not occur every time wave fronts crossed such a region; rather, it only occurred when the wave front was very close behind the previous wave in the cycle and was therefore propagating into relatively inexcitable tissue. Close wave front spacing resulted from nonstationary reentrant propagation. Thus, uneven surface curvature and nonstationary reentrant propagation worked in concert to produce wave front fragmentation and complex activation patterns. None of the factors previously thought to be necessary for local propagation block (e.g., heterogeneous refractory period, steep action potential duration restitution) were present. We conclude that the complex geometry of the heart may be an important determinant of VF activation patterns.
TL;DR: This work discusses the development of exact reaction-diffusion equations (RDEs) describing mesoscale spatial pattern formation in surface reactions and their applications in atomistic lattice-gas models for surface reactions.
Abstract: Atomistic lattice-gas models for surface reactions can accurately describe spatial correlations and ordering in chemisorbed layers due to adspecies interactions or due to limited mobility of some adspecies. The primary challenge in such modeling is to describe spatiotemporal behavior in the physically relevant “hydrodynamic” regime of rapid diffusion of (at least some) reactant adspecies. For such models, we discuss the development of exact reaction-diffusion equations (RDEs) describing mesoscale spatial pattern formation in surface reactions. Formulation and implementation of these RDEs requires detailed analysis of chemical diffusion in mixed reactant adlayers, as well as development of novel hybrid and parallel simulation techniques.
TL;DR: The nonlinear evolution of the shelves is described in terms of exact solutions to the KdV-Whitham equations with natural boundary conditions for the Riemann invariants, which describe the generation of small "secondary" solitary waves in the trailing shelves.
Abstract: We study the long-time evolution of the trailing shelves that form behind solitary waves moving through an inhomogeneous medium, within the framework of the variable-coefficient Korteweg–de Vries equation. We show that the nonlinear evolution of the shelf leads typically to the generation of an undular bore and an expansion fan, which form apart but start to overlap and nonlinearly interact after a certain time interval. The interaction zone expands with time and asymptotically as time goes to infinity occupies the whole perturbed region. Its oscillatory structure strongly depends on the sign of the inhomogeneity gradient of the variable background medium. We describe the nonlinear evolution of the shelves in terms of exact solutions to the KdV–Whitham equations with natural boundary conditions for the Riemann invariants. These analytic solutions, in particular, describe the generation of small “secondary” solitary waves in the trailing shelves, a process observed earlier in various numerical simulations.
TL;DR: This paper suggests that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geometry of a recurrence plot of those data, providing not only a simple way to locate stable periodic orbits in chaotic time-series data, but also a potentially effective way to use aRecurrence plot to identify a dynamical system.
Abstract: A recurrence plot is a two-dimensional visualization technique for sequential data. These plots are useful in that they bring out correlations at all scales in a manner that is obvious to the human eye, but their rich geometric structure can make them hard to interpret. In this paper, we suggest that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geometry of a recurrence plot of those data. This provides not only a simple way to locate unstable periodic orbits in chaotic time-series data, but also a potentially effective way to use a recurrence plot to identify a dynamical system.
TL;DR: A control method to create Hopf bifurcations in discrete-time nonlinear systems using the center manifold method, normal form technique and the Iooss's Hopfbifurcation theory are employed in the derivation of the control gain.
Abstract: Bifurcation characteristics of a nonlinear system can be manipulated by small controls. In this paper, we present a control method to create Hopf bifurcations in discrete-time nonlinear systems. The critical conditions for the Hopf bifurcations are discussed. The center manifold method, normal form technique and the Iooss’s Hopf bifurcation theory are employed in the derivation of the control gain. Numerical demonstration is provided.
TL;DR: The interaction of waves and obstacles is simulated by adding heterogeneities to a FitzHugh-Nagumo model and a cellular automata model, which give insights concerning spiral wave formation in heterogeneous excitable media.
Abstract: The interaction of waves and obstacles is simulated by adding heterogeneities to a FitzHugh–Nagumo model and a cellular automata model The cellular automata model is formulated to account for heterogeneities by modelling the interaction between current sources and current sinks In both models, wave fronts propagate if the size of the heterogeneities is small, and block if the size of the heterogeneities is large For intermediate values, wave fronts break up into numerous spiral waves The theoretical models give insights concerning spiral wave formation in heterogeneous excitable media
TL;DR: In this article, the authors studied the dynamics of pacing in one-and two-dimensional models of anatomical reentry and found that the existence of inhomogeneity in the reentry circuit is essential for successful termination of tachycardia under certain conditions.
Abstract: Reentry around nonconducting ventricular scar tissue, a cause of lethal arrhythmias, is typically treated by rapid electrical stimulation from an implantable cardioverter defibrillator. However, the dynamical mechanisms of termination (success and failure) are poorly understood. To elucidate such mechanisms, we study the dynamics of pacing in one- and two-dimensional models of anatomical reentry. In a crucial realistic difference from previous studies of such systems, we have placed the pacing site away from the reentry circuit. Our model-independent results suggest that with such off-circuit pacing, the existence of inhomogeneity in the reentry circuit is essential for successful termination of tachycardia under certain conditions. Considering the critical role of such inhomogeneities may lead to more effective pacing algorithms.
TL;DR: A map is derived that approximates the response of the Fenton-Karma model to stimulation in zero spatial dimensions, describing theaction potential duration as a function of the previous diastolic interval and the previous action potential duration.
Abstract: The Fenton–Karma model is a simplification of complex ionic models of cardiac membrane that reproduces quantitatively many of the characteristics of heart cells; its behavior is simple enough to be understood analytically. In this paper, a map is derived that approximates the response of the Fenton–Karma model to stimulation in zero spatial dimensions. This map contains some amount of memory, describing the action potential duration as a function of the previous diastolic interval and the previous action potential duration. Results obtained from iteration of the map and numerical simulations of the Fenton–Karma model are in good agreement. In particular, the iterated map admits different types of solutions corresponding to various dynamical behavior of the cardiac cell, such as 1:1 and 2:1 patterns.