Experimental studies on long-wavelength instability and spiral breakup in a reaction-diffusion system

Eckhaus selection: The mechanism of pattern persistence in a reaction-diffusion system

In this work, we show theoretically and numerically that a one-dimensional reaction-diffusion system, near the Turing bifurcation, produces different number of stripes when, in addition to random noise, the Fourier mode of a prepattern used to initialize the system changes. We also show that the Fourier modes that persist are inside the Eckhaus stability regions, while those outside this region follow a wave number selection process not predicted by the linear analysis. To test our results, we use the Brusselator reaction-diffusion system obtaining an excellent agreement between the weakly nonlinear predictions of the real Ginzburg-Landau equations and the numerical solutions near the bifurcation. Although the persistence of patterns is not relevant as a simple generating mechanism of self-organization, it is crucial to understand the formation of patterns that occurs in multiple stages. In this work, we discuss the relevance of our results on the robustness and diversity of solutions in multiple-steps mechanisms of biological pattern formation and auto-organization in growing domains.

Nonlinear physics of electrical wave propagation in the heart: a review

The beating of the heart is a synchronized contraction of muscle cells (myocytes) that is triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media with applications to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact for cardiac arrhythmias.

Three-dimensional superdiffusive chemical waves in a precipitation system

We report and characterize new three-dimensional precipitation circular targets and spiral waves with superdiffusive dynamics.

Emitting waves from heterogeneity by a rotating electric field

In a generic model of excitable media, we simulate wave emission from a heterogeneity (WEH) induced by an electric field. Based on the WEH effect, a rotating electric field is proposed to terminate existed spatiotemporal turbulence. Compared with the effects resulted by a periodic pulsed electric field, the rotating electric field displays several improvements, such as lower required intensity, emitting waves on smaller obstacles, and shorter suppression time. Furthermore, due to rotation of the electric field, it can automatically source waves from the boundary of an obstacle with small curvature.

Selection of multiarmed spiral waves in a regular network of neurons

Formation and selection of multiarmed spiral wave due to spontaneous symmetry breaking are investigated in a regular network of Hodgkin-Huxley neuron by changing the excitability and imposing spatial forcing currents on the neurons in the network. The arm number of the multiarmed spiral wave is dependent on the distribution of spatial forcing currents and excitability diversity in the network, and the selection criterion for supporting multiarmed spiral waves is discussed. A broken spiral segment is measured by a short polygonal line connected by three adjacent points (controlled nodes), and a double-spiral wave can be developed from the spiral segment. Multiarmed spiral wave is formed when a group of double-spiral waves rotate in the same direction in the network. In the numerical studies, a group of controlled nodes are selected and spatial forcing currents are imposed on these nodes, and our results show that l-arm stable spiral wave (l = 2, 3, 4,...8) can be induced to occupy the network completely. It is also confirmed that low excitability is critical to induce multiarmed spiral waves while high excitability is important to propagate the multiarmed spiral wave outside so that distinct multiarmed spiral wave can occupy the network completely. Our results confirm that symmetry breaking of target wave in the media accounts for emergence of multiarmed spiral wave, which can be developed from a group of spiral waves with single arm under appropriate condition, thus the potential formation mechanism of multiarmed spiral wave in the media is explained.

Noise-induced spatiotemporal patterns in Hodgkin-Huxley neuronal network

The effect of noise on the pattern selection in a regular network of Hodgkin-Huxley neurons is investigated, and the transition of pattern in the network is measured from subexcitable to excitable media. Extensive numerical results confirm that kinds of travelling wave such as spiral wave, circle wave and target wave could be developed and kept alive in the subexcitable network due to the noise. In the case of excitable media under noise, the developed spiral wave and target wave could coexist and new target-like wave is induced near to the border of media. The averaged membrane potentials over all neurons in the network are calculated to detect the periodicity of the time series and the generated traveling wave. Furthermore, the firing probabilities of neurons in networks are also calculated to analyze the collective behavior of networks.

TRANSITION OF ORDERED WAVES IN NEURONAL NETWORK INDUCED BY DIFFUSIVE POISONING OF ION CHANNELS

Normal physiological activities are often affected by some drugs, and some ion channels are blocked due to the katogene of drugs. This paper investigates the propagation of ordered waves in neuronal networks induced by diffusive poisoning, where the process is measured by increasing the number of neurons in the poisoned area of the networks. A coefficient of poisoning K is defined to measure the time units from one poisoned site to the adjacent site, a smaller K means that more neurons are poisoned in a certain period (a higher poisoning speed). A statistical factor of synchronization R in the two-dimensional array is defined to detect the transition of spiral waves induced by ion channel blocking. It is confirmed that the evolution of the spiral waves depends on the coefficient of poisoning K and number of poisoned neurons. Furthermore, breakup of the spirals is observed when weak channel noise is considered. Finally, the formation of the spiral wave induced by blocking the target wave with line defects is briefly discussed.

Scroll wave meandering induced by phase difference in a three-dimensional excitable medium

We investigated scroll waves in an inhomogeneous excitable 3D system with gradient of excitability. The gradient promotes twisting of the scroll waves. Sufficiently large excitability gradient enhances the twisting and causes simple scroll waves to transition to meandering scroll waves. For the twist-induced instability of scroll waves, we analyzed the stability of 2D spiral waves sliced from the twisted scroll in the vertical direction. The 3D problem is simplified by taking into account the diffusive coupling in the third direction as a time-delayed perturbation to the 2D spiral wave. An additional "negative mass" term measuring the twist thus arises in the 2D system and induces the transition from simple rotation to meandering. A further increase in the gradient ruins partially the unity of the meandering scrolls and generates semiturbulence, the analogs of which were observed in the Belousov-Zhabotinski reaction. We also generated the phase diagram in the parameter space by adjusting the threshold for excitation of the media.

Spontaneous suppression of spiral turbulence based on feedback strategy

Control of defect-mediated turbulence or spiral turbulence has been a subject of pattern dynamics study in the last two decades. As a result, many control strategies has been proposed in theory and tested in computer simulation. Here we propose a self-regulation system that can spontaneously eliminate topological defects in the reaction-diffusion system that suffers from the Doppler instability. The core of the self-regulation scheme is a feedback introduction of local inhomogeneities around spiral tips. We used modified FitzHugh-Nagumo model in computer simulations and light-sensitive Belousov-Zhabotinsky reaction in experiments to test the functionality of the proposed system. We found that if the inhomogeneities were induced on the spiral tips once they were identified, the spiral can be exterminated thus the spiral turbulence can be controlled. Compared with the studies on anatomic obstacles in heart, this self-regulation model may provide a possible mechanism for spontaneous termination of cardiac fibrillation.

Spiral instabilities in media supporting complex oscillations under periodic forcing

The periodically forced Brusselator model displays temporal mixed-mode and quasiperiodic oscillations, period doubling, and chaos. We explore the behavior of such media as reaction-diffusion systems for investigating spiral instabilities. Besides near-core breakup and far-field breakup resulting from unstable modes in the radial direction or Doppler-induced instability (destabilization of the core's location), the observed complex phenomena include backfiring, spiral regeneration, and amplitude modulation from line defects. Amplitude modulation of spirals can evolve to chambered spirals resembling those found in nature, such as pine cones and sunflowers. When the forcing amplitude is increased, the spiral-tip meander evolves from simple rotation to complex petals, corresponding to transformation of the local dynamics from simple oscillations to mixed-mode, period-2, and quasiperiodic oscillations. The number of petals is related to the complexity of the mixed-mode oscillations. Spiral turbulence, standing waves, and homogeneous synchronization permeate the entire system when the forcing amplitude is further increased.

Arm splitting and backfiring of spiral waves in media displaying local mixed-mode oscillations

The behavior of spiral waves is investigated in a model of reaction-diffusion media supporting local mixed-mode oscillations for a range of values of a control parameter. This local behavior is accompanied by the formation of nodes, at which the arms of the simple spiral waves begin to split. With further parameter changes, this nodal structure loses stability, becoming quite irregular, eventually evolving into turbulence, while the local dynamics increases in complexity. The breakup of the spiral waves arises from a backfiring instability of the nodes induced by the arm splitting. This process of spiral breakup in the presence of mixed-mode oscillations represents an alternative to previously described scenarios of instability of line defects and superspirals in media with period-doubling and quasiperiodic oscillations, respectively.

Defect-mediated turbulence in the Belousov-Zhabotinsky reaction

Statistical properties of topological defects in defect-mediated turbulence due to the Doppler instability are examined experimentally in the Belousov-Zhabotinsky reaction. By applying the phase space reconstruction approach, processes of defect creation, annihilation, and defect movement are analyzed. The defect dynamics can be well interpreted within the framework of stochastic Markovian process. In contrast to previous studies that made direct measure of the gain and loss rates, which is practically difficult, we demonstrate that the rates can be obtained directly from the analysis of the time series of defects, and the shape of the probability distribution function can be reproduced in a simple way.

Breakup of propagating waves through the development of a transient unexcitable regime

Spontaneous breakup of propagating waves was investigated in this paper with a ferroin-catalyzed Belousov-Zhabotinsky reaction modified with the inclusion of a second substrate, 1,4-cyclohexanedione (CHD). The presence of CHD, which forms a separate chemical oscillator with acidic bromate, led to the development of a scattered unexcitable regime in the studied medium, where as waves passed through these unexcitable media, the propagation was disrupted, causing the creation of free ends. It thus presents a new avenue through which a chemical wave breaks up to form spirals. By manipulating the concentrations of CHD, sulfuric acid, and bromate, unexcitable regimes of different sizes with different survival times were obtained. Kinetic data on wave speed prior to and after the unexcitable window illustrates that the occurrence of an unexcitable regime is not due to depletion of components needed for wave propagation.

Control of scroll wave turbulence in a three-dimensional reaction-diffusion system with gradient

In this paper, we summarize our recent experimental and theoretical works on observation and control of scroll wave (SW) turbulence. The experiments were conducted in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system with chemical concentration gradients in one dimension. A spatially homogeneous external forcing was used in the experiments as a control; it was realized by illuminating white light on the light sensitive reaction medium. We observed that, in the oscillatory regime of the system, SW can appear automatically in the gradient system, which will be led to spatiotemporal chaos under certain conditions. A suitable periodic forcing may stabilize inherent turbulence of SW. The mechanism of the transition to SW turbulence is due to the phase twist of SW in the presence of chemical gradients, while modulating the phase twist with a proper periodic forcing can delay this transition. Using the FitzHugh-Nagumo model with an external periodic forcing, we confirmed the control mechanism with numerical simulation. Moreover, we also show in the simulation that adding temporal external noise to the system may have the same control effect. During this process, we observed a new state called "intermittent turbulence," which may undergo a transition into a new type of SW collapse when the noise intensity is further increased. The intermittent state and the collapse could be explained by a random process.

Control of spiral turbulence by periodic forcing in a reaction-diffusion system with gradients

We report an experimental result on successfully controlling spiral turbulence in a reaction-diffusion system. The control is realized by periodic forcing in a three-dimensional Belousov-Zhabotinsky reaction-diffusion system, which has chemical concentration gradients in the third dimension. We observe that, in the oscillatory regime of the system, a suitable periodic forcing may stabilize scroll waves (SWs), which otherwise undergo a transition to spiral turbulence. Relating the spiral phase shift due to gradients and the forcing frequency, the mechanism of the control can be well understood by modulating the phase twist of SWs. We use the FitzHugh-Nagumo model to demonstrate this mechanism.

Local wave grouping in a parameter-gradient system and its formation mechanism

In a ferroin-catalyzed Belousov-Zhabotinsky (BZ) reaction-diffusion system with reagent concentration gradients, we observed in the experiment a type of spirals with local waves forming groups. Here, we propose an interpretation of the wave grouping phenomenon. The wave grouping mechanism can be well explained in terms of the cooperation of the excitability gradient and the Doppler effect induced by spiral tip's meandering. In the simulation based on three-dimensional reaction-diffusion system using Oregonator model, spiral patterns analogous to the experiment observation are well reproduced when the parameter gradient in the z direction is introduced.

Renormalized phase dynamics in oscillatory media

Based on the complex Ginzburg-Landau equation (CGLE), a new mapping model of oscillatory media is proposed. The present dynamics is fully determined by an effective phase field renormalized by amplitude. The model exhibits phase turbulence, amplitude turbulence, and a frozen state reported in the CGLE. In addition, we find a state in which the phase and amplitude have spiral structures with opposite rotational directions. This state is found to be observed also in the CGLE. Thus, one concludes that the behaviors observed in the CGLE can be described by only the phase dynamics appropriately constructed.

Control of spiral breakup by an alternating advective field

The control of spiral breakup due to Doppler instability is investigated. It is found that applying an alternating advective field with suitable amplitude and period can prevent the breakup of spiral waves. Further numerical simulations show that the growing meandering behavior of a spiral tip caused by decreasing the excitability of the medium can be efficiently suppressed by the alternating advective field, which inhibits the breakup of spiral waves eventually.

Spiral turbulence developed through the formation of superimposed target waves in an oscillatory reaction-diffusion medium

An approach leading to the development of spiral turbulence is reported here in an oscillatory reaction-diffusion medium, which is through the spontaneous formation of targetlike waves near the core of a spiral wave. The newly formed target wave emerges with its own characteristic frequency and propagates on top of the original spiral wave, which eventually leads to the breakup of the spiral at a location far from the spiral center. The radius of the surviving spiral segment decreases rapidly with the bifurcation control parameter. Calculation of power spectra suggests that the meandering of the spiral tip is responsible for the onset of the superimposed target and the phase desynchronization of the superimposed target waves.

Line-defects-mediated complex-oscillatory spiral waves in a chemical system

In this paper, we summarize our experimental observations on complex-oscillatory spiral waves that arise in a Belousov-Zhabotinsky (BZ) reaction-diffusion system. The observed wave structures generically bear line defects across which the phase of local oscillation changes by a multiple of 2 pi. The local oscillation at every spatial point along a line defect of period-2 (P-2) oscillatory media is period-1 (P-1) oscillatory. For the homogeneous BZ reaction can be excitable, simply periodic, complex periodic, or chaotic as the control parameters are tuned, a number of different complex wave states are revealed. A two-dimensional phase diagram, which includes domains of P-2 oscillatory spirals, intermittently breathing spirals, period-3 (P-3) oscillatory spirals, two different types of mixed-mode periodic spirals, and line-defect-mediated turbulence, is constructed. Several different transitions among different dynamic states are described systematically. In all cases, line defects are found to play an important role.

X-wave solutions of complex Ginzburg-Landau equations

A solution in the form of X-wave patterns of the complex Ginzburg-Landau equation with a harmonic background inhomogeneity is obtained. The pattern can be attributed to the effects of the harmonic potential and the boundary configuration and size. By varying the harmonic of the background potential, the competition among three types of wave patterns: spiral, X, and target, is investigated by following the evolution of the Fourier modes.

Curvature effects on spiral spectra: generation of point eigenvalues near branch points

It is shown that spiral waves may possess many isolated point eigenvalues that appear near branch points of the linear dispersion relation. These eigenvalues are created by the same mechanism that leads to infinitely many bound states for selfadjoint Schrödinger operators with sufficiently weakly decaying long-range potentials. For spirals, the weak decay of the potential is due to the curvature effects on the profile of the spiral in an intermediate spatial range that separates the spiral core from the far field.

Wave grouping of a meandering spiral induced by Doppler effects and oscillatory dispersion

A new type of meandering spiral pattern, in which the dense waves form groups while the sparse waves keep evenly spaced, is observed in a spatial open reactor using a ferroin-catalyzed Belousov-Zhabotinsky reaction. Such a phenomenon is related to both the Doppler effect of a meandering spiral and the oscillatory dispersion relation of the system. Simulation in the two-dimensional Oregonator reaction-diffusion model with an oscillatory dispersion relation gives very similar results.

Spontaneous scroll ring creation and scroll instability in oscillatory medium with gradients

Scroll waves in a quasi-three-dimensional reaction-diffusion medium with a gradient in the third dimension are studied by numerical simulations using the Fitzhugh-Nagumo model. Under a simple initial condition with only one straight filament, we show a spontaneous creation of twisted scroll rings surrounding the initial filament when the control parameter is increased across a threshold. We find that due to the presence of the gradient, the difference of oscillation frequencies in the third dimension is the underlying cause of the phenomenon. Further increase of the control parameter will lead to the spiral breakups, as a result of the interaction of filaments. Observations in the experiments conducted in the same type of medium with the Belousov-Zhabotinsiky reaction qualitatively agree with our simulation results.

Effect of noise on defect chaos in a reaction-diffusion model

The influence of noise on defect chaos due to breakup of spiral waves through Doppler and Eckhaus instabilities is investigated numerically with a modified Fitzhugh-Nagumo model. By numerical simulations we show that the noise can drastically enhance the creation and annihilation rates of topological defects. The noise-free probability distribution function for defects in this model is found not to fit with the previously reported squared-Poisson distribution. Under the influence of noise, the distributions are flattened, and can fit with the squared-Poisson or the modified-Poisson distribution. The defect lifetime and diffusive property of defects under the influence of noise are also checked in this model.

Spiral breakup due to mechanical deformation in excitable media

To address the problem of how cardiac muscle contraction affects the dynamics of rotating spiral waves, spiral breakup induced by mechanical deformation in excitable media is studied in two partial-differential-equation models. It is shown that spirals begin to break up at omega=0.5 omega(0) when we increase the amplitude of the mechanical deformation gradually. Our numerical results point to a new mechanism of transition from spirals to spatiotemporal chaos, in which the anisotropic time-dependent diffusion coefficient is essential.

Tracking waves and vortex nucleation in excitable systems with anomalous dispersion

We report experimental results obtained from a chemical reaction-diffusion system in which wave propagation is limited to a finite band of wavelengths and in which no solitary pulses exist. Wave patterns increase their size through repeated annihilation events of the frontier pulse that allow the succeeding pulses to advance farther. A related type of wave dynamics involves a stable but slow frontier pulse that annihilates subsequent waves in front-to-back collisions. These so-called merging dynamics give rise to an unexpected form of spiral wave nucleation. All of these phenomena are reproduced by a simple, three-species reaction-diffusion model that reveals the importance of the underlying anomalous dispersion relation.

Attraction of spiral waves by localized inhomogeneities with small-world connections in excitable media

Trapping and untrapping of spiral tips in a two-dimensional homogeneous excitable medium with local small-world connections are studied by numerical simulation. In a homogeneous medium which can be simulated with a lattice of regular neighborhood connections, the spiral wave is in the meandering regime. When changing the topology of a small region from regular connections to small-world connections, the tip of the spiral waves is attracted by the small-world region, where the average path length declines with the introduction of long distant connections. The "trapped" phenomenon also occurs in regular lattices where the diffusion coefficient of the small region is increased. The above results can be explained by the eikonal equation, the Luther equation, and the relation between the core radius and the diffusion coefficient.

Chemical waves with line defects in the Belousov-Zhabotinsky reaction

We report our experimental observations of line-defected chemical waves in a quasi-two-dimensional reaction-diffusion system of Belousov-Zhabotinsky reaction. The observed line defects are explicit, which can be directly monitored in real time. In the parameter space, the state of the chemical waves with line defects is located between two regimes of the defect-mediated turbulence. The line defects appear in target waves as well as in spiral waves. We demonstrate that the line defects come out in traveling waves as the later reorganize their spatial topologies to adapt to the change in the local dynamics from simple to complex oscillations or vice versa.

Mechanism for spiral wave breakup in excitable and oscillatory media

We study spiral wave breakup using a Fitzhugh-Nagumo-type system. We find that spiral wave breakup can occur near the core or far from it in both excitable and oscillatory regimes. There is a faraway breakup scenario in both excitable and oscillatory media that depends on long wavelength modulation modes. We observed three distinct scenarios, including one that involves breakup that does not develop into turbulence. However, we find that the mechanisms behind these three scenarios are the same: they are caused by the interaction between the dispersion relation and the asymptotic behavior of the modulation mode. The difference in phenomenology is due to the asymptotic behavior of the modulation mode.

Doppler effect of nonlinear waves and superspirals in oscillatory media

Nonlinear waves emitted from a moving source are studied. A meandering spiral in a reaction-diffusion medium provides an example in which waves originate from a source exhibiting a back-and-forth movement in a radial direction. The periodic motion of the source induces a Doppler effect that causes a modulation in wavelength and amplitude of the waves ("superspiral"). Using direct simulations as well as numerical nonlinear analysis within the complex Ginzburg-Landau equation, we show that waves subject to a convective Eckhaus instability can exhibit monotonic growth or decay as well as saturation of these modulations depending on the perturbation frequency. Our findings elucidate recent experimental observations concerning superspirals and their decay to spatiotemporal chaos.

Transition from spiral waves to defect-mediated turbulence induced by gradient effects in a reaction-diffusion system

The transition from spiral waves to defect-mediated turbulence was studied in a spatial open reactor using Belousov-Zhabotinsky reaction. The experimental results show a new mechanism of the transition from spirals to spatiotemporal chaos, in which the gradient effects in the three-dimensional system are essential. The transition scenario consists of two stages: first, the effects of gradients in the third dimension cause a splitting of the spiral tip and a deletion of certain wave segments, generating new wave sources; second, the waves sent by the new wave sources undergo a backfire instability, and the back waves are laterally unstable. As a result, defects are automatically generated and fill all over the system. The result of numerical simulation using the FitzHugh-Nagumo model essentially agrees with the experimental observation.

Pattern formation on the edge of chaos: experiments with CO oxidation on a Pt(110) surface under global delayed feedback

Experiments with catalytic oxidation of carbon monoxide on Pt(110) show that chemical turbulence in this system can be suppressed by application of appropriate global delayed feedback. Different spatiotemporal patterns, seen near the transition from turbulence to uniform oscillations, are investigated. Such patterns include intermittent turbulence, oscillatory standing waves, cellular structures, and phase clusters. Using a method based on the Hilbert transform, spatial distributions of local phase and amplitude in these patterns are reconstructed from the experimental data.

Analytical approach to the drift of the tips of spiral waves in the complex Ginzburg-Landau equation

In this paper, we investigate the motion of spiral waves in the complex Ginzburg-Landau equation (CGLE) analytically and numerically. We find that the tip of the spiral wave drifts primarily in the direction of the electric field and there is a smaller component of the drift that is perpendicular to the field when a uniform field is applied to the system. The velocity of the tip is uniform and its component along the electric field is equal to the strength of the field. When the CGLE system is driven by white noise, a diffusion law for the vortex core of the spiral wave is derived at long time explicitly. The diffusion constant is found to be D=T/C(2), in which T is the noise strength and C is the core asymptotic factor of the spiral wave. When the external force is a simple oscillation we find that the tip of the spiral wave drifts if the frequency of the external force is the same as that of the system. Our analytical results are verified using numerical simulations.

Relation between the wave front and the tip movement of spirals

The relation between the movement of spiral waves and their tip for a simple spiral is experimentally investigated in a spatial open reactor using the Belousov-Zhabotinsky reaction. In the quasi-two dimensional regime, our results agree with the combustion model proposed by Lázár et al. [Chaos 5, 443 (1995)] which states that the speed of spiral tip movement is proportional to the speed of spiral waves, c(tip) infinity c(front), when expressed in the radius of the spiral tip trajectory R and the spiral wavelength lambda, R infinity lambda. The measurement shows that the slope of the linear R-lambda curve is about 0.13, independent of the control parameter. This observation is also consistent with the combustion model.

Formation of a spiraling line defect and its meandering transition in a period-2 medium

The instability of a period-1 spiral wave resulting in a period-2 spiral wave with a line defect is investigated for the first time in a laboratory system. At the very onset the transition proceeds by an emergence of a spiraling line defect, "breathing" intermittently while retaining its symmetry of a period-1 spiral wave. With a further change in a control parameter, the line defect undergoes a meandering transition producing a compound tip trajectory, following a dynamic shape transition. The observed transitions have a strong analogy to the phase synchronization transition of two coupled nonlinear oscillators and the meandering transition of a period-1 spiral wave.

Unusual spiral wave tip trajectories in a parametrically forced nonequilibrium system

The effect of parametric modulations on spiral waves that form in a thin layer of liquid crystal under rotating magnetic field is studied in laboratory experiments and numerical simulations. Parametric forcings that are sinusoidal in time make a simply rotating spiral tip to meander, rendering a compound tip trajectory that is composed of a main circular orbit and a satellite orbit with an unusual crescent shape. No evidence of frequency locking is found. The underlying mechanism for the unusual shape of the satellite orbit is elucidated, and the dependence of the rotation frequency of the main circular orbit f(o) on the perturbation parameters are discussed.

Superspiral structures of meandering and drifting spiral waves

The spatiotemporal superstructure of meandering and drifting spiral waves is explained analytically. It is also demonstrated that the Hopf eigenmode that causes the transition to meandering waves is weakly exponentially localized at onset but grows exponentially slightly before onset.