Stability Of Homogeneous States Research Articles

Robust network of globally coupled heterogeneous limit cycle oscillators due to inertia

We investigate the phase transition from macroscopic oscillatory state to stable homogeneous steady state in a heterogeneous network of globally coupled Stuart–Landau limit cycle oscillators in the presence of the inertial effect. The phase transition, known as aging transition, onsets above a critical fraction of inactive constituents in the mixed population of active and inactive units. We show that even a feeble increase in the inertial strength increases the critical fraction of inactive units significantly for the onset of the phase transition to the macroscopic steady state thereby resulting in a more robust network, in general. In contrast, a large coupling strength, in the case of a homogeneous network, facilitates the manifestation of the phase transition even for a small fraction of inactive oscillators leading to a more fragile network. Nevertheless, a large coupling strength, in the case of a heterogeneous network, increases the resilience of the network by facilitating the phase transition at a large fraction of inactive oscillators. Furthermore, a larger standard deviation of the natural frequencies always leads to a more fragile network. We derive the macroscopic evolution equations for the order parameters and the stability curve using the first-order moment expansion around the mean-field. In addition, we also deduce the critical fraction of inactive units and the critical inertial strength analytically that matches with the simulation results. Interestingly, we find that the critical inertial strength is reciprocally related to the square of the mean frequency of the network.

Effects of noise on the critical points of Turing instability in complex ecosystems.

Noise is ubiquitous in natural and artificial systems. In a noisy environment, the interactions among nodes may fluctuate randomly, leading to more complicated interactions. In this paper we focus on the effects of noise and network topology on the Turing pattern of ecological networks with activator-inhibitor structure, which may be interpreted as prey-predator interactions. Based on the stability theory of stochastic differential equations, a sufficient condition for the uniform state is derived. The analytical results indicate that noise is beneficial for the uniform state. When the ratio between the diffusion coefficients of the predator and prey increases, the ecosystems can exhibit a transition from a uniform stable state to a Turing pattern, while when the ratio decreases, the ecosystems transit from a Turing pattern to a uniform stable state. There are two crucial critical points in Turing patterns, forward and backward. We find that both forward and backward critical points increase as the noise intensity increases. This means that noise favors a stable homogeneous state compared to a state with a heterogeneous pattern, which is consistent with the analytical results. In addition, noise can weaken the hysteresis phenomenon and even eliminate it in some cases. Furthermore, we report that network topology plays an important role in modulating the uniform state of ecosystems, such as the size of prey-predator systems, the network connectivity, and the strength of interaction.

General conditions for Turing and wave instabilities in reaction -diffusion systems

Necessary and sufficient conditions are provided for a diffusion-driven instability of a stable equilibrium of a reaction–diffusion system with n components and diagonal diffusion matrix. These can be either Turing or wave instabilities. Known necessary and sufficient conditions are reproduced for there to exist diffusion rates that cause a Turing bifurcation of a stable homogeneous state in the absence of diffusion. The method of proof here though, which is based on study of dispersion relations in the contrasting limits in which the wavenumber tends to zero and to infty , gives a constructive method for choosing diffusion constants. The results are illustrated on a 3-component FitzHugh–Nagumo-like model proposed to study excitable wavetrains, and for two different coupled Brusselator systems with 4-components.

Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction–diffusion equations

We consider a class of singularly perturbed 2-component reaction–diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow–fast–slow, interfaces between stable homogeneous rest states. In many example systems, such as models of desertification fronts in dryland ecosystems, such fronts can exhibit an instability by which the interface destabilizes into fingering patterns. Motivated by the appearance of such patterns, we propose two versions of a 2D stability criterion for (transversal) long wavelength perturbations along the interface of these traveling slow–fast–slow fronts. The fronts are constructed using geometric singular perturbation techniques by connecting slow orbits on two distinct normally hyperbolic slow manifolds through a heteroclinic orbit in the fast problem. The associated stability criteria are expressed in terms of the nonlinearities of the system and the slow–fast–slow structure of the fronts. We illustrate and further elaborate on the general set-up by explicitly working out the existence and transversal (in)stability of traveling fronts in a number of example systems/models. We analytically establish the instability of invading bare soil/vegetation interfaces against transversal long wavelength perturbations in several dryland ecosystem models and numerically recover fingering vegetation patterns counter-invading an invading desertification front.

Localized structures formed through domain wall locking in cavity-enhanced second-harmonic generation.

We analyze the formation of localized structures in cavity-enhanced second-harmonic generation. We focus on the phase-matched limit, and consider that fundamental and generated waves have opposite signs of group velocity dispersion. We show that these states form due to the locking of domain walls connecting two stable homogeneous states of the system, and undergo collapsed snaking. We study the impact of temporal walk-off on the stability and dynamics of these localized states.

Loss of elastic stability and formation of inhomogeneous states at phase transitions in thin films on substrates

Within the Landau-like approach we study stability of homogeneous states near phase transitions in thin films on substrates. The order parameter is electrically neutral in order to exclude the effects of the depolarizing fields and concentrate on the elasticity effects. We consider the case where the first-order transition in free crystal would convert into a second order when in the film on a substrate if the system remained homogeneous. Limit of stability of the homogeneous state is found analytically, though approximately. Numerical simulations provide qualitatively similar results and reveal temperature evolution of the arising inhomogeneities.

Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect

A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

Transition from homogeneous to inhomogeneous limit cycles: Effect of local filtering in coupled oscillators.

We report an interesting symmetry-breaking transition in coupled identical oscillators, namely, the continuous transition from homogeneous to inhomogeneous limit cycle oscillations. The observed transition is the oscillatory analog of the Turing-type symmetry-breaking transition from amplitude death (i.e., stable homogeneous steady state) to oscillation death (i.e., stable inhomogeneous steady state). This novel transition occurs in the parametric zone of occurrence of rhythmogenesis and oscillation death as a consequence of the presence of local filtering in the coupling path. We consider paradigmatic oscillators, such as Stuart-Landau and van der Pol oscillators, under mean-field coupling with low-pass or all-pass filtered self-feedback and through a rigorous bifurcation analysis we explore the genesis of this transition. Further, we experimentally demonstrate the observed transition, which establishes its robustness in the presence of parameter fluctuations and noise.

Turing Patterns in the Lengyel–Epstein System with Superdiffusion

Turing instability and pattern formation in the Lengyel–Epstein (L–E) model with superdiffusion are investigated in this paper. The effects of superdiffusion on the stability of the homogeneous steady state are studied in detail. In the presence of superdiffusion, instability will occur in the stable homogeneous steady state and more complex dynamics will exist. As a result of Turing instability, some patterns are formed. Through stability analysis of the system at the equilibrium point, conditions ensuring Turing and Hopf bifurcations are obtained. To further explore pattern selection, the weakly nonlinear analysis and multiple scale analysis are employed to derive amplitude equations of the stationary patterns. Then complex dynamics of amplitude equations, such as the existence of homogeneous solutions, stripe and hexagon patterns, mixed structure patterns, their stability, interaction and transition between them, are analyzed. Then different patterns occur immediately. Finally, the numerical simulations are presented to show the effectiveness of theoretical analysis and patterns are identified numerically. Whereas in the existing results of such model with normal diffusion, no amplitude equations are derived and patterns are only identified through numerical simulations.

Spatiotemporal complexity of a discrete space-time predator−prey system with self- and cross-diffusion

In this research, we investigate the spatiotemporal dynamics of a discrete space-time predator−prey system with self- and cross-diffusion. Through stability analysis and bifurcation analysis, Turing pattern formation conditions are derived and two nonlinear mechanisms of pattern formation are found, i.e., pure Turing instability and Hopf-Turing instability. Numerical simulations reveal rich dynamics of the discrete predator−prey system. In spatially homogeneous case, stable homogeneous stationary states, homogeneous periodic, quasiperiodic and chaotic oscillating states are exhibited; in spatially heterogeneous case, a surprising variety of prey and predator patterns are described, including spotted, striped, labyrinth, gapped, spiral, circled patterns and many intermediate patterns. Moreover, sensitivity of spatiotemporal pattern formation to initial conditions is predicted along with Hopf-Turing instability, suggesting the self-organization of diverse patterns under identical parametric conditions. In comparison with former results in literature, the discrete version of reaction-diffusion model developed in this research capture more complicated and richer nonlinear dynamical behaviors, contributing to a new comprehending on the complex pattern formation of spatially extended discrete predator−prey systems.

Reaction, diffusion and non-local interaction.

Recent years have seen the introduction of non-local interactions in various fields. A typical example of a non-local interaction is where the convolution kernel incorporates short-range activation and long-range inhibition. This paper presents the relationship between non-local interactions and reaction-diffusion systems in the following sense: (a) the relationship between the instability induced by non-local interaction and diffusion-driven instability; (b) the realization of non-local interactions by reaction-diffusion systems. More precisely, it is shown that the non-local interaction of a Mexican-hat kernel destabilizes the stable homogeneous state and that this instability is related to diffusion-driven instability. Furthermore, a reaction-diffusion system that approximates the non-local interaction system with any even convolution kernel is shown to exist.

Flexible dynamics of two quorum-sensing coupled repressilators.

Genetic oscillators play important roles in cell life regulation. The regulatory efficiency usually depends strongly on the emergence of stable collective dynamic modes, which requires designing the interactions between genetic networks. We investigate the dynamics of two identical synthetic genetic repressilators coupled by an additional plasmid which implements quorum sensing (QS) in each network thereby supporting global coupling. In a basic genetic ring oscillator network in which three genes inhibit each other in unidirectional manner, QS stimulates the transcriptional activity of chosen genes providing for competition between inhibitory and stimulatory activities localized in those genes. The "promoter strength", the Hill cooperativity coefficient of transcription repression, and the coupling strength, i.e., parameters controlling the basic rates of genetic reactions, were chosen for extensive bifurcation analysis. The results are presented as a map of dynamic regimes. We found that the remarkable multistability of the antiphase limit cycle and stable homogeneous and inhomogeneous steady states exists over broad ranges of control parameters. We studied the antiphase limit cycle stability and the evolution of irregular oscillatory regimes in the parameter areas where the antiphase cycle loses stability. In these regions we observed developing complex oscillations, collective chaos, and multistability between regular limit cycles and complex oscillations over uncommonly large intervals of coupling strength. QS coupling stimulates the appearance of intrachaotic periodic windows with spatially symmetric and asymmetric partial limit cycles which, in turn, change the type of chaos from a simple antiphase character into chaos composed of pieces of the trajectories having alternating polarity. The very rich dynamics discovered in the system of two identical simple ring oscillators may serve as a possible background for biological phenotypic diversification, as well as a stimulator to search for similar coupling in oscillator arrays in other areas of nature, e.g., in neurobiology, ecology, climatology, etc.

Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system

The spatiotemporal dynamics of a space- and time-discrete predator–prey system is investigated in this research. The conditions for stable homogeneous stationary state of the system are derived via stability analysis. By using center manifold theorem and bifurcation theory, critical parameter values for flip bifurcation, Hopf bifurcation and Turing bifurcation are determined, respectively. Based on the bifurcation analysis, pattern formation conditions are also provided. Numerical simulations are performed not only to illustrate the theoretical results, but also to exhibit new and complex dynamical behaviors, including period-doubling cascade, invariant circles, periodic windows, chaotic dynamics, and pattern formation. Maximum Lyapunov exponents are calculated to distinguish chaos from regular behaviors. In the routes from bifurcation to chaos, flip-Turing instability and Hopf-Turing instability emerge, capturing the formation of diverse complex patterns, such as mosaic, circle, spiral, spatiotemporal chaotic patterns, and so on. The analysis and results in this research contribute to a new understanding on the relationship among bifurcation, chaos and pattern formation.

Chemical Oscillations and Spatial Structures in Polymerisation Reactions

This article discusses the possibility of coupling a polymerization reaction to an oscillatory kinetic model, complemented by diffusion which can lead to spatial structure. We used three well–known mathematical models of oscillators: a variant of the Rossler multivibrator, a model proposed by Edelstein, and the Oregon Oscillator. One or some of the terms in the equations of these models come from a polymerization reaction, while the other terms of these equations will come from collateral processes. So, almost any reaction could become oscillatory and/or with dissipative structure, adding the adequate collateral processes. The propagation stages are considered as invariants and initiation reactions of order α = 0, 1, or 2, and termination reactions of order β = 1 or 2 are assumed. Except in the case α = 1, β = 2, all six remaining reactions combinations (α, β) can be coupled to least one, and often to several of the models. The effects of destabilization a stable homogeneous steady state by the presence of diffusion is also discussed, which is always be possible.

Pattern analysis in a benthic bacteria-nutrient system.

We study steady states in a reaction-diffusion system for a benthic bacteria-nutrient model in a marine sediment over 1D and 2D domains by using Landau reductions and numerical path following methods. We point out how the system reacts to changes of the strength of food supply and ingestion. We find that the system has a stable homogeneous steady state for relatively large rates of food supply and ingestion, while this state becomes unstable if one of these rates decreases and Turing patterns such as hexagons and stripes start to exist. One of the main results of the present work is a global bifurcation diagram for solutions over a bounded 2D domain. This bifurcation diagram includes branches of stripes, hexagons, and mixed modes. Furthermore, we find a number of snaking branches of stationary states, which are spatial connections between homogeneous states and hexagons, homogeneous states and stripes as well as stripes and hexagons in parameter ranges, where both corresponding states are stable. The system under consideration originally contains some spatially varying coefficients and with these exhibits layerings of patterns. The existence of spatial connections between different steady states in bistable ranges shows that spatially varying patterns are not necessarily due to spatially varying coefficients. The present work gives another example, where these effects arise and shows how the analytical and numerical observations can be used to detect signs that a marine bacteria population is in danger to die out or on its way to recovery, respectively. We find a type of hexagon patches on a homogeneous background, which seems to be new discovery. We show the first numerically calculated solution-branch, which connects two different types of hexagons in parameter space. We check numerically for bounded domains whether the stability changes for hexagons and stripes, which are extended homogeneously into the third dimension. We find that stripes and one type of hexagons have the same stable range over bounded 2D and 3D domains. This does not hold for the other type of hexagons. Their stable range is shorter for the bounded 3D domain, which we used here. We find a snaking branch, which bifurcates when the hexagonal prisms loose their stability. Solutions on this branch connects spatially between hexagonal prisms and a genuine 3D pattern (balls).

Fractional derivation stabilizing virtue-induced quenching phenomena in coupled oscillators

We investigate quenching oscillations phenomena in a system of two diffusively and mutually coupled identical fractional-order Stuart-Landau oscillators. We first consider the uncoupled unit and find that the stabilizing virtue of the fractional derivative yields suppression of oscillations via a Hopf bifurcation. The oscillatory solutions of the fractional-order Stuart-Landau equation are provided as well. Quenching phenomena are then investigated in the coupled system. It is found that the fractional derivatives enhance oscillation death by widening its domain of existence in coupling strength space and initial conditions space, leading to oscillation death dominance. A region of stable homogeneous steady state appears where the uncoupled oscillators are resting and not oscillating as usually accepted for the realization of amplitude death.

Restoration of rhythmicity in diffusively coupled dynamical networks.

Oscillatory behaviour is essential for proper functioning of various physical and biological processes. However, diffusive coupling is capable of suppressing intrinsic oscillations due to the manifestation of the phenomena of amplitude and oscillation deaths. Here we present a scheme to revoke these quenching states in diffusively coupled dynamical networks, and demonstrate the approach in experiments with an oscillatory chemical reaction. By introducing a simple feedback factor in the diffusive coupling, we show that the stable (in)homogeneous steady states can be effectively destabilized to restore dynamic behaviours of coupled systems. Even a feeble deviation from the normal diffusive coupling drastically shrinks the death regions in the parameter space. The generality of our method is corroborated in diverse non-linear systems of diffusively coupled paradigmatic models with various death scenarios. Our study provides a general framework to strengthen the robustness of dynamic activity in diffusively coupled dynamical networks.

Analysis and numerics of traveling waves for asymmetric fractional reaction-diffusion equations

We consider a scalar reaction-diffusion equation in one spatial dimension with bistable nonlinearity and a nonlocal space-fractional diffusion operator of Riesz-Feller type. We present our analytical results on the existence, uniqueness (up to translations) and stability of a traveling wave solution connecting two stable homogeneous steady states. Moreover, we review numerical methods for the case of reaction-diffusion equations with fractional Laplacian and discuss possible extensions to our reaction-diffusion equations with Riesz-Feller operators. In particular, we present a direct method using integral operator discretization in combination with projection boundary conditions to visualize our analytical results about traveling waves.

Homogeneous self-aligned liquid crystals on wrinkled-wall poly(dimethylsiloxane) via localised ion-beam irradiation.

We demonstrate self-aligned liquid crystals (LCs) using a wrinkled-wall polydimethylsiloxane (PDMS) wrinkle structure, which is a key factor to obtain a stable homogeneous alignment state with positive LCs. We constructed the wrinkled walls via localised surface exposure to IB radiation, which passed through a long length localised pattern mask. The creation of the wrinkled wall helped to align the LC molecules homogeneously because the wrinkled wall acted as a guide for the arrangement of positive LC molecules. In addition, we confirmed the stability of the alignment state as the width of the wrinkled wall was changed. Although this wrinkled-wall method is a non-contact method, LC alignment is achieved via an anisotropic topographical guide, which provides the LC molecules with stable homogeneous alignment.

The ballistic transport instability in Saturn's rings - II. Non-linear wave dynamics

The ejecta discharged by impacting meteorites can redistribute a planetary ring's mass and angular momentum. This `ballistic transport' of ring properties instigates a linear instability that could generate the 100--1000-km undulations observed in Saturn's inner B-ring and in its C-ring. We present semi-analytic results demonstrating how the instability sustains steadily travelling nonlinear wavetrains. At low optical depths, the instability produces approximately sinusoidal waves of low amplitude, which we identify with those observed between radii 77,000 and 86,000 km in the C-ring. On the other hand, optical depths of 1 or more exhibit hysteresis, whereby the ring falls into multiple stable states: the homogeneous background equilibrium or large-amplitude wave states. Possibly the `flat zones' and `wave zones' between radii 93,000 and 98,000 km in the B-ring correspond to the stable homogeneous and wave states, respectively. In addition, we test the linear stability of the wavetrains and show that only a small subset are stable. In particular, stable solutions all possess wavelengths greater than the lengthscale of fastest linear growth. We supplement our calculations with a weakly nonlinear analysis that suggests the C-ring reproduces some of the dynamics of the complex Ginzburg--Landau equation. In the third paper in the series, these results will be tested and extended with numerical simulations.