Slow-fast Dynamics Research Articles

Genetic Networks with Stochastic Fluctuations

Stochastic fluctuations may not only affect the dynamics of biological systems but also be exploited by living organisms to actively facilitate certain functions. Explicitly considering all variables and chemical reactions in a cell is unrealistic for a gene regulatory network from modeling, analysing and computing viewpoint. However, in a cell, many different time scales characterize the gene regulatory processes, which can be exploited to reduce the complexity of the mathematical models. For instance, the transcription and translation processes generally evolve on a time scale that is much slower than that of phosphorylation, dimerization or binding reactions of transcription factors. In addition, there are also many conservation conditions, such as total DNA number in a cell, which are important in modeling the gene networks. In this paper, we focus to model the gene regulatory networks with stochastic noise by using the property of fast-slow dynamics, and further investigate the biological implications of noise.

RD-CNN for driving a 2-D conveyor belt via memory shape alloys

The paper proposes a two-dimensional conveyor belt controlled by using an analog neural processing approach. A particular Cellular Neural Networks (CNN), named Reaction–Diffusion CNN (RD-CNN), is adopted to generate wave propagation phenomena. These waves can propagate on the conveyor belt plane (an elastic membrane) moving an object on it. The adopted actuator is a shape memory alloy such as nitinol. A neural network identification approach was adopted to characterize the membrane model. It is shown how both thermal and timing nitinol behaviors are very similar to the slow–fast dynamics exhibited by a RD-CNN allowing an appropriate driver system to be designed.

H∞ Control Theory Applied to Xenon Control for Load-Following Operation of a Nuclear Reactor

A robust controller is designed by applying the H∞ optimal control theory to the xenon control for the load-following operation of a nuclear reactor. The set of reactor model equations for controller design is a stiff system. This singularly perturbed system arises from the interaction of slow dynamics modes (iodine and xenon concentrations) and fast dynamics modes (neutron density, fuel and coolant temperatures). The singular perturbation technique is used to overcome this stiffness problem. The design specifications are incorporated by the frequency weights using the mixed-sensitivity problem approach. The robustness of H∞ control is demonstrated by comparing it with linear quadratic Gaussian (LQG) control in the case of a measurement delay of the power measurement system.Since the gains and phase margins of H∞ control are larger than those of LQG control, the H∞ control is expected to provide excellent stability robustness and performance robustness against external disturbances and noises, model parameter variations, and modeling errors as well as hardware failures. It may also provide a practical design method because the design specifications can be easily implemented by the frequency weights.

Self-induced slow-fast dynamics and swept bifurcation diagrams in weakly desynchronized systems.

In systems close to the state of phase synchronization, the fast timescale of oscillations interacts with the slow timescale of the phase drift. As a result, "fast" dynamics is subjected to a slow modulation, due to which an autonomous system under fixed parameter values can imitate repeated bifurcational transitions. We demonstrate the action of this general mechanism for a set of two coupled autonomous chaotic oscillators and for a chaotic system perturbed by a periodic external force. In both cases, the Poincaré sections of phase portraits resemble bifurcation diagram of a logistic mapping with time-dependent parameter.

Homeostasis and Aging. Slow-Fast Dynamics of Senescence and Death

Abstract Homeostasis, the well known physiological concept, is used to investigate aging. Slow age-wise decrease in homeostatic ability modulates metabolic processes in the organism and its resources gradually fall down, being maintained with weakening homeostatic mechanisms. Death occurs when the resources become exhausted. Multitime-scale methods are used to model slow-fast dynamics of homeostasis, and a simple two-time-scale model is presented describing slow-fast oxygen dynamics. Drosophila Subobscura is used as an example. As the model is a mathematical correlate of the oxygen damage concept and the Rate-of-Living theory, the results of the modeling may be regarded as supporting these theories.

On the stock–recruitment relationships in fish population models

The framework of this article is modeling applied to fisheries. It aims at relating some classical stock–recruitment curves to a dynamic structured model, consisting of the adults stages of the stock and a pre-recruits stage (i.e., eggs, larvae, juveniles). The main result is that no functional stock–recruitment relationship can be found, unless further restrictive assumptions are added to the model (slow–fast dynamics and uniformity). Among the curves thus obtained is the Beverton–Holt curve, but not Ricker's.

Visualizing the dynamical behavior of Wonderland

The article concentrates on the visualization of a four dimensional continuous dynamical system, the Wonderland model. This system poses some interesting analysis challenges. Its temporal evolution, for example, is characterized by slow-fast dynamics. This means that some of the system variables vary much faster than others. The velocity varies by two magnitudes and thus makes visualization difficult. Due to the underlying slow-fast dynamics, some equilibrium surfaces play an essential role in the model's analysis. We consider equilibrium surfaces on which the derivative of the fast system variable equals zero. Due to their definition, these equilibrium surfaces do not describe stream surfaces, which means that trajectories intersect and cross them. For these surfaces we cannot use visualization techniques designed for stream surfaces, which are based on the assumption that a flow starting on the surface will stay within it. Econometrics research typically uses only elementary visualization techniques such as still images, simple line drawings, and 2D projections or cross-sections of higher dimensional phase spaces. Color use is scarce. We collaborated with econometricians to apply more sophisticated visualization techniques and adaptations to the Wonderland model using standard visualization software (AVS by Advanced Visual Systems). The visualizations provided new insights into the system's slow-fast dynamics. Using a commercial rendering package (Autodesk's 3D Studio), we also turned the visualizations into high quality images for presentations.

Slow-fast dynamics in a model of population and resource growth.

"Models of the interaction of population, the economy, and the environment often contain nonlinear functional relationships and variables that move at different speeds. These properties foster apparent unpredictabilities in system behaviour. Using a simple deterministic model of demographic, economic and environmental interactions we illustrate the usefulness of geometric singular perturbation theory in environmental population economics. In contrast to local stability analysis, the theory of slow-fast dynamics helps to gain new insights into the global behaviour of the system. In particular, the knowledge of the basins of attraction of the stationary states enables one to determine the regions of sustainable future paths of resources and population." (SUMMARY IN FRE)

Slow-fast dynamics in Wonderland

Taking Wonderland—a simple model of demographic, economic and environmental interactions—as our artificial world, we illustrate the use of geometric singular perturbation theory in environmental demoeconomics. The theory of slow-fast dynamics helps us to gain new insights into the system's behaviour and enables one to reduce the inherent unpredictability of a “natural catastrophe” in Wonderland. Though we cannot predict the exact date of such an “environmental crash”, we can state the specific demographic, economic and environmental constellations of our artificial world at which the sustainability of nature becomes endangered.

Fast-slow dynamics for parabolic perturbations of conservation laws

We study the {open_quote}large time{close_quote} behaviour of parabolic perturbations of hyperbolic systems having on linearly degenerate field, the others being genuinely non linear fields. The initial data is periodic. For {open_quote}moderated time{close_quote}, the perturbation can often be ignored. For {open_quote}large time{close_quote}, the oscillations in non liner modes should be damped whereas the linear one behaves as a travelling wave. Because of the coupling, all the modes are modulated by the slow time. The purpose of the article is three-fold. First we give a mathematical description of this problem by means of an asymptotic expansion. We then formally describe the slow evolution for the Navier-Stokes equations of a compressible viscous heat conductive fluid. Finally, we justify the asymptotic development for model problem, arising in elasticity theory: the Keyfitz-Kranzer`s system. 8 refs.

Noise and slow-fast dynamics in a three-wave resonance problem.

Recent research on the dynamics of certain fluid-dynamical instabilities shows that when there is a slow invariant manifold subject to fast time-scale instability the dynamics are extremely sensitive to noise. The behavior of such systems can be described in terms of a one-dimensional map, and previous work has shown how the effect of noise can be modeled by a simple adjustment to the map. Here we undertake an in-depth investigation of a particular set of equations, using the methods of stochastic integration. We confirm the prediction of the earlier studies that the noise becomes important when \ensuremath{\mu}\ensuremath{\Vert}ln\ensuremath{\epsilon}\ensuremath{\Vert}=O(1), where \ensuremath{\mu} is the small time-scale ratio and \ensuremath{\epsilon} is the noise level. In addition, we present detailed information about the statistics of the solution when the noise is a dominant effect; the analytical results show excellent agreement with numerical simulations.

Existence of periodic travelling wave solutions of reaction-diffusion systems with fast-slow dynamics

On demontre des theoremes d'existence et de non existence de solutions periodiques pour des systemes de reaction-diffusion