Frequency domain analysis and identification of block-oriented nonlinear systems

Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener-Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.     

Morphogenetic action through flux-limited spreading

A central question in biology is how secreted morphogens act to induce different cellular responses within a group of cells in a concentration-dependent manner. Modeling morphogenetic output in multicellular systems has so far employed linear diffusion, which is the normal type of diffusion associated with Brownian processes. However, there is evidence that at least some morphogens, such as Hedgehog (Hh) molecules, may not freely diffuse. Moreover, the mathematical analysis of such models necessarily implies unrealistic instantaneous spreading of morphogen molecules, which are derived from the assumptions of Brownian motion in its continuous formulation. A strict mathematical model considering Fick?s diffusion law predicts morphogen exposure of the whole tissue at the same time. Such a strict model thus does not describe true biological patterns, even if similar and attractive patterns appear as results of applying such simple model. To eliminate non-biological behaviors from diffusion models we introduce flux-limited spreading (FLS), which implies a restricted velocity for morphogen propagation and a nonlinear mechanism of transport. Using FLS and focusing on intercellular Hh-Gli signaling, we model a morphogen gradient and highlight the propagation velocity of morphogen particles as a new key biological parameter. This model is then applied to the formation and action of the Sonic Hh (Shh) gradient in the vertebrate embryonic neural tube using our experimental data on Hh spreading in heterologous systems together with published data. Unlike linear diffusion models, FLS modeling predicts concentration fronts and the evolution of gradient dynamics and responses over time. In addition to spreading restrictions by extracellular binding partners, we suggest that the constraints imposed by direct bridges of information transfer such as nanotubes or cytonemes underlie FLS. Indeed, we detect and measure morphogen particle velocity in such cell extensions in different systems.     

State and parameter estimation in nonlinear systems as an optimal tracking problem

In verifying and validating models of nonlinear processes it is important to incorporate information from observations in an efficient manner. Using the idea of synchronization of nonlinear dynamical systems, we present a framework for connecting a data signal with a model in a way that minimizes the required coupling yet allows the estimation of unknown parameters in the model. The need to evaluate unknown parameters in models of nonlinear physical, biophysical, and engineering systems occurs throughout the development of phenomenological or reduced models of dynamics. Our approach builds on existing work that uses synchronization as a tool for parameter estimation. We address some of the critical issues in that work and provide a practical framework for finding an accurate solution. In particular, we show the equivalence of this problem to that of tracking within an optimal control framework. This equivalence allows the application of powerful numerical methods that provide robust practical tools for model development and validation.     

Forecasting with imperfect models, dynamically constrained inverse problems, and gradient descent algorithms

This paper is part of a larger study investigating the meaning of, and appropriate procedures for, forecasting with imperfect models. (In the author’s opinion there is currently no satisfactory general theory and practice for doing so with complex nonlinear systems.) The focus of this paper is on initialisation of the forecast. At the heart of every forecasting scheme there is an inverse problem that translates observations of reality into an initial state, or ensemble of states, of the model. Inverse problems are divided into two classes depending on whether the underlying model of reality is that of a stochastic process or a (deterministic) dynamical process. The two classes have quite different formulations of their inverse problems and consequent solutions methods. This paper considers dynamical process models and their inverse problems, which will be referred to as the dynamically constrained inverse problem (DCIP) line. The interpretation and solutions of the DCIP line are investigated and new algorithms for solving them are presented. The new algorithms are modifications of classical gradient descent algorithms. The new algorithms are applied to a low-dimensional chaotic system and a high-dimensional operational weather forecasting model. Our examination of DCIP shows that gradient descent algorithms are an effective way of solving the inverse problem for complex nonlinear system given an imperfect dynamical model.     

A splitting approach for the fully nonlinear and weakly dispersive Green–Naghdi model

The fully nonlinear and weakly dispersive Green–Naghdi model for shallow water waves of large amplitude is studied. The original model is first recast under a new formulation more suitable for numerical resolution. An hybrid finite volume and finite difference splitting approach is then proposed, which could be adapted to many physical models that are dispersive corrections of hyperbolic systems. The hyperbolic part of the equations is handled with a high-order finite volume scheme allowing for breaking waves and dry areas. The dispersive part is treated with a classical finite difference approach. Extensive numerical validations are then performed in one horizontal dimension, relying both on analytical solutions and experimental data. The results show that our approach gives a good account of all the processes of wave transformation in coastal areas: shoaling, wave breaking and run-up.     

An insight into the viscosity prediction of ternary alloys with limited solubility

Geometrical models have been successfully applied into the calculation of viscosity of alloys. However, traditional geometrical (TG) models are feasible only in systems with a complete solubility area. Otherwise, they will not work. In this paper, a new method has been introduced for the calculation of alloy systems with limited solubility. How the new method overcomes the drawbacks of the TG models is discussed and analysed. The viscosity of two alloy systems with limited solubility is calculated by the present model. Comparisons between the experimental viscosity and the calculated values by different models show that our model gives the best results, especially for the data nearby the limited solubility area. The introduction of this model provides a way to solve the calculation problems of ternary alloys with limited solubility, which will extend geometrical models to more practical systems.     

Mathematical modeling of intracellular signaling pathways

Dynamic modeling and simulation of signal transduction pathways is an important topic in systems biology and is obtaining growing attention from researchers with experimental or theoretical background. Here we review attempts to analyze and model specific signaling systems. We review the structure of recurrent building blocks of signaling pathways and their integration into more comprehensive models, which enables the understanding of complex cellular processes. The variety of mechanisms found and modeling techniques used are illustrated with models of different signaling pathways. Focusing on the close interplay between experimental investigation of pathways and the mathematical representations of cellular dynamics, we discuss challenges and perspectives that emerge in studies of signaling systems.     

A medium with optical activity depending on the incoherent light intensity in a ring interferometer: Models of processes

As part of the program of developing perspective incoherent optics systems in which synergy phenomena are realizable, a complex of models of processes in a nonlinear ring interferometer with an optically active medium whose rotary power depends on the incoherent radiation intensity in the nonlinear ring interferometer is constructed. The light intensity depends nonlinearly on the angle of polarization plane rotation. The analogy of the constructed models with the well-known model of the nonlinear ring interferometer with the Kerr medium and laser radiation is established. The (non)linearity of the models and the application of the corresponding nonlinear ring interferometers are discussed. The results obtained are important for improvement in methods of natural light field processing. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 26–34, June, 2007.     

The thermodynamic characteristics of multicomponent reaction mixtures in the sub- and supercritical states

The paper presents mathematical models and calculation methods for solving particular research problems related to the thermodynamic characteristics of multicomponent and multiphase mixtures. The special features of chemical and phase equilibria in such mixtures are considered in the ideal gas approximation and taking nonideality into account. The conditions of equilibrium phase stability are studied for multiphase systems. The results of calculations of characteristic phase diagrams and binodal and spinodal are given for model systems with a fixed chemical composition, and a new interpretation of the mathematical model for localizing the critical point of a multicomponent mixture with a given composition is presented. A new interpretation of the well-known classic homotopy method is suggested for solving complex nonlinear systems of equations. Some anomalies of phase portraits and critical curves that are necessary to take into account in selecting (planning) experimental conditions and calculating chemical processes and reaction parameters are considered separately. The possibility of calculating thermodynamic and thermophysical properties (entropy, enthalpy, heat capacity, heat effects of reactions, and adiabatic heating) is demonstrated for the example of particular multicomponent nonideal mixtures. The conclusion is drawn that cubic equations of state can be used for predicting the deviations of these properties from the ideal gas state and their anomalies in the vicinity of the critical points of mixtures.     

BOR-FDTD analysis of nonlinear Fiber Bragg grating and distributed Bragg resonator

Recently, nonlinear materials have attracted a great deal of attention because of their importance in designing new devices to meet a need range of optical systems. An intense investigation of the possibility of using these materials for all optical ultrafast applications is achieved by allowing their dielectric characteristics to be varied in such a way that a periodic perturbation of their refractive index along the length of the waveguide will be formed. The Finite-Difference Time-Domain (FDTD) method, on the other hand, has been proven to be one of the most powerful numerical techniques that are usefully applied to a wide range of optical devices. In this paper, a FDTD technique, developed for nonlinear structures, is used to analyze a nonlinear waveguide and periodic nonlinear structures that exhibit attractive properties that make them suitable for novel devices with wavelength tunable characteristics. More specifically, the Bodies of Revolution (BOR) FDTD numerical simulation method will be used to model the fiber Bragg Grating (FBG) and the direct integration method will be employed to include the effect of Self Phase Modulation (SPM) in this model. The combination of these techniques will result in a model that is used to analyze two different types of periodic nonlinear structure, FBG and Distributed Bragg Resonator (DBR). The nonlinear effect provides the designer an added degree of design flexibility for devices with wavelength tunable characteristics, for example, in the design of tunable filters, WDM systems and optical sensors.     

Exploiting dynamical coherence: A geometric approach to parameter estimation in nonlinear models

Parameter estimation in nonlinear models is a common task, and one for which there is no general solution at present. In the case of linear models, the distribution of forecast errors provides a reliable guide to parameter estimation, but in nonlinear models the facts that predictability may vary with location in state space, and that the distribution of forecast errors is expected not to be Normal, means that parameter estimation based on least squares methods will result in systematic errors. A new approach to parameter estimation is presented which focuses on the geometry of trajectories of the model rather than the distribution of distances between model forecast and the observation at a given lead time. Specifically, we test a number of candidate trajectories to determine the duration for which they can shadow the observations, rather than evaluating a forecast error statistic at any specific lead time(s). This yields insights into both the parameters of the dynamical model and those of the observational noise model. The advances reported here are made possible by extracting more information from the dynamical equations, and thus improving the balance between information gleaned from the structural form of the equations and that from the observations. The technique is illustrated for both flows and maps, applied in 2-, 3-, and 8-dimensional dynamical systems, and shown to be effective in a case of incomplete observation where some components of the state are not observed at all. While the demonstration of effectiveness is strong, there remain fundamental challenges in the problem of estimating model parameters when the system that generated the observations is not a member of the model class. Parameter estimation appears ill defined in this case.     

Completely integrable models of nonlinear optics

The models of the nonlinear optics in which solitons appeared are considered. These models are of paramount importance in studies of nonlinear wave phenomena. The classical examples of phenomena of this kind are the self-focusing, self-induced transparency and parametric interaction of three waves. At present there are a number of theories based on completely integrable systems of equations, which are, both, generations of the original known models and new ones. The modified Korteweg-de Vries equation, the nonlinear Schrödinger equation, the derivative nonlinear Schrödinger equation. Sine-Gordon equation, the reduced Maxwell-Bloch equation. Hirota equation, the principal chiral field equations, and the equations of massive Thirring model are some soliton equations, which are usually to be found in nonlinear optics theory.     

Bosonization for disordered and chaotic systems

Using a supersymmetry formalism, we reduce exactly the problem of electron motion in an external potential to a new supermatrix model valid at all distances. All approximate nonlinear sigma models obtained previously for disordered systems can be derived from our exact model using a coarse-graining procedure. As an example, we consider a model for a smooth disorder and demonstrate that using our approach does not lead to a "mode-locking" problem. As a new application, we consider scattering on strong impurities for which the Born approximation cannot be used. Our method provides a new calculational scheme for disordered and chaotic systems.     

Suppression of nonlinear effects by phase alternation in strongly dispersion-managed optical transmission

The nonlinear effects of amplitude jitter and ghost pulse generation, which are present in strongly dispersion-managed optical communication systems can be suppressed by alternation of the phase of the bits. A physical explanation for this effect is given that shows that with suitably chosen phase modulations the processes that give rise to the nonlinear effects will counteract each other.     

Chaos in symmetric phase oscillator networks

Phase-coupled oscillators serve as paradigmatic models of networks of weakly interacting oscillatory units in physics and biology. The order parameter which quantifies synchronization so far has been found to be chaotic only in systems with inhomogeneities. Here we show that even symmetric systems of identical oscillators may not only exhibit chaotic dynamics, but also chaotically fluctuating order parameters. Our findings imply that neither inhomogeneities nor amplitude variations are necessary to obtain chaos; i.e., nonlinear interactions of phases give rise to the necessary instabilities.     

Line shapes in nonlinear spectroscopy

The shape of the spectral lines of an optically active system interacting with one or more strong radiation fields in the presence of a perturbing bath is studied. A method based on the statistics of the fluctuation of the interaction between the radiator and the perturbing environment (the model Markov microfield theory) is used. This method permits the foundations of line shape theory in modern statistical mechanics to be seen clearly. Multiphoton processes and homogeneous, inhomogeneous, and power broadening mechanisms are included in the analysis. The correlations between radiative and collisional processes which arise in nonlinear spectroscopy are included explicitly. A discussion of the new information that is obtained from these correlations in nonlinear spectroscopy is also presented. Several model systems are presented as illustrative examples of the theory.     

The modular systems biology approach to investigate the control of apoptosis in Alzheimer's disease neurodegeneration

Apoptosis is a programmed cell death that plays a critical role during the development of the nervous system and in many chronic neurodegenerative diseases, including Alzheimer's disease (AD). This pathology, characterized by a progressive degeneration of cholinergic function resulting in a remarkable cognitive decline, is the most common form of dementia with high social and economic impact. Current therapies of AD are only symptomatic, therefore the need to elucidate the mechanisms underlying the onset and progression of the disease is surely needed in order to develop effective pharmacological therapies. Because of its pivotal role in neuronal cell death, apoptosis has been considered one of the most appealing therapeutic targets, however, due to the complexity of the molecular mechanisms involving the various triggering events and the many signaling cascades leading to cell death, a comprehensive understanding of this process is still lacking. Modular systems biology is a very effective strategy in organizing information about complex biological processes and deriving modular and mathematical models that greatly simplify the identification of key steps of a given process. This review aims at describing the main steps underlying the strategy of modular systems biology and briefly summarizes how this approach has been successfully applied for cell cycle studies. Moreover, after giving an overview of the many molecular mechanisms underlying apoptosis in AD, we present both a modular and a molecular model of neuronal apoptosis that suggest new insights on neuroprotection for this disease.     

Nonlinear Electrostatic Structures in Collisional Dusty Plasmas

A review of some problems of electrostatic waves in dusty plasmas is presented. It is concluded that in the most models of waves in dusty plasmas, the charge numbers of the ions and dusty grains are supposed to be constant. Besides most of the studies are elated to linear waves in collisionless systems. It is shown that even if the dynamics of dusty grains is not considered in the models, the physical processes causing wave dissipation have to be taken into account. The existing nonlinear models are mostly one‐dimensional ones. It is summarized that in dusty plasmas various types of nonlinear stationary structures may exist, and that these structures differ from the nonlinear structures found in ionospheric plasmas without dust. The nonlinear electrostatic stuctures seem to be observable in ionospheric, solar or interplanetary plasmas.