Stationary and oscillatory localized patterns, and subcritical bifurcations

Stationary and oscillatory localized patterns (oscillons) are found in the Belousov-Zhabotinsky reaction dispersed in Aerosol OT water-in-oil microemulsion. The experimental findings are analyzed in terms of subcritical Hopf instability, subcritical Turing instability, and their combination.     

Nonlinear analysis of a reaction-diffusion system: Amplitude equations

A reaction-diffusion system with a nonlinear diffusion term is considered. Based on nonlinear analysis, the amplitude equations are obtained in the cases of the Hopf and Turing instabilities in the system. Turing pattern-forming regions in the parameter space are determined for supercritical and subcritical instabilities in a two-component reaction-diffusion system.     

Competition of Spatial and Temporal Instabilities underTime Delay near Codimension-Two Turing-Hopf Bifurcations

Competition of spatial and temporal instabilities under time delay near the codimension-two Turing-Hopf bifurcations is studied in a reaction-diffusion equation. The time delay changes remarkably the oscillation frequency, the intrinsic wave vector, and the intensities of both Turing and Hopf modes.
The application of appropriate time delay can control the competition between the Turing and Hopf modes. Analysis shows that individual or both feedbacks can realizethe control of the transformation between the Turing and Hopf patterns.Two-dimensional numerical simulations validate the analytical results.     

Numerical simulation and analysis of complex patterns in a two-layer coupled reaction diffusion system

The resonance interaction between two modes is investigated using a two-layer coupled Brusselator model. When two different wavelength modes satisfy resonance conditions, new modes will appear, and a variety of superlattice patterns can be obtained in a short wavelength mode subsystem. We find that even though the wavenumbers of two Turing modes are fixed, the parameter changes have influences on wave intensity and pattern selection. When a hexagon pattern occurs in the short wavelength mode layer and a stripe pattern appears in the long wavelength mode layer, the Hopf instability may happen in a nonlinearly coupled model, and twinkling-eye hexagon and travelling hexagon patterns will be obtained. The symmetries of patterns resulting from the coupled modes may be different from those of their parents, such as the cluster hexagon pattern and square pattern. With the increase of perturbation and coupling intensity, the nonlinear system will convert between a static pattern and a dynamic pattern when the Turing instability and Hopf instability happen in the nonlinear system. Besides the wavenumber ratio and intensity ratio of the two different wavelength Turing modes, perturbation and coupling intensity play an important role in the pattern formation and selection. According to the simulation results, we find that two modes with different symmetries can also be in the spatial resonance under certain conditions, and complex patterns appear in the two-layer coupled reaction diffusion systems.     

Controlling the transition between Turing and antispiral patterns by using time-delayed-feedback

The controllable transition between Turing and antispiral patterns is studied by using a time-delayed-feedback strategy in a FitzHugh-Nagumo model.We treat the time delay as a perturbation and analyse the effect of the time delay on the Turing and Hopf instabilities near the Turing-Hopf codimension-two phase space.Numerical simulations show that the transition between the Turing patterns(hexagon,stripe,and honeycomb),the dual-mode antispiral,and the antispiral by applying appropriate feedback parameters.The dual-mode antispiral pattern originates from the competition between the Turing and Hopf instabilities.Our results have shown the flexibility of the time delay on controlling the pattern formations near the Turing-Hopf codimension-two phase space.     

Turing instability in reaction-diffusion systems with nonlinear diffusion

The Turing instability is studied in two-component reaction-diffusion systems with nonlinear diffusion terms, and the regions in parametric space where Turing patterns can form are determined. The boundaries between super- and subcritical bifurcations are found. Calculations are performed for one-dimensional brusselator and oregonator models.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

Pattern formation in an N+Q component reaction-diffusion system

A general N+Q component reaction-diffusion system is analyzed with regard to pattern forming instabilities (Turing bifurcations). The system consists of N mobile species and Q immobile species. The Q immobile species form in response to reactions between the N mobile species and an immobile substrate and allow the Turing instability to occur. These results are valid both for bifurcations from a spatially uniform state and for systems with an externally imposed gradient as in the experimental systems in which Turing patterns have been observed. It is shown that the critical wave number and the location of the instability in parameter space are independent of the substrate concentration. It is also found that the system necessarily undergoes a Hopf bifurcation as the total substrate concentration is decreased. Further, in the case that all the mobile species diffuse at identical rates we show that if the full system is at a point of Turing bifurcation then the N component mobile subsystem is at transition from an unstable focus to an unstable node, and the critical wave number is simply related to the degenerate positive eigenvalue of the mobile subsystem. A sequence of bifurcations that occur in the eigenspectra as the total substrate concentration is decreased to zero is also discussed.     

Pattern formation in forced reaction diffusion systems with nearly degenerate bifurcations

The existence and stability of stable standing-wave patterns in an assembly of spatially distributed generic oscillators governed by a couple of complex Ginzburg-Landau equations, subjected to parametric forcing, are reported. The mechanism of a dispersion-induced pattern in dissipative oscillators parametrically forced near the degenerate Turing-Hopf bifurcation is also illustrated. We show that, when excitation occurs just after the Turing bifurcation and before the Hopf instability, the system exhibits a new type of stable standing-wave structures, namely the mixed-mode solutions. The Brussellator-model, parametrically forced below the threshold of oscillations, is analyzed as an example of calculation.     

Bifurcation characteristics and flame dynamics of a ducted non-premixed flame with finite rate chemistry

The influence of system parameters such as the flame location, Peclet number and Damköhler number on the bifurcation characteristics and flame dynamics of a ducted non-premixed flame with finite rate chemistry is presented in this paper. In the bifurcation plot with flame location as the bifurcation parameter, subcritical Hopf bifurcation is found for lower values of flame location and supercritical Hopf bifurcation for higher values of flame location, for all the Damköhler numbers used in this study. The flame shapes are captured at eight different phases of a cycle of time series data of acoustic velocity at both the fold and Hopf points for bifurcation with flame location as the parameter. We find that the range of flame height variations at the Hopf point is more than the range of flame height variations obtained at the fold point. We also find that the flame oscillates in the same phase as pressure fluctuation but in a phase different from both velocity and heat release rate fluctuations in the region of hysteresis for bifurcation with flame location. The non-dimensional hysteresis width is plotted as a function of Damköhler number for variation of flame location in the subcritical region. An inverse power law relation is found between the non-dimensional hysteresis width and the Damköhler number. The bifurcation plot with Peclet number as parameter shows a subcritical Hopf bifurcation.     

Oscillatory Turing patterns in reaction-diffusion systems with two coupled layers

A model reaction-diffusion system with two coupled layers yields oscillatory Turing patterns when oscillation occurs in one layer and the other supports stationary Turing structures. Patterns include "twinkling eyes," where oscillating Turing spots are arranged as a hexagonal lattice, and localized spiral or concentric waves within spot-like or stripe-like Turing structures. A new approach to generating the short-wave instability is proposed.     

Turing patterns beyond hexagons and stripes

The best known Turing patterns are composed of stripes or simple hexagonal arrangements of spots. Until recently, Turing patterns with other geometries have been observed only rarely. Here we present experimental studies and mathematical modeling of the formation and stability of hexagonal and square Turing superlattice patterns in a photosensitive reaction-diffusion system. The superlattices develop from initial conditions created by illuminating the system through a mask consisting of a simple hexagonal or square lattice with a wavelength close to a multiple of the intrinsic Turing pattern's wavelength. We show that interaction of the photochemical periodic forcing with the Turing instability generates multiple spatial harmonics of the forcing patterns. The harmonics situated within the Turing instability band survive after the illumination is switched off and form superlattices. The square superlattices are the first examples of time-independent square Turing patterns. We also demonstrate that in a system where the Turing band is slightly below criticality, spatially uniform internal or external oscillations can create oscillating square patterns.     

双层耦合非对称反应扩散系统中的超点阵斑图

通过线性耦合Brusselator模型和Lengyel-Epstein模型,数值研究了双层耦合非对称反应扩散系统中图灵模之间的相互作用以及斑图的形成机理.模拟结果表明,合适的波数比以及相同的对称性是两个图灵模之间达到空间共振的必要条件,而耦合强度则直接影响了图灵斑图的振幅大小.为了保证对称性相同,两个图灵模的本征值高度要位于一定的范围内.只有失稳模为长波模时,才能对另一个图灵模产生调制作用,并形成多尺度时空斑图.随着波数比的增加,短波模子系统依次经历黑眼斑图、白眼斑图以及时序振荡六边形斑图的转变.研究表明失稳图灵模与处于短波不稳定区域的高阶谐波模之间的共振是产生时序振荡六边形的主要原因.     

Turing Patterns in a Reaction-Diffusion System

We have further investigated Turing patterns in a reaction-diffusion system by theoretical analysis and numerical simulations. Simple Turing patterns and complex superlattice structures are observed. We find that the shape and type of Turing patterns depend on dynamical parameters and external periodic forcing, and is independent of effective diffusivity rate σ in the Lengyel-Epstein model. Our numerical results provide additional insight into understanding the mechanism of development of Turing patterns and predicting new pattern formations.     

Hopf Bifurcation in a New Four-Dimensional Hyperchaotic System

In this paper, the Hopf bifurcation in a new hyperchaotic system is studied. Based on the first Lyapunov coefficient theory and symbolic computation, the conditions of supercritical and subcritical bifurcation in the new hyperchaotic system are obtained. Numerical simulations are used to illustrate some main results.     

Excitable dynamics in the presence of time delay

The spiking properties of a subcritical Hopf oscillator with a time delayed nonlinear feedback is investigated. Finite time delay is found to significantly affect both the statistics and the fine structure of the spiking behavior. These dynamical changes are explained in terms of the fundamental modifications occurring in the bifurcation scenario of the system. Our mathematical model can find useful applications in understanding the dynamical behavior of various real life excitable systems where propagation delay effects are ubiquitous.     

Pattern formation in a reaction-diffusion-advection system with wave instability

In this paper, we show by means of numerical simulations how new patterns can emerge in a system with wave instability when a unidirectional advective flow (plug flow) is added to the system. First, we introduce a three variable model with one activator and two inhibitors with similar kinetics to those of the Oregonator model of the Belousov-Zhabotinsky reaction. For this model, we explore the type of patterns that can be obtained without advection, and then explore the effect of different velocities of the advective flow for different patterns. We observe standing waves, and with flow there is a transition from out of phase oscillations between neighboring units to in-phase oscillations with a doubling in frequency. Also mixed and clustered states are generated at higher velocities of the advective flow. There is also a regime of "waving Turing patterns" (quasi-stationary structures that come close and separate periodically), where low advective flow is able to stabilize the stationary Turing pattern. At higher velocities, superposition and interaction of patterns are observed. For both types of patterns, at high velocities of the advective field, the known flow distributed oscillations are observed.     

Prediction of the two-tone suppression and intermodulation performance of auditory systems

Using a model based on subcritical Hopf amplifiers, closed form expressions are obtained for two-tone suppression and the third-order intermodulation performance of the human auditory system. The proposed model approximates the Hopf amplifier characteristic using Fourier-series expansion. The parameters of the Fourier-series approximation can be easily obtained without recourse to sophisticated curve-fitting techniques. The results are compared with previously reported observations.     

Hopf Bifurcation in a New Four-Dimensional Hyperchaotic System

In this paper, the Hopf bifurcation in a new hyperchaotic system is studied. Based on the first Lya-punov coefficient theory and symbolic computation, the conditions of supercritical and subcritical bifurcation in the new hyperchaotic system are obtained. Numerical simulations are used to illustrate some main results.     

Nonlinear analysis of a maglev system with time-delayed feedback control

This paper undertakes a nonlinear analysis of a model for a maglev system with time-delayed feedback. Using linear analysis, we determine constraints on the feedback control gains and the time delay which ensure stability of the maglev system. We then show that a Hopf bifurcation occurs at the linear stability boundary. To gain insight into the periodic motion which arises from the Hopf bifurcation, we use the method of multiple scales on the nonlinear model. This analysis shows that for practical operating ranges, the maglev system undergoes both subcritical and supercritical bifurcations, which give rise to unstable and stable limit cycles respectively. Numerical simulations confirm the theoretical results and indicate that unstable limit cycles may coexist with the stable equilibrium state. This means that large enough perturbations may cause instability in the system even if the feedback gains are such that the linear theory predicts that the equilibrium state is stable.