International Journal of Bifurcation and Chaos

Diffusive instabilities provide the engine for an ever increasing number of dissipative structures. In this class autocatalytic chemical systems are prone to generate temporal and spatial self-organization phenomena. The development of open spatial reactors and the subsequent discovery in 1989 of the stationary reaction–diffusion patterns predicted by Turing [1952] have triggered a large amount of research. This review aims at a comparison between theoretical predictions and experimental results obtained with various type of reactors in use. The differences arising from the use of reactions exhibiting either bistability of homogeneous steady states or a single one in a CSTR are emphasized.

In this paper we survey some recent results on the behavior of solutions of parabolic equations subjected to nonlinear boundary conditions.

The results range from local existence and regularity of solutions, to global existence, dissipativeness and existence of attractors, and to blow-up in finite time.

Some applications are given to some singular perturbation problems and to pattern formation from boundary flux.

Brief descriptions of spatio-temporal structures that arose in the author's explorations of theoretical immunology are provided. Topics range from singular perturbation theory for a biological assay and a model for skin rashes (reaction–diffusion patterns), through immunotherapy for allergy and selection of appropriate antibody types (compartment models), to a somewhat abstract "shape space".

Over the years, Manuel Velarde and myself shared often a common interest in very varied topics. In this hommage to him, I go back to our very first foray in research, that is to a fundamental question of nonequilibrium thermodynamics. With our common mentor, Pierre Résibois, Manuel wrote one of his first papers on some quantum effects in kinetic theory [Résibois & Velarde, 1971; Velarde & Résibois, 1973]. Below I examine a question where quantum effects and thermal fluctuations are of fundamental importance too, that is the coherence time of a Bose–Einstein condensate at finite temperature. Surprisingly, although this question could have been looked at long ago, I have not found it treated in the literature, at least following the simple and straightforward approach developed below.

We study a one-dimensional ring chain of length L with N particles interacting via Morse potentials and influenced by dissipative forces (passive and active friction). We show that by negative friction the system can be driven far from the thermodynamic equilibrium states. For over-critical pumping with free energy several types of coherent motions including uniform rotations, optical oscillations and waves emerge in the ring. We also show the existence of a critical particle density n_c = N/L_c, below that the particles spontaneously organize into clusters which can actively rotate. Additionally, the influence of white noise on the clustering is discussed.

We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula

The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by

This also establishes the leading large-n behavior that may be expected on general grounds, where N_n is the number of sites on the nth generation tower and is the spectral dimension of the fractal.

The spreading of the position and momentum probability distributions for the stable free oscillations of a circular membrane of radius l is analyzed by means of the associated Boltzmann–Shannon information entropies in the correspondence principle limit (n → ∞, m fixed), where the numbers (n, m), n ∈ ℕ and m ∈ ℤ, uniquely characterize an oscillation of this two-dimensional system. This is done by solving the short-wave asymptotics of the physical entropies in the two complementary spaces, which boils down to the calculation of the asymptotic behavior of certain entropic integrals of Bessel functions. It is rigorously shown that the position and momentum entropies behave as 2ln(l) + ln(4π) - 2 and ln(n) - 2ln(l) + ln(2π³) when n → ∞, respectively. So the total entropy sum has a logarithmic dependence on n and it does not depend on the membrane radius. The former indicates that the ordering of short-wavelength oscillations is exactly identical for the entropic sum and the single-particle energy. The latter holds for all oscillations of the membrane because of the uniform scaling invariance of the entropy sum.

Initial value problems as well as stationary solitary and periodic waves are investigated for a perturbed KdV equation including the Hilbert transform; u_t + uu_x + βu_xxx + η(ℋu_x - u_xx) = 0 (η > 0). Multi-hump stationary solitary and periodic wave solutions are numerically identified. Furthermore, the close relation between the structure of the stationary waves and the behavior of the temporal evolutions is discussed in comparison with other perturbed KdV equations with different instability and dissipation terms. The results support some general features common to this type of nonlinear evolution equations.

The interaction of a strongly nonlinear solitary wave with an external force is studied using the extended Korteweg–de Vries equation as a model. This equation has several different families of nonlinear wave solutions: solitons, the so-called "thick" solitons, algebraic solitons and breathers, depending upon the sign of the cubic nonlinear term. A simple nonlinear dynamical system of the second order for the amplitude and position of the solitary wave is derived, and used to study the interaction. Its solutions are investigated in the phase plane. The conditions for the capture or reflection of a solitary wave by a single localized external force are obtained, with an emphasis on the role of the cubic nonlinear term.

We study the propagation of nonlinear waves in a three-component reaction–diffusion system. The problem of the existence of the stationary pulse-like solutions is reduced to the analysis of homoclinic trajectories of a fourth-order system of nonlinear ODEs. We have obtained the parameter set corresponding to the homoclinic bifurcations that defines the velocity spectra of the traveling pulses. We have shown that the pulses behave like autowaves annihilating in head-on collision and like dissipative solitons crossing each other, reflecting at boundaries. We have provided a qualitative explanation for such a behavior.

Generalized Wave Equations containing dispersion, dissipation and energy-production (GDWE) are considered in lieu of dissipative NEE as more suitable models for two-way interaction of localized waves. The quasi-particle behavior and the long-time evolution of localized solutions upon take-over and head-on collisions are investigated numerically by means of an adequate difference scheme which represents faithfully the balance/conservation laws. It is shown that in most cases the balance between energy production/dissipation and nonlinearity plays a similar role to the classical Boussinesq balance between dispersion and nonlinearity, namely it can create and support localized solutions which behave as quasi-particles upon collisions and for a reasonably long time after that.

The aim of this paper is to review some of the mechanisms which leads to stable localized patterns in nonequilibrium systems.

It is shown that pulses in the complete quintic one-dimensional Ginzburg–Landau equation with complex coefficients appear through a saddle-node bifurcation which is determined analytically through a suitable approximation of the explicit form of the pulses. The results are in excellent agreement with direct numerical simulations.

The estimation of the domain of stability of fixed points of nonlinear differential systems constitutes a practical problem of much interest in engineering. The procedures based on Lyapunov's second method configures an alternative worth consideration. It has the appeal of reducing calculation complexity and is time-saving with respect to the direct, computer crunching approach which requires a systematic numerical integration of the evolution equations from a gridlike pattern of initial conditions. However, it is not devoid of problems inasmuch as the Lyapunov function itself is problem-dependent and relies too much on presumptions. Additionally, the evaluation of its corresponding domain is produced in terms of a nonlinear programming problem with inequality constraints the resolution of which may sometimes require a large investment in computer time. These problems are in part avoided by restricting to quadratic Lyapunov functions, with the possible obvious consequence of limiting the estimation of the domain. In order to simplify the estimation of domains we exploit here a novel formulation of the issue of stability of invariant surfaces within Lyapunov's direct method [Díaz-Sierra et al., 2001]. The resulting method addresses directly the optimization problem associated to the evaluation of the stability domain. The problem is recast in a new, simpler form by playing both on the Lyapunov function itself and on the constraints. The gains from the procedure permit to conceive increased returns in the application of Lyapunov's direct method once it is realized that it is not prohibitive from a computational point of view to depart from the limited quadratic Lyapunov functions.

An analytical approach to determine critical parameter values of homoclinic bifurcations in three-dimensional systems is reported. The homoclinic orbit is supposed to be a limit of a unique periodic orbit. Hence, the multiple scales perturbation method is performed to construct an approximation of the periodic solution and its frequency. Then, two simple criteria are used. The first criterion is based on the collision between the periodic and the hyperbolic fixed point involved in the bifurcation. The second uses the infinity condition of the period of the periodic orbit. For illustration a specific system is investigated.

Nonpotential effects in nonlinear evolution of Marangoni convection patterns are investigated analytically and numerically. Three manifestations of nonpotential effects are considered: (i) spatial modulations of hexagonal patterns; (ii) interaction between a short-scale hexagonal pattern and a long-scale slow deformational (Goldstone) mode; (iii) generation of the mean flow by the free-surface deformation in a large-scale Marangoni convection with poorly conducting boundaries. Nonpotential effects are shown to cause various secondary instabilities leading to skewed hexagonal structures, coexisting u- and d-hexagons, oscillating hexagonal patterns, spatially irregular cellular patterns, etc.

The onset of convection in systems that are heated via current dissipation in the lower boundary or that lose heat from the top boundary via Newton's law of cooling is formulated as a bifurcation problem. The Rayleigh number as usually defined is shown to be inappropriate as a bifurcation parameter since the temperature difference across the layer depends on the amplitude of convection and hence changes as convection evolves at fixed external parameter values. A modified Rayleigh number is introduced that does remain constant even when the system is evolving, and solutions obtained with the standard formulation are compared with those obtained via the new one. Near the 1 : 2 spatial resonance in low Prandtl number fluids these effects open up intervals of Rayleigh number with no stable solutions in the form of steady convection or steadily traveling waves. Direct numerical simulations in two dimensions show that in such intervals the dynamics typically take the form of a nearly heteroclinic modulated traveling wave. This wave may be quasiperiodic or chaotic.

This paper reviews the studies that have been carried out over the years to analyze the electroconvection of an insulating plane liquid layer subjected to unipolar charge injection from one of the electrodes. We consider in detail the motion near the onset of nonlinear instability that for brevity we call chaotic electroconvection. We also present some recent advances in the characterization of this chaotic motion. In particular, we have computed the maximal Lyapunov exponents for time series obtained from numerical simulation, for weak as well as strong injection regimes, and we discuss the relevance of these results for the characterization of this EHD chaos.

A weakly nonlinear analysis of an overdriven detonation is carried out in the neighborhood of the instability threshold. The result leads to a nonlinear integral-differential equation describing the dynamics of the cellular front and exhibiting "diamond" patterns similar to those observed in experiments. An unexpected outcome of the analysis is a self-sustained mean streaming motion associated with the nonlinear dynamics of a weakly unstable detonation front.

The flashback or propagation of premixed flames against the flow of a reacting mixture, along the low velocity region near a cold wall, is investigated numerically. The analysis, carried out using the constant density approximation for an Arrhenius overall reaction, accounts for the effects of the Lewis number of the limiting reactant. Flame front propagation and flashback are only possible for values of the near wall velocity gradient below a critical value. The flame propagation becomes chaotic for small values of the Lewis number.

The coexistence of different regimes in a class of atmospheric flow models is analyzed from the standpoint of irreversible thermodynamics. The entropy production, measuring the degree of dissipation within the system is evaluated across the bifurcation point. Some interesting trends are brought out, regarding the relative importance of dissipation in two qualitatively different circulation regimes referred to as zonal and blocked flows. It is shown that in the presence of stochastic disturbances entropy production exhibits critical behavior both in the vicinity of the bifurcation point and in parameter values where the two flow regimes are equally probable.

We start by reviewing the basic physical properties of cavity solitons, embedding this phenomenon in the general framework of optical instabilities and optical pattern formation. Next, we focus on the case of semiconductor microcavities and consider a first principle model. Its homogeneous stationary solutions and their stability are evaluated analytically. With the help of numerical simulations, we discuss two case studies, one in the passive and one in the active configuration.

Replication and differentiation of spots in a class of reaction–diffusion equations are studied by extending the Gray–Scott model with self-replicating spots so that it includes many chemical species. By examining many possible reaction networks, the behavior of this model is categorized into three types: replication of homogeneous fixed spots, replication of oscillatory spots, and differentiation from "multipotent spots". These multipotent spots either replicate or differentiate into other types of spots with different fixed-point dynamics, and as a result, an inhomogeneous pattern of spots is formed. This differentiation process of spots is analyzed in terms of the loss of chemical diversity and decrease of the local Kolmogorov–Sinai entropy. Initial condition dependence and robustness of a pattern against macroscopic perturbation are also analyzed. Relevance of the results to developmental cell biology is also discussed.

A delayed response tracer experiment is suggested, based on the following constraints: (1) The kinetics of the process can be expressed by local evolution equations without delays, for example by the mass action law. (2) The kinetic isotope effect can be neglected, that is, the rate coefficients for labeled and unlabeled chemicals are the same. (3) The total fluxes of the various chemicals are generally time dependent, but are not modified by the presence of the labeled compounds. (4) The experiment consists in the measurement of the time dependence of the fractions β_u, u = 1, 2,… of labeled chemicals in the output fluxes as functionals of the time dependence of the fractions α_u, u = 1, 2,… of labeled chemicals in the input fluxes, which are controlled by the researcher. We show that the output fluxes are related to the input fluxes by a linear delayed superposition theorem: β_u(t) = ∑_u′ ∫ χ_uu′(t,t′)α_u′(t′)dt′, where χ_uu′(t,t′), is a delayed susceptibility function, which is related to the probability density of the transit time, that is, the time necessary for a molecular fragment to cross the system. This linear superposition law is not the result of a linearization procedure and holds even if the underlying kinetic equations are highly nonlinear. We establish a relationship between the transit time probability densities and the lifetime distributions of the various species in the system. The law permits extracting information about the mechanism and kinetics of chemical processes from response experiments.

This paper studies phase-locking in a network of N neurons. The dynamic variables are the phases of the axonal pulses as well as the dendritic currents. The coupling via synaptic strengths may be arbitrary, up to a specific constraint. Arbitrary time delays of pulses and dendritic currents are included. The class of integrate and fire models treated here is characterized by a great variety of dendritic response functions (Green's functions). We determine the phase-locked state, the pulse interval, and present the stability equation for the case in which the dendritic response function is Rall's α-function, but still arbitrary couplings and delays are admitted.

We have developed a simple scheme to visualize the string composition of long DNA sequences and applied it to all available prokaryote complete genomes. Each species has a specific "portrait" and most clearly seen patterns in these portraits are determined by short avoided and under-represented strings. The "spatial" patterns in the space of short strings might reflect some "temporal" events in the evolutionary history.

We experimentally study stochastic resonance (SR) in the human brain through a noise effect for entrainment dynamics of the alpha (α) wave. The measurement has been carried out under the following conditions in order to obtain clear evidence of the SR phenomenon in the central nervous system. The periodic and noisy stimuli are respectively applied to the right and the left eyes of the subject independently. When only periodic and constant (but weak enough) stimulus is applied to the right eye, it does not induce any global entrainment of α-oscillators to the stimulus frequency. In this situation increasing the amplitude of the noise stimulus (frequency bandwidth from 15 to 60 Hz), harmonic entrainment occurs for a certain range of the noise amplitude and shows a sharp peak in the power spectrum of brain waves. Plotting the amplitude of the peak against the noise amplitude, a bell-type shape is seen, i.e. the maximum signal amplitude at an optimum noise. It clearly indicates the SR phenomenon. Taking account of our experimental conditions and set-up, the present SR phenomenon occurs not on the retina but on the visual processing area in the central nervous systems beyond the optic chiasma.

The behavior of a system of coupled ordinary differential equations is studied in order to characterize the CA3 region of the hippocampus. Clustering and synchronization behavior in a one-dimensional array of cells modeled by a modified Morris–Lecar model is analyzed in terms of a time delay included in the model. The random formation of phase dislocations whose number increases with the time delay seems to be responsible for complex spatiotemporal patterns that have been observed. Alterations to the transmission time between cells have been simulated by adding some noise to the system.