Comprehending ecological and economic sustainability

Universite´Libre de Bruxelles, Service de Chimie-Physique, Case Postale 231, Boulevard du Triomphe, 1050 Bruxelles, Belgium~Received 28 May 1996!The numerical resolution of the Langevin equations, with specific internal noises deduced from masterequations, exhibits two qualitatively different behaviors for the reaction-diffusion wave fronts associated witheither a cubic or a quadratic chemical rate. In the case of a wave front between two stable stationary states,illustrated by the Schlo¨gl model, the effect of fluctuations in the vicinity of a bifurcation induces perturbativedeviations from the deterministic predictions on observable properties, like the propagation velocity, the profilewidth, and the value of the highest plateau. These deviations obey power laws that are determined. For wavefronts propagating into an unstable stationary state, such as in the Fisher model, a nonperturbative fluctuationeffect on velocity and profile width is observed, in relation to the selection, in the presence of noise, of aparticular solution in the continuum of linearly stable deterministic solutions. @S1063-651X~96!03010-3#PACS number~s!: 47.70.Fw, 82.20.2w, 05.40.1jI. INTRODUCTION

The subject of the present chapter is the front propagation in reaction-diffusion systems. We start with the consideration of the basic model, Fisher-Kolmogorov equation, and discuss the linear and nonlinear criteria of the velocity selection for fronts between a stable and an unstable stationary states. Then we consider the dynamics of plane and curved fronts between stable stationary states. The last section of this chapter contains the description of combustion fronts and their stability.

We study the dynamics of two charged point particles settling in a Stokes flow.We find what ranges of initial relative positions and what ranges of system parameters lead to formation of stable doublets.The system is parameterized by the ratio of radii, ratio of masses and the ratio of electrostatic to gravitational force.We focus on opposite charges.We find a new class of stationary states with the line of the particle centers inclined with respect to gravity and demonstrate that they are always locallyasymptotically stable. Stability properties of stationary states with the vertical line of the particle centers are also discussed.We find examples of systems with multiple stable stationary states.We show that the basin of attraction for each stable stationary state has infinite measure, so that particles can capture one another even when they are very distant, and even if their charge is very small. This behavior is qualitatively different from the uncharged case where there only exists a bounded set of periodic relative trajectories.We determine the range of ratios of Stokes velocities and ratio masses which give rise to non-overlapping stable stationary states (given the appropriate ratio of electrostatic to gravitational force). For non-overlapping stable inclined or vertical stationary states the larger particle is always above the smaller particle. The non-overlapping stable inclined stationary states existonly if the larger particle has greater Stokes velocity, but there are non-overlapping stable vertical stationary states where the larger particle has higher or lower Stokes velocity.

The average rate of entropy production in a homogenous chemical system is investigated in oscillating periodic and chaotic modes as well as in coexisting stationary states. The simulations are based on an abstract model of a chemical reaction system with three freely varying concentrations. Five concentrations are assumed to be kept constant by suitable flows across the boundary. A fixed concentration is used as a control parameter. Second order mass action kinetics with reverse reaction is used. An unexpected result is that periodic modes in some windows in the chaotic interval have higher average rate of entropy production than the surrounding chaotic modes. A chaotic mode coexists with a stable stationary state with smaller entropy production. A unique (unstable) stationary state produces more entropy than the corresponding oscillating mode.

General principles of the theory of stability of solutions to differential equations are considered. The stability of equations describing the dynamics of changes in reagent concentrations in polyenzymatic biochemical chains is analyzed. Various mechanisms of formation of stable and unstable stationary states are considered, and unbalanced regimes and collapse are analyzed. The influence of systems of toxins and drugs on stability is studied. An interpretation of pathological processes based on stability theory is given.

The unstable branch of stationary states in an illuminated thermochemical bistable system is stabilized with an external feedback loop which does not otherwise alter the autonomous system. The technique allows the measurement of the entire set of stationary states available to the system. Trajectories of transition between branches of stationary states (relaxation) are investigated theoretically and experimentally. Critical slowing down in the speed of the trajectories is predicted and measured as the points of marginal stability are approached. Trajectories are observed to spread as they depart from the unstable stationary state and then to coalesce as they approach a stable stationary state. These results are obtained due to experimental spread in initial conditions and not from noise along the relaxation trajectory.

The eikonal approximation (instanton technique) is applied to the problem of large fluctuations of the number of species in spatially homogeneous chemical reactions with the probability density distribution described by a master equation. For both autocatalytic and nonautocatalytic reactions, the analysis of the distribution about a stable stationary state and of the transitions between coexisting stable states comes, to logarithmic accuracy, to the analysis of Hamiltonian dynamics of an auxiliary dynamical system. The latter can be done explicitly in a few cases, including one-species systems, systems with detailed balance, and systems close to the bifurcation points where the number of the stable states changes. In the last case, the fluctuations display universal features, and, for saddle-node bifurcation points, the logarithm of the probability of escape from the metastable state (per unit time) is proportional to the distance to the bifurcation point (in the parameter space) raised to the power 3/2. We compare the eikonal approximation for the stationary distribution of a master equation to Monte Carlo numerical solutions for two chemical two-variable systems with multiple stationary states, where none of the cited restrictions exists. For one of the systems in the pattern of optimal paths we observe caustics emanating from the saddle point.

A model based on five independent composition variables has been tested by simulating the complex bursting patterns observed experimentally for a Belousov–Zhabotinsky system in a flow reactor. If the system is in a region of constraint space where all stationary states are locally unstable, and if malonic acid concentration in the feed is decreased at constant flow of bromide ion, simple limit-cycle oscillations persist until the system reaches the Hopf bifurcation which generates a stable stationary state with low concentration of bromide ion. If bromide ion concentration in the feed is increased at constant flow of malonic acid, more and more small-amplitude oscillations are interspersed between trains of fewer and fewer large-amplitude excursions until a Hopf bifurcation leads to a stable stationary state with a high concentration of bromide ion. Patterns may be quite complex but usually repeat regularly. If a firing number is defined as ΣS/(ΣS+ΣL), where the summations represent the numbers of small- and large-amplitude oscillations in a complete pattern, then this firing number increases monotonically with increasing bromide ion in the feed. Narrow regions of chaotic mixing of patterns are also observed. The computations agree spectacularly with the careful experimental observations of Maselko and Swinney. Two separate curves are generated by the Hopf bifurcations which limit local stabilities of stationary states containing low and high concentrations of bromide ion, and the intersection of these curves provides a rationale for the ‘‘cross-shaped phase diagrams’’ which define regions of bistability and of oscillations in constraint spaces for such systems.

Semiconductor superlattices are periodic nanostructures consisting of epitaxially grown quantum wells and barriers. For thick barriers, the quantum wells are weakly coupled and the main transport mechanism is a sequential resonant tunneling of electrons between wells. We review quantum transport in these materials, and the rate equations for electron densities, currents, and the self-consistent electric potential or field. Depending on superlattice configuration, doping density, temperature, voltage bias, and other parameters, superlattices behave as excitable systems, and can respond to abrupt dc bias changes by large transients involving charge density waves before arriving at a stable stationary state. For other parameters, the superlattices may have self-sustained oscillations of the current through them. These oscillations are due to repeated triggering and recycling of charge density waves, and can be periodic in time, quasiperiodic, and chaotic. Modifying the superlattice configuration, it is possible to attain robust chaos due to wave dynamics. External noise of appropriate strength can generate time-periodic current oscillations when the superlattice is in a stable stationary state without noise, which is called the coherence resonance. In turn, these oscillations can resonate with a periodic signal in the presence of sufficient noise, thereby displaying a stochastic resonance. These properties can be exploited to design and build many devices. Here, we describe detectors of weak signals by using coherence and stochastic resonance and fast generators of true random sequences useful for safe communications and storage.

Traditional cellular neural networks are arrays of coupled processing cells, where every uncoupled cell has two stationary stable states. In this paper an oscillatory function is used to define a cellular neural network whose uncoupled cells have a number of stable stationary states larger than two, so the output image has three or more grey levels. The proposed dynamics is applied to the decomposition of images with multiple gray levels.

The laser as nonlinear element for an optical associative memory

The Kuroshio Extension (KE) jet transitions between stable and unstable states on interannual time scales. Cross‐jet eddy mixing in the two states is contrasted in the KE region ( ), using a global eddying configuration of the Parallel Ocean Program with online numerical particles. The 4 year period chosen (June 1994 to May 1998) covers a full cycle of the stable state, unstable state and the transition period. Large values of cross‐jet eddy diffusivities within the KE jet are concentrated in the upper 1000 m. In the upper ocean, elevated cross‐jet mixing within the KE jet is mainly concentrated in the downstream part of the KE jet, where the jet is weak but eddy activity is strong. The simulated time‐mean KE jet is more intense and extends further east in the stable state than in the unstable state. Consequently, strong cross‐jet mixing within the KE jet is located west of E during June 1996 to May 1997 (a typical unstable state), but east of during June 1995 to May 1996 (a typical stable state). However, average mixing within the KE jet is indistinguishable in the typical stable and unstable states. In the deep ocean, mixing is strongly influenced by topography, and thus their horizontal structures have less inter‐annual variability than in the upper ocean. One caveat is that results here cover one representative cycle of the two states. To obtain the climate mean mixing structures for the stable or unstable state, one would need numerical output covering a period much longer than 4 years.

Results concerning the middle-latitude atmospheric flow in barotropic and quasi-geostrophic approximations are presented. With no dissipation, two semi-infinite regions of stability bound a finite region of instability, ensuring that the solutions are limited. In the presence of dissipation, above a critical value of forcing below which there is one stationary stable state, there are three stationary states, one unstable and two stable. Numerically we find that the two basins of attraction in phase space are strongly asymmetric, the one, corresponding to the solution resonant with the mountain, being much larger than the other. The model used for this study is the Charney-DeVore one.

In the last few decades, the stabilization in stationary states has emerged as a new auspicious campaigner in chaos theory and found a celebrated place through various control techniques such as predictive control, delayed feedback control, constant proportional feedback control and oscillating feedback control system. Generally, it is accepted that the superiority of control systems is not only to quash the irregular distribution of stationary states, but also to illustrate its basin of attractions as large as possible depending on the numerical as well as analytical observance. In this article, the universal stabilization in unstable stationary states is studied through superior fixed point feedback control system for a family of one-dimensional maps. Further, it is interesting to know that the novel system provides freedom in the control parameter γ due to which the stabilization increases more rapidly for the lager range of parameter γ in [0,1]. The analytical as well as numerical simulations are demonstrated to examine the behavior of parameter γ for which the unstable stationary state admits universal stability.

<title>Intracavity laser spectroscopy using dispersive and dynamic resonators</title>