Slow-fast Dynamics Research Articles

Slow and fast dynamics model of a Malaria with Sickle-Cell genetic disease with multi-stage infections of the mosquitoes population

Malaria, which is caused by Plasmodium, is a common disease in tropical areas. There are three types of Plasmodium i.e. Plasmodium Vivax, Plasmodium Malariae, and Plasmodium Falciparum. The most dangerous cases of the Malaria are mainly caused by the Plasmodium Falciparum. One of the important characteristics for the Plasmodium infection is due to the immunity of erythrocyte that contains HbS (Haemoglobin Sickle-cell) genes. The individuals who has the HbS gene has better immunity against the disease. In this paper, we consider a model that shows the spread of malaria involving the interaction between the mosquitos population, the human who has HbS genes population and the human with normal gene population. We do some analytical and numerical simulation to study the basic reproduction ratio and the slow-fast dynamics of the phase-portrait. The slow dynamics in our model represents the response of the human population with HbS gene to the Malaria disease while the fast dynamics show the response of the human population with the normal gene to the disease. The slow and fast dynamics phenomena are due to the fact that the population of the individuals who have HbS gene is much smaller than the individuals who has normal genes.

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow–fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

Singularly Perturbed Networked Control Systems

We study networked control systems (NCSs) where the controller is given by a state-feedback law and the plant is modeled by a dynamical system evolving on two time-scales, representing a characterization by some slow and fast dynamics. When using the stability analysis frameworks for NCSs from the literature, this time-scale separation is ignored and, as a result, the slow dynamics are in general updated at the same rate as the fast dynamics, leading to many redundant transmissions of the slow dynamics. Therefore, we assume in this paper that the slow dynamics and fast dynamics can be transmitted separately over the network, allowing us to use techniques inspired by singular perturbation methods in the stability analysis. That is, we show by means of a Lyapunov-based proof how to obtain conditions on the transmission rates (expressed in maximal allowable transmission intervals (MATIs)) for the slow and fast dynamics separately such that stability of the NCS is guaranteed, based only on approximated models of the slow and the fast dynamics.

Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics

We consider a smooth planar system having slow-fast motion, where the slow motion takes place near a curve γ. We explore the idea of replacing the original smooth system with a system with discontinuous right-hand side (DRHS system for short), whereby the DRHS system coincides with the smooth one away from a neighborhood of γ. After this reformulation, in the region of phase-space where γ is attracting for the DRHS system, we will obtain sliding motion on γ and numerical methods apt at integrating for sliding motion can be applied. Moreover, we further bypass resolving the sliding motion and monitor entries (transversal) and exits (tangential) on the curve γ, a fact that can be done independently of resolving for the motion itself. The end result is a method free from the need to adopt stiff integrators or to worry about resolving sliding motion for the DRHS system. We illustrate the performance of our method on a few problems, highlighting the feasibility of using simple explicit Runge-Kutta schemes, and that we obtain much the same orbits of the original smooth system.

A latencies tolerant model predictive control approach to damp Inter-area oscillations in delayed power systems

Automatic voltage regulators and power system stabilizers have been employed successfully to deal with fast dynamics associated to local oscillations phenomena. Whereas, inter-area oscillations damping in large power systems requires remote feedback controllers fed by Wide Area Monitoring Systems (WAMS). Although these improvements in the inter-area oscillations behavior have been effective, a new challenge emerges: reaching stability with a closed loop control despite latencies due to measurements taken far away from the control centre. The research was motivated by the need for modernization of power systems capable of dealing with control difficulties in centralised WAMS for damping inter-area oscillations in power systems caused by delays in the communication system. Herein, time delay and control problems are addressed separately. The time delay problem is solved by a database based time compensation solution relying on the most updated available state of the system. The control problem is solved by a Model Predictive Control (MPC) with terminal cost and constraint set to handle complexities due to nonlinearities of the power system, the large scale of the problem and the parametric variations. Both solutions work in a coordinated way with local controllers to implement a decentralised coordinated strategy that manages slow global dynamics and fast local dynamics as well. The integrated proposed approach is called Time-Delay-Tolerant Model Predictive Control (TDT-MPC). Coordinated and coherent performance of the two TDT-MPC components (Kalman compensator and MPC) is achieved thanks to unifying power system reference models for both strategies. The approach has been tested on the IEEE 39 system and validated with time domain nonlinear simulations, obtaining post-fault damped oscillations and a good tracking of new power references when tripping tie lines.

Optimal Management of Ecosystem Services with Pollution Traps: The Lake Model Revisited

AbstractIn this paper, optimal management of the lake model and common-property outcomes are reconsidered when the lake model is extended with the slowly changing variable. New optimal trajectories are found that were hidden in the simplified analysis. Furthermore, it is shown that two Nash equilibria may exist with the one leading to the steady state with a high level of ecological services dominating the other one. For larger initial states, only the Nash equilibrium with a low level of ecological services exists, which implies that the users of the lake can be trapped in the bad Nash equilibrium. Finally, it is shown that implementing the optimal phosphorus loadings from the simplified version into the full lake model can have considerable welfare losses, as the lake can end up in the wrong basin of attraction, but in most cases welfare losses are small because of the fast-slow dynamics.

Systems Biology Approach and Mathematical Modeling for Analyzing Phase-Space Switch During Epithelial-Mesenchymal Transition.

In this report, we aim at presenting a viable strategy for the study of Epithelial-Mesenchymal Transition (EMT) and its opposite Mesenchymal-Epithelial Transition (MET) by means of a Systems Biology approach combined with a suitable Mathematical Modeling analysis. Precisely, it is shown how the presence of a metastable state, that is identified at a mesoscopic level of description, is crucial for making possible the appearance of a phase transition mechanism in the framework of fast-slow dynamics for Ordinary Differential Equations (ODEs).

A parametrized stock-recruitment relationship derived from a slow-fast population dynamic model

The Beverton–Holt, Ricker and Deriso functions are three distinct descriptions of the link between a parental population size and subsequent offspring that may survive to become part of the fish population.This paper presents a model consisting of a system of ordinary differential equations, which couples a stage of young fish with several adult stages. The slow-fast dynamics captures the different time scales of the dynamics of the population and leads to a singular perturbation problem.The novelty of the model presented here is its capability to replicate a rich class of the stock-recruitment relationship, including the Beverton–Holt, Ricker and Deriso dynamics. The results are explained using geometric singular perturbation theory and illustrated by numerical simulations.

Bifurcation Analysis of Hysteretic Systems with Saddle Dynamics

Abstract This paper is devoted to the analysis of bidimensional piecewise linear systems with hysteresis coming from a reduction of symmetric 3D systems with slow-fast dynamics. We concentrate our attention on the saddle dynamics cases, determining the existence of periodic orbits as well as their stability, and possible bifurcations. Dealing with reachable saddles not in the central hysteresis band, we show the existence of subcritical/supercritical heteroclinic bifurcations as well as saddle-node bifurcations of periodic orbits.

The impact of inter-annual rainfall variability on African savannas changes with mean rainfall

Savannas are mixed tree-grass ecosystems whose dynamics are predominantly regulated by resource competition and the temporal variability in climatic and environmental factors such as rainfall and fire. Hence, increasing inter-annual rainfall variability due to climate change could have a significant impact on savannas. To investigate this, we used an ecohydrological model of stochastic differential equations and simulated African savanna dynamics along a gradient of mean annual rainfall (520–780 mm/year) for a range of inter-annual rainfall variabilities. Our simulations produced alternative states of grassland and savanna across the mean rainfall gradient. Increasing inter-annual variability had a negative effect on the savanna state under dry conditions (520 mm/year), and a positive effect under moister conditions (580–780 mm/year). The former resulted from the net negative effect of dry and wet extremes on trees. In semi-arid conditions (520 mm/year), dry extremes caused a loss of tree cover, which could not be recovered during wet extremes because of strong resource competition and the increased frequency of fires. At high mean rainfall (780 mm/year), increased variability enhanced savanna resilience. Here, resources were no longer limiting and the slow tree dynamics buffered against variability by maintaining a stable population during ‘dry’ extremes, providing the basis for growth during wet extremes. Simultaneously, high rainfall years had a weak marginal benefit on grass cover due to density-regulation and grazing. Our results suggest that the effects of the slow tree and fast grass dynamics on tree-grass interactions will become a major determinant of the savanna vegetation composition with increasing rainfall variability.

Controllable biomimetic birdsong.

Birdsong is the product of the controlled generation of sound embodied in a neuromotor system. From a biophysical perspective, a natural question is that of the difficulty of producing birdsong. To address this, we built a biomimetic syrinx consisting of a stretched simple rubber tube through which air is blown, subject to localized mechanical squeezing with a linear actuator. A large static tension on the tube and small dynamic variations in the localized squeezing allow us to control transitions between three states: a quiescent state, a periodic state and a solitary wave state. The static load brings the system close to threshold for spontaneous oscillations, while small dynamic loads allow for rapid transitions between the states. We use this to mimic a variety of birdsongs via the slow-fast modulated nonlinear dynamics of the physical substrate, the syrinx, regulated by a simple controller. Finally, a minimal mathematical model of the system inspired by our observations allows us to address the problem of song mimicry in an excitable oscillator for tonal songs.

Simplified Decoupler-Based Multivariable Controller With a Gain Scheduling Strategy for the Exhaust Gas Recirculation and Variable Geometry Turbocharger Systems in Diesel Engines

This paper presents a simplified decoupler-based multivariable controller with a gain scheduling strategy in order to deal with strong nonlinearities and cross-coupled characteristics for exhaust gas recirculation (EGR) and variable geometry turbocharger (VGT) systems in diesel engines. A feedback controller is designed with the gain scheduling strategy, which updates control gains according to engine operating conditions. The gain scheduling strategy is implemented by using a proposed scheduling variable derived from indirect measurements of the EGR mass flow, such as the pressure ratio of the intake, exhaust manifolds, and the exhaust air-to-fuel ratio. The scheduling variable is utilized to estimate static gains of the EGR and VGT systems; it has a large dispersion in various engine operating conditions. Based on the estimated static gains of the plant, the Skogestad internal model control (SIMC) method determines appropriate control gains. The dynamic decoupler is designed to deal with the cross-coupled effects of the EGR and VGT systems by applying a simplified decoupler design method. The simplified decoupler is beneficial for compensating for the dynamics difference between two control loops of the EGR and VGT systems, for example, slow VGT dynamics and fast EGR dynamics. The proposed control algorithm is evaluated through engine experiments. Step test results of set points reveal that root-mean-square (RMS) error of the gain-scheduled feedback controller is reduced by 47% as compared to those of the fixed gain controller. Furthermore, the designed simplified decoupler decreased the tracking error under transients by 14–66% in various engine operating conditions.

Erratum to: A slow-fast dynamic decomposition links neutral and non-neutral coexistence in interacting multi-strain pathogens

Understanding the dynamics of multi-type microbial ecosystems remains a challenge, despite advancing molecular technologies for diversity resolution within and between hosts. Analytical progress becomes difficult when modelling realistic levels of community richness, relying on computationally-intensive simulations and detailed parametrisation. Simplification of dynamics in polymorphic pathogen systems is possible using aggregation methods and the slow-fast dynamics approach. Here, we develop one new such framework, tailored to the epidemiology of an endemic multi-strain pathogen. We apply Goldstone’s idea of slow dynamics resulting from spontaneously broken symmetries to study direct interactions in co-colonization, ranging from competition to facilitation between strains. The slow-fast dynamics approach interpolates between a neutral and non-neutral model for multi-strain coexistence, and quantifies the asymmetries that are important for the maintenance and stabilisation of diversity.

Semigroup estimates and fast-slow dynamics in parabolic-hyperbolic systems

Abstract We present a general procedure to describe slow dynamics in parabolic-hyperbolic systems, under suitable assumptions on the terms appearing in the equations. In particular, our strategy relies in semigroup estimates for the evolution system associated to the family of time-dependent linearized operators around an interfaced configuration. We finally apply the general strategy to the explicit example of an hyperbolic system with relaxation.

Generalized Play Hysteresis Operators in Limits of Fast-Slow Systems

Hysteresis operators appear in many applications, such as elasto-plasticity and micromagnetics, and can be used for a wider class of systems, where rate-independent memory plays a role. A natural approximation for systems of evolution equations with hysteresis operators are fast-slow dynamical systems, which---in their used approximation form---do not involve any memory effects. Hence, viewing differential equations with hysteresis operators in the nonlinearity as a limit of approximating fast-slow dynamics involves subtle limit procedures. In this paper, we give a proof of Netushil's “observation” that broad classes of planar fast-slow systems with a two-dimensional critical manifold are expected to yield generalized play operators in the singular limit. We provide two proofs of this “observation” based upon the fast-slow systems paradigm of decomposition into subsystems. One proof strategy employs suitable convergence in function spaces, while the other considers a geometric strategy via local linearization and patching adapted originally from problems in stochastic analysis. We also provide an illustration of our results in the context of oscillations in forced planar nonautonomous fast-slow systems. The study of this example also strongly suggests that new canard-type mechanisms can occur for two-dimensional critical manifolds in planar systems.

Multiscale gentlest ascent dynamics for saddle point in effective dynamics of slow-fast system

Here we present a multiscale method to calculate the saddle point associated with the effective dynamics arising from a stochastic system which couples slow deterministic drift and fast stochastic dynamics. This problem is motivated by the transition states on free energy surfaces in chemical physics. Our method is based on the gentlest ascent dynamics which couples the position variable and the direction variable and has the local convergence to saddle points. The dynamics of the direction vector is derived in terms of the covariance function with respective to the equilibrium distribution of the fast stochastic process. We apply the multiscale numerical methods to efficiently solve the obtained multiscale gentlest ascent dynamics, {and discuss the acceleration techniques based on the adaptive idea.} The examples of stochastic ordinary and partial differential equations are presented.

Bursting and Synchronization in a Two-Compartment Model with Current-Feedback Control

Based on the fast–slow dynamics and bifurcation analysis, two different types of bursting, that is, the “subHopf/homoclinic” and the “circle/fold cycle” types of bursting, are presented and analyzed in a two-compartment neuron model with current feedback control due to totally different generation mechanisms. The synchronization transition process from burst synchronization to nearly complete synchronization is considered in two electrically coupled nonidentical neurons, and fast–slow analysis can be extended to explore the bursting behavior of nearly complete synchronization. The analysis of bursting types and multitime-scale synchronization transition may help us better understand the information encoding and transmission in neural systems.

A slow-fast dynamic decomposition links neutral and non-neutral coexistence in interacting multi-strain pathogens

Understanding the dynamics of multi-type microbial ecosystems remains a challenge, despite advancing molecular technologies for diversity resolution within and between hosts. Analytical progress becomes difficult when modelling realistic levels of community richness, relying on computationally-intensive simulations and detailed parametrisation. Simplification of dynamics in polymorphic pathogen systems is possible using aggregation methods and the slow-fast dynamics approach. Here, we develop one new such framework, tailored to the epidemiology of an endemic multi-strain pathogen. We apply Goldstone’s idea of slow dynamics resulting from spontaneously broken symmetries to study direct interactions in co-colonization, ranging from competition to facilitation between strains. The slow-fast dynamics approach interpolates between a neutral and non-neutral model for multi-strain coexistence, and quantifies the asymmetries that are important for the maintenance and stabilisation of diversity.

Adaptive model parameter identification for large capacity Li-ion batteries on separated time scales

The accurate identification of battery model parameters is critical to the development of the battery management system (BMS). For large capacity Li-ion batteries, different internal processes happen inside the cell during charging and discharging, which introduce the complex dynamics that occur on different time scales. The multi time-scaled effect of the battery dynamics imposes difficulties on the design of an accurate parameter identification algorithm. As an original contribution, we propose a novel adaptive identification algorithm of the model parameters on separated time scales. The battery dynamics are described with a second-order ECM (equivalent circuit model), where the slow dynamics and fast dynamics are described separately. The parameter identification algorithm is composed of two separated modules, of which one is for the identification of slow dynamics and the other is for the identification of fast dynamics. The two modules are executed on separated time scales. The identification module for slow dynamics is based on extended Kalman filtering (EKF) while the module for fast dynamics is based on recursive least squares (RLS). The coupling of the two modules is through the voltage response of the slow dynamics. To make the algorithm more adaptive, the operation time scale of the slow identification module is not constant, but dependent on current profiles. Validation with experimental results shows that the proposed identification strategy performs better than the traditional RLS based identification methods.

A Lumped-Parameter Model of Multiscale Dynamics in Steam Supply Systems

This paper focuses on multiscale dynamics occurring in steam supply systems. The dynamics of interest are originally described by a distributed-parameter model for fast steam flows over a pipe network coupled with a lumped-parameter model for slow internal dynamics of boilers. We derive a lumped-parameter model for the dynamics through physically relevant approximations. The derived model is then analyzed theoretically and numerically in terms of existence of normally hyperbolic invariant manifold in the phase space of the model. The existence of the manifold is a dynamical evidence that the derived model preserves the slow–fast dynamics, and suggests a separation principle of short-term and long-term operations of steam supply systems, which is analog to electric power systems. We also quantitatively verify the correctness of the derived model by comparison with brute-force simulation of the original model.