Slow-fast Decomposition Research Articles

Stability Analysis of Some Types of Singularly Perturbed Time-Delay Differential Systems: Symmetric Matrix Riccati Equation Approach

Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficiently small value of the parameter of singular perturbation, are analyzed. For the stability analysis in the linear case, a partial exact slow–fast decomposition of the original system and an application of the Symmetric Matrix Riccati Equation method are proposed. Such an analysis yields parameter-free conditions, providing the asymptotic stability of the considered linear singularly perturbed time-delay differential systems for any sufficiently small value of the parameter of singular perturbation. Using the asymptotic stability results for the considered linear systems and the method of asymptotic stability in the first approximation, parameter-free conditions, guaranteeing the asymptotic stability of the trivial solution to the considered nonlinear systems for any sufficiently small value of the parameter of singular perturbation, are derived. Illustrative examples are presented.

Pairing cellular and synaptic dynamics into building blocks of rhythmic neural circuits. A tutorial.

In this study we focus on two subnetworks common in the circuitry of swim central pattern generators (CPGs) in the sea slugs, Melibe leonina and Dendronotus iris and show that they are independently capable of stably producing emergent network bursting. This observation raises the question of whether the coordination of redundant bursting mechanisms plays a role in the generation of rhythm and its regulation in the given swim CPGs. To address this question, we investigate two pairwise rhythm-generating networks and examine the properties of their fundamental components: cellular and synaptic, which are crucial for proper network assembly and its stable function. We perform a slow-fast decomposition analysis of cellular dynamics and highlight its significant bifurcations occurring in isolated and coupled neurons. A novel model for slow synapses with high filtering efficiency and temporal delay is also introduced and examined. Our findings demonstrate the existence of two modes of oscillation in bicellular rhythm-generating networks with network hysteresis: i) a half-center oscillator and ii) an excitatory-inhibitory pair. These 2-cell networks offer potential as common building blocks combined in modular organization of larger neural circuits preserving robust network hysteresis.

Quasi-neutral dynamics in a coinfection system with N strains and asymmetries along multiple traits.

Understanding the interplay of different traits in a co-infection system with multiple strains has many applications in ecology and epidemiology. Because of high dimensionality and complex feedback between traits manifested in infection and co-infection, the study of such systems remains a challenge. In the case where strains are similar (quasi-neutrality assumption), we can model trait variation as perturbations in parameters, which simplifies analysis. Here, we apply singular perturbation theory to many strain parameters simultaneously and advance analytically to obtain their explicit collective dynamics. We consider and study such a quasi-neutral model of susceptible-infected-susceptible (SIS) dynamics among N strains, which vary in 5 fitness dimensions: transmissibility, clearance rate of single- and co-infection, transmission probability from mixed coinfection, and co-colonization vulnerability factors encompassing cooperation and competition. This quasi-neutral system is analyzed with a singular perturbation method through an appropriate slow-fast decomposition. The fast dynamics correspond to the embedded neutral system, while the slow dynamics are governed by an N-dimensional replicator equation, describing the time evolution of strain frequencies. The coefficients of this replicator system are pairwise invasion fitnesses between strains, which, in our model, are an explicit weighted sum of pairwise asymmetries along all trait dimensions. Remarkably these weights depend only on the parameters of the neutral system. Such model reduction highlights the centrality of the neutral system for dynamics at the edge of neutrality and exposes critical features for the maintenance of diversity.

AC conductivity, dielectric and thermal properties of three-phase hybrid composite: PVA/Bi2O3/cotton microfiber composite

The impact of bismuth (III) oxide (Bi2O3) on the characteristics of the cellulose/polyvinyl alcohol (PVA) blend was reported for a high weight ratio of the oxide (15wt%). Composite samples were made with 15wt% oxide and 2:1 weight ratio of PVA to cellulose using a hot hydraulic press technique (5 MPa and 175 °C), which led to samples in the form of a disk. The thermal stability of the composite was illustrated using the thermal gravitational analysis (TGS) at a heating rate of 10 °C min−1 in N2 environment. The results show that the thermal stability of the composite sample was greater than that of the blended sample in the high-temperature region. The blend and composite samples exhibited two weight-loss stages throughout the thermal decomposition process. These two stages correspond to the slow decomposition (200 to 400 °C) and fast decomposition stages (400 to 450 °C for blend and from 430 to 460 °C for composite). Only 5% mass loss for both samples was detected due to heating from 50 °C to 200 °C. Dielectric spectroscopy (from 100 Hz to 1 MHz) was used to investigate the effects of Bi2O3 on the relaxation and conduction mechanisms of the composite samples at different temperatures. Dielectric permittivity, AC conductivity, electrical modulus, and complex impedance were investigated. Jonscher’s equation was applied to the blend and composite samples. The modified Jonscher’s equation fit well at low temperatures. As the temperature increases, the deviation from the normal Jonscher equation decreases. The activation energies of the blend and composite were calculated by determining the bulk resistance (RB) from the Nyquist plots. The activation energy of the blend was increased by adding the filler (Bi2O3).

Efficient hydrogen evolution by liquid phase plasma irradiation over Sn doped ZnO/CNTs photocatalyst

In this study, the liquid phase plasma (LPP) was irradiated over pure zinc oxide (ZnO), strontium (Sn) doped ZnO, and Sn doped ZnO/CNTs photocatalysts for hydrogen evolution from pure water and from aqueous solution of water-methanol. The possible relationship between hydrogen evolution and optical emissions from LPP for activation of ZnO based photocatalysts was revealed. The role of carbon nanotubes (CNTs) as a support material for improved photocatalytic hydrogen evolution was also investigated in this study. The photocatalytic hydrogen evolution from water mixed methanol under LPP irradiation was compared with pure water splitting. The photolysis produced negligible amount of hydrogen due to minimal photodecomposition of water molecules under LPP irradiation. The plasma born reactive species also played crucial role in photolysis. However, the hydrogen evolution rate increased significantly in the presence of ZnO photocatalyst. Further improvement in hydrogen evolution rate was noticed on Sn doping of ZnO and compositing with CNTs. The highest hydrogen evolution rate of 11.46 mmh−1g−1 from water mixed methanol was achieved with Sn doped ZnO/CNTs photocatalyst. This hydrogen evolution rate from water-methanol solution was 9 times higher than from the splitting of pure water. This hydrogen evolution rate is attributed to excessive production of hydroxyl radicals, red shift in optical band gap of Sn doped ZnO/CNTs photocatalyst, slow electron-hole recombination and fast decomposition of methanol as sacrificial reagent.

Hyperbolicity of velocity–stress–electromagnetic field equations for waves in anisotropic magnetoelectroelastic solids with hexagonal symmetry

This paper reports on a theoretical framework to analyse coupled wave propagation in magnetoelectroelastic solids of hexagonal symmetry. The constitutive equations contain the effective properties of micromechanics to model piezoelectric–piezomagnetic composites. The Mori–Tanaka scheme is used. The governing equations include the elastodynamic equations of motions and unique forms of the Maxwell equations for electromagnetics. The result is a set of fifteen first-order, fully coupled, hyperbolic partial differential equations with velocities, elastic stress components, and electromagnetic fields as the unknowns and as the components of the state vector. The structural analysis of the introduced equation set is performed, and the hyperbolicity is formally proved. The physics of coupled acoustic–electromagnetic wave propagation are fully described by the eigenstructure of matrices included in the considered system of equations. In particular, the eigenvalues of the main matrix pencil are the wave speeds, and a part of the left eigenvectors represents the coupled-wave polarization. The spectrum of the main matrix pencil shows the two-scale structure with the gap between mainly electromagnetic and mainly acoustic coupled modes. The analysis of the two-scale system dynamics is also performed. The small parameter is identified, and the slow–fast decomposition is obtained. The slow subsystem in the two-time-scale model is identified as the so-called quasi-static approximation of the magnetoelectroelastic continuum dynamics.

Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay

<p style='text-indent:20px;'>A singularly perturbed linear time-dependent controlled system with a point-wise nonsmall (of order of <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>) delay in the input (the control variable) is considered. Sufficient conditions of the complete Euclidean space controllability for this system, robust with respect to the parameter of singular perturbation, are derived. This derivation is based on an asymptotic analysis of the controllability matrix for the considered system and on such an analysis of the determinant of this matrix. However, this derivation does not use a slow-fast decomposition of the considered system. The theoretical result is illustrated by an example.

Extended dissipative synchronization for singularly perturbed semi-Markov jump neural networks with randomly occurring uncertainties

This paper concentrates on the synchronization problem for singularly perturbed neural networks with semi-Markov jump parameters and randomly occurring uncertainties. A continuous-time semi-Markov process is utilized to model the stochastic switching of the parameters. An independent singularly perturbed parameter is separated through the use of singularly perturbed slow-fast decomposition method. Some sufficient conditions are deduced to ensure that the error system is synchronized and meets the extended dissipative property. In particular, the uncertainty of the networks is considered to occur randomly, which is more realistic than the existing work. Moreover, the efficiency of the presented method is demonstrated by a numerical example.

Exact nonlinear model reduction for a von Kármán beam: Slow-fast decomposition and spectral submanifolds

We apply two recently formulated mathematical techniques, Slow-Fast Decomposition (SFD) and Spectral Submanifold (SSM) reduction, to a von Kármán beam with geometric nonlinearities and viscoelastic damping. SFD identifies a global slow manifold in the full system which attracts solutions at rates faster than typical rates within the manifold. An SSM, the smoothest nonlinear continuation of a linear modal subspace, is then used to further reduce the beam equations within the slow manifold. This two-stage, mathematically exact procedure results in a drastic reduction of the finite-element beam model to a one-degree-of freedom nonlinear oscillator. We also introduce the technique of spectral quotient analysis, which gives the number of modes relevant for reduction as output rather than input to the reduction process.

Exact model reduction by a slow–fast decomposition of nonlinear mechanical systems

We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the softer degrees of freedom. This slow–fast decomposition (SFD) enslaves exponentially fast the stiffer degrees of freedom to the softer ones as all oscillations converge to the reduced model defined on a slow manifold. We obtain an expression for the domain boundary beyond which the reduced model ceases to be relevant due to a generic loss of stability of the slow manifold. We also find that near equilibria, the SFD gives a mathematical justification for two modal reduction methods used in structural dynamics: static condensation and modal derivatives. These formal reduction procedures, however, are also found to return incorrect results when the SFD conditions do not hold. We illustrate all these results on mechanical examples.

A Novel Approach to Exact Slow-Fast Decomposition of Linear Singularly Perturbed Systems with Small Delays

A linear time-invariant singularly perturbed system with multiple pointwise and distributed small time delays is considered. A novel (direct) approach to exact slow-fast decomposition of this system is proposed. In contrast with the existing method, this approach uses neither a preliminary transformation of the original differential system to an integral one nor rather complicated integral manifold and operator techniques. Moreover, the approach of the present paper does not assume the exponential stability of the fast subsystem. Based on this decomposition, an exact slow-fast decomposition of the spectrum of a singularly perturbed system with a single pointwise small delay is carried out. Using the theoretical results, the stability of a multilink single-sink optical network is analyzed.

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.

A novel method for fault detection in singularly perturbed systems via the finite frequency strategy

This paper addresses the fault detection synthesis for a class of linear singularly perturbed systems with unknown disturbances. The observer is designed via a novel method, in which singularly perturbed systems could be processed within a uniform framework instead of using the classical slow–fast decomposition. The problem of robust fault detection can be converted into a standard H∞ model-matching. Based on the generalized Kalman–Yakubovich–Popov lemma and parameter-dependent Lyapunov functions, a full-order observer is designed such that the corresponding error dynamic system is asymptotically stable and satisfies a prescribed finite-frequency H_/H∞ performance index. A novel three-step design procedure is then proposed to extract the fault feature from strong background disturbances. An illustrative example is given to demonstrate the validity and applicability of the proposed approaches in the simulation part.

Study on the Congestion Controller for Timedelay Networked Control Systems with External Disturbances

Abstract A successive approximation approach (SAA) is developed to obtain a new congestion controller for the singularly perturbed time-delay networked control systems affected by external disturbances. Based on the slow-fast decomposition theory of singular perturbations, the system is first decomposed into a fast non-delay subsystem and a slow time-delay subsystem with disturbances. Then, the perturbation approach is proposed to solve the slow-time scale time-delay optimal control problem, and the feedforward compensation technique is used to reject the external disturbances. We obtain the conditions of existence and uniqueness of the feedforward and feedback composite control (FFCC) law. The FFCC law consists of linear analytic functions and a time-delay compensation term which is a series sum of adjoint vectors. The linear analytic functions can be found by solving a Riccati matrix equation and a Sylvester equation respectively. The compensation term can be approximately obtained by an iterative formula of adjoint vector equations. A reduced-order disturbance observer is constructed to make the FFCC law physically realizable. Numerical examples are presented to illustrate the effectiveness and robustness of the proposed design approach.

ASYMMETRIC TRANSITION AND TIME-SCALE SEPARATION IN INTERLINKED POSITIVE FEEDBACK LOOPS

In [Brandman et al., 2005] it was proposed that interlinked fast and slow positive feedback loops are a frequent motif in biological signaling, because such a device can allow for a rapid response to an external stimulus (sensitivity) along with a certain noise-buffering capacity (robustness), as soon as the two loops operate on different time scales. Here we explore the properties of the nonlinear system responsible for this behavior. We argue that (a) the noise buffering is not linked to the stochastic nature of the stimulus, but only to the time scale of the stimulus variation compared to the intrinsic time scales of the system, and (b) this buffering of stimulus variations follows from the stabilization of a region of the state space away from the equilibrium branches of the system. Our analysis is based on a slow-fast decomposition of the dynamics. We analyze the strength of this buffering as a function of the time scales involved and the Boolean logic of the coupling between dynamic variables, as well as of the amplitude of the stimulus variations. We underline that such a nonequilibrium regime is universal as soon as the stimulus time scale is smaller than the larger time scale of the system, preventing the prediction of the behavior from the features of the bifurcation diagram or using a linear analysis.

Interactions of persistent sodium and calcium-activated nonspecific cationic currents yield dynamically distinct bursting regimes in a model of respiratory neurons.

The preBötzinger complex (preBötC) is a heterogeneous neuronal network within the mammalian brainstem that has been experimentally found to generate robust, synchronous bursts that drive the inspiratory phase of the respiratory rhythm. The persistent sodium (NaP) current is observed in every preBötC neuron, and significant modeling effort has characterized its contribution to square-wave bursting in the preBötC. Recent experimental work demonstrated that neurons within the preBötC are endowed with a calcium-activated nonspecific cationic (CAN) current that is activated by a signaling cascade initiated by glutamate. In a preBötC model, the CAN current was shown to promote robust bursts that experience depolarization block (DB bursts). We consider a self-coupled model neuron, which we represent as a single compartment based on our experimental finding of electrotonic compactness, under variation of g (NaP), the conductance of the NaP current, and g (CAN), the conductance of the CAN current. Varying these two conductances yields a spectrum of activity patterns, including quiescence, tonic activity, square-wave bursting, DB bursting, and a novel mixture of square-wave and DB bursts, which match well with activity that we observe in experimental preparations. We elucidate the mechanisms underlying these dynamics, as well as the transitions between these regimes and the occurrence of bistability, by applying the mathematical tools of bifurcation analysis and slow-fast decomposition. Based on the prevalence of NaP and CAN currents, we expect that the generalizable framework for modeling their interactions that we present may be relevant to the rhythmicity of other brain areas beyond the preBötC as well.

Neuronal bursting: interactions of the persistent sodium and CAN currents

The pre-Botzinger complex (pBC) is a heterogeneous neuronal network within the mammalian brainstem and has been experimentally found to generate robust, synchronous bursts [1]. Significant modeling research has been conducted on characterizing the dynamics of individual neurons within the pBC. [2, 3] It is well known that the persistent sodium current (INaP) contributes to square-wave bursting seen in the pBC [4]. Recent experimental work within the pBC identified a signaling cascade that starts with presynaptic glutamate and ends with the release of intracellular calcium that activates a nonspecific cationic current (ICAN) [5]. A subsequent model demonstrated that ICAN may contribute to bursts within the pBC that exhibit depolarization block [6]. With these two mechanisms for generating bursts present within the pBC, an open question is how do they combine to generate the robust bursts seen in the network? The present work seeks to analyze the result of including both INaP and ICAN within the same model. We consider the effects of heterogeneity in the conductance gNaP of INaP and the conductance gCAN of ICAN; with this heterogeneity in mind, the model cell may be quiescent, tonically active, have only square-wave bursts, have only depolarization-block exhibiting bursts, or may show both types of bursting. Using the mathematical tools of bifurcation analysis and slow-fast decomposition, we illuminate the mechanisms underlying the transitions of a model cell between the types of dynamics listed above. Our results show that, in cases where gCAN is relatively high, increasing gNaP increases the range of gCAN where the resultant cell has depolarization-block exhibiting bursts. On the other hand, when gCAN is relatively low, increasing gNaP may cause the cell to transition from quiescence, to square wave bursting, to tonic activity, to square wave bursts with high duty cycles, and finally further increase of gNaP causes the cell to again be tonically active. The latter two transitions do not occur if ICAN is absent. The interactions of ICAN and INaP are relevant to many systems beyond the pBC. Individually, ICAN and INaP have been focused on as important to rhythmic burst generation in other systems such as the entorhinal cortex [7]; however, it is likely that both currents are present in these systems. Thus, a detailed account for the interaction of ICAN and INaP may help explain the rhythm generation encountered in other systems beyond the pBC.

Exact slow-fast decomposition of the singularly perturbed matrix differential Riccati equation

In this paper the Hamiltonian matrix formulation of the Riccati equation is used to derive the reduced-order pure-slow and pure-fast matrix differential Riccati equations of singularly perturbed systems. These pure-slow and pure-fast matrix differential Riccati equations are obtained by decoupling the singularly perturbed matrix differential Riccati equation of dimension n 1 + n 2 into the pure-slow regular matrix differential Riccati equation of dimension n 1 and the pure-fast stiff matrix differential Riccati equation of dimension n 2 . A formula is derived that produces the solution of the original singularly perturbed matrix differential Riccati equation in terms of solutions of the pure-slow and pure-fast reduced-order matrix differential Riccati equations and solutions of two reduced-order initial value problems. In addition to its theoretical importance, the main result of this paper can also be used to implement optimal filtering and control schemes for singularly perturbed linear time-invariant systems independently in pure-slow and pure-fast time scales.

On the dynamics of chaotic spiking-bursting transition in the Hindmarsh–Rose neuron

The paper considers the neuron model of Hindmarsh-Rose and studies in detail the system dynamics which controls the transition between the spiking and bursting regimes. In particular, such a passage occurs in a chaotic region and different explanations have been given in the literature to represent the process, generally based on a slow-fast decomposition of the neuron model. This paper proposes a novel view of the chaotic spiking-bursting transition exploiting the whole system dynamics and putting in evidence the essential role played in the phenomenon by the manifolds of the equilibrium point. An analytical approximation is developed for the related crucial elements and a subsequent numerical analysis signifies the properness of the suggested conjecture.

Bumpless Transfer for Adaptive Switching Controls

In this paper, a new bumpless transfer method is introduced based on slow-fast decomposition of the controller. The method is especially well-suited to situations in which the plant model is poor or yet to be identified, as may be the case in adaptive switching control. Simulation results are presented.