Multiple Time Scale Behavior Research Articles

Dynamic Community Detection Decouples Multiple Time Scale Behavior of Complex Chemical Systems.

Although community or cluster identification is becoming a standard tool within the simulation community, traditional algorithms are challenging to adapt to time-dependent data. Here, we introduce temporal community identification using the Δ-screening algorithm, which has the flexibility to account for varying community compositions, merging and splitting behaviors within dynamically evolving chemical networks. When applied to a complex chemical system whose varying chemical environments cause multiple time scale behavior, Δ-screening is able to resolve the multiple time scales of temporal communities. This computationally efficient algorithm is easily adapted to a wide range of dynamic chemical systems; flexibility in implementation allows the user to increase or decrease the resolution of temporal features by controlling parameters associated with community composition and fluctuations therein.

A geometric analysis of the SIR, SIRS and SIRWS epidemiological models

We study fast–slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behaviour is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast–slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov’s method, we are able to not only analyse the stability of the endemic equilibria but also to show that in some of the models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model, for which we provide a detailed description of its dynamics by combining analytic and numerical techniques.

On Timescale Separation in Networked Systems With Intermittent Communication

This paper studies the multiple timescale behavior that is induced by dynamic coupling between continuous-time and discrete-time systems, and that arises naturally in distributed networked systems. An order reduction method is proposed that establishes a mathematically rigorous separation principle between the fast evolution of the continuous-time dynamics and the slow updates of the discrete-time dynamics. Quantitative conditions on the discrete update rate are then derived that ensure the stability of the coupled dynamics based on the behavior of the isolated systems. The results are illustrated for a distributed network of satellites whose attitudes evolve continuously while communicating intermittently over the network.

Advances made by Belgrade's group in research of oscillatory reactions

Oscillatory dynamic states as one form of selforganization of nonlinear systems can be found in almost all sciences, like mechanics, physical chemistry or biomedicine. Although origin of these oscillations is different, computational challenges in modelling oscillatory phenomena remain similar in all fields. Since 1979 researchers from Belgrade's group perform systematic examinations of oscillatory reactions. As stability of steady states is the central point in modelling oscillatory reactions, in last 10 years they have adapted and improved powerful tool of the Stoichiometric Network Analysis for this goal. Moreover, bifurcations of few types were identified in several models of oscillatory reactions. Even very complex chaotic motions in phase space were characterized and quantified by several numerical techniques. Multiple time scale behaviour is found within the core of the complex dynamic behaviour of mixed-mode oscillations. Analytical applications were developed, too.

Greedy Extremum Seeking Control with Applications to Biochemical Processes

Extremum seeking control is a subclass of adaptive control, aimed at steady-state optimization. In this paper we apply ideas from extremum seeking control to optimize the transient performance of processes displaying multiple time-scale behavior. The main motivation is the need to optimize biochemical reactors where the biomass growth is significantly slower than the metabolism and where it is of interest to optimize the substrate conversion during the extended periods of net biomass growth or decay. Essentially, by employing singular perturbations, we design a controller that optimizes the fast boundary layer of the system ensuring that the process output is maintained near its maximum during transients towards the steady-state. Similar to greedy methods in optimization theory, we show that the local optimization of the fast layer under certain conditions will provide convergence to the overall optimal steady-state. In particular, this will apply to the type of biochemical reactors that are of main concern in this paper. The proposed controller is demonstrated by application to a model of the CANON process used for nitrogen removal in wastewater treatment.

Multiple Time Scale Behavior and Network Dynamics in Liquid Methanol

Multiple Time Scale Behavior and Network Dynamics in Liquid Methanol

Diffusional anomaly and network dynamics in liquid silica

The present study applies the power spectral analysis technique to understand the diffusional anomaly in liquid silica, modeled using the Beest-Kramer-van Santen (BKS) potential. Molecular-dynamics simulations have been carried out to show that power spectrum of tagged particle potential energy of silica shows a regime with 1f(alpha) dependence on frequency f which is the characteristic signature of multiple time scale behaviour in networks. As demonstrated earlier in the case of water [J. Chem. Phys. 122, 104507 (2005)], the variations in the mobility associated with the diffusional anomaly are mirrored in the scaling exponent alpha associated with this multiple time scale behavior. Our results indicate that in the anomalous regime, as the local tetrahedral order decreases with temperature or pressure, the coupling of local modes to network reorganizations increases and so does the diffusivity. This symmetry-dependence of the vibrational couplings is responsible for the connection between the structural and diffusional anomalies.

Effect of Ionic Solutes on the Hydrogen Bond Network Dynamics of Water: Power Spectral Analysis of Aqueous NaCl Solutions

To understand the modifications of the hydrogen bond network of water by ionic solutes, power spectra as well as static distributions of the potential energies of tagged solvent molecules and solute ions have been computed from molecular dynamics simulations of aqueous NaCl solutions. The key power spectral features of interest are the presence of high-frequency peaks due to localized vibrational modes, the existence of a multiple time scale or 1/falpha frequency regime characteristic of networked liquids, and the frequency of crossover from 1/falpha type behavior to white noise. Hydrophilic solutes, such as the sodium cation and the chloride anion, are shown to mirror the multiple time scale behavior of the hydrogen bond network fluctuations, unlike hydrophobic solutes which display essentially white noise spectra. While the power spectra associated with tagged H2O molecules are not very sensitive to concentration in the intermediate frequency 1/falpha regime, the crossover to white noise is shifted to lower frequencies on going from pure solvent to aqueous alkali halide solutions. This suggests that new and relatively slow time scales enter the picture, possibly associated with processes such as migration of water molecules from the hydration shell to the bulk or conversion of contact ion pairs into solvent-separated ion pairs which translate into variations in equilibrium transport properties of salt solutions with concentration. For anions, cations, and solvent molecules, the trends in the alpha exponents of the multiple time scale region and the self-diffusivities are found to be strongly correlated.

Multiple Time-Scale Behavior of the Hydrogen-Bond Network in Water

Multiple Time-Scale Behavior of the Hydrogen-Bond Network in Water

Multiple Time-Scale Behavior of the Hydrogen-Bond Network in Water

The temperature-dependent changes in the hydrogen-bond network of SPC/E water have been examined using power spectral analysis of fluctuations in tagged-molecule potential energies and local tetrahedral order parameters. The clear signatures of multiple time-scale or 1/fα behavior in the power spectra are shown to depend sensitively on the strength of hydrogen bonding. The analysis focuses on three specific power spectral features: the frequency of crossover to white noise behavior, the exponent in the 1/fα regime, and the librational peak. The exponent of the tagged-particle potential-energy fluctuations is shown to be strongly correlated with the diffusivity in the temperature range of 230 to 300 K. This correlation is strongest in the temperature−density regimes where the mechanism for diffusion is likely to be dominated by translational−rotational coupling, suggesting that the value of the exponent is a measure of the efficiency of the coupling of librational modes with network vibrations. The temperature dependence of all power spectral features was found to be strongest along the 0.9-g cm-3 isochore, which corresponds closely to the density of minimum diffusivity for the temperature range studied here. The static distributions of the tagged-particle quantities were examined to determine the degree of heterogeneity of the local molecular environment and its relationship with power spectral features.

Control-Oriented Modeling of Transcritical Vapor Compression Systems

This paper presents a methodology for developing a low order dynamic model of a transcritical air-conditioning system, specifically suited for multivariable controller design. An 11th-order nonlinear dynamic model of the system is derived using first principles. Analysis indicates that the system exhibits multiple time scale behavior, and that model reduction is appropriate. Model reduction using singular perturbation techniques yields physical insight as to which physical phenomena are relatively fast/slow, and a 5th-order dynamic model appropriate for multivariable controller design. Although all results shown are for a transcritical cycle, the methodology presented can easily be extended to subcritical cycles.

Signatures of multiple time-scale behaviour in the power spectra of water

Power spectra associated with fluctuations in the tagged particle potential and kinetic energies are analysed for bulk SPC/E water for a range of temperatures along the 1.0 g/cm 3 isochore. Fluctuations in the tagged particle potential energies give rise to 1/ f α noise, indicative of multiple time-scale behaviour, over a temperature-dependent frequency regime. In contrast, the tagged particle centre-of-mass and rotational kinetic energies, which are indicative of the magnitude of local thermal fluctuations, do not show any evidence of 1/ f α behaviour.

Monte Carlo investigations of Coulombic correlations in lattice gas models

Monte Carlo simulations were performed on lattice gas systems with Coulombic interactions. Emphasis was placed on two lattice gases. The first consists of both mobile anions and cations while the second is composed of mobile anions and a random distribution of fixed cations. Comparisons are made to a strictly repulsive lattice gas. The addition of attractive forces is shown to significantly retard particle motion relative to the repulsive system. In the mobile-anion, mobile-cation system, at temperatures high enough to suppress ion clustering, the effect of backward correlations on the particle diffusivity is found to be similar to that for the strictly repulsive system. In the mobile-anion, fixed-cation system, however, backward correlations are much stronger due to the presence of immobile Coulomb traps. Both systems deviate from Nernst–Einstein behavior. The mobile-anion, mobile-cation system exhibits diminished conductivity (Haven ratio ≳1) due to the migration of neutral ion pairs, whereas the mobile-anion, fixed-cation system exhibits slightly enhanced conductivity (Haven ratio <1) due to the mutual repulsions between mobile charge carriers. The results of these simulations are discussed in terms of multiple timescale behavior, specifically including Funke’s jump relaxation model, and observations on relaxation times are reported.

Hierarchical aggregation of linear systems with multiple time scales

In this paper we carry out a detailed analysis of the multiple time scale behavior of singularly perturbed linear systems of the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}^{\epsilon}(t) = A(\epsilon)x^{\epsilon}(t)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">A(\epsilon)</tex> is analytic in the small parameter ε. Our basic result is a uniform asymptotic approximation to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\exp A(\epsilon)t</tex> that we obtain under a certain multiple semistability condition. This asymptotic approximation gives a complete multiple time scale decomposition of the above system and specifies a set of reduced order models valid at each time scale. Our contribution is threefold. 1) We do not require that the state variables be chosen so as to display the time scale structure of the system. 2) Our formulation can handle systems with multiple ( > 2) time scales and we obtain uniform asymptotic expansions for their behavior on [ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0, \infty</tex> ]. 3) We give an aggregation method to produce increasingly simplified models valid at progressively slower time scales.

Secular Behavior in an Analysis of Damped Standing Waves in a Gas

By considering multiple time scale behavior in the viscous damping of a one-dimensional wave, it has been shown how secularities (terms growing unbounded in time) present in the Hilbert solution of this problem may be eliminated. The method of time scales and the accompanying freedom to remove secular terms overcomes in part the major disadvantage of the Hilbert procedure which requires a periodic renormalization of the lowest order initial conditions. This method, which has proved most useful in nonequilibrium statistical mechanics, provides a link between the Hilbert and Chapman-Enskog treatment of a simple hydrodynamic problem.