The past decade has evidenced the increased prevalence of irregularly spaced longitudinal data in social sciences. Clearly lacking, however, are modeling tools that allow researchers to fit dynamic models to irregularly spaced data, particularly data that show nonlinearity and heterogeneity in dynamical structures. We consider the issue of fitting multivariate nonlinear differential equation models with random effects and unknown initial conditions to irregularly spaced data. A stochastic approximation expectation–maximization algorithm is proposed and its performance is evaluated using a benchmark nonlinear dynamical systems model, namely, the Van der Pol oscillator equations. The empirical utility of the proposed technique is illustrated using a set of 24-h ambulatory cardiovascular data from 168 men and women. Pertinent methodological challenges and unresolved issues are discussed.

Nonlinear random coefficient models (NRCMs) for continuous longitudinal data are often used for examining individual behaviors that display nonlinear patterns of development (or growth) over time in measured variables. As an extension of this model, this study considers the finite mixture of NRCMs that combine features of NRCMs with the idea of finite mixture (or latent class) models. The efficacy of this model is that it allows the integration of intrinsically nonlinear functions where the data come from a mixture of two or more unobserved subpopulations, thus allowing the simultaneous investigation of intra-individual (within-person) variability, inter-individual (between-person) variability, and subpopulation heterogeneity. Effectiveness of this model to work under real data analytic conditions was examined by executing a Monte Carlo simulation study. The simulation study was carried out using an R routine specifically developed for the purpose of this study. The R routine used maximum likelihood with the expectation–maximization algorithm. The design of the study mimicked the output obtained from running a two-class mixture model on task completion data.

The longitudinal process that leads to university student dropout in STEM subjects can be described by referring to (a) inter-individual differences (e.g., cognitive abilities) as well as (b) intra-individual changes (e.g., affective states), (c) (unobserved) heterogeneity of trajectories, and d) time-dependent variables. Large dynamic latent variable model frameworks for intensive longitudinal data (ILD) have been proposed which are (partially) capable of simultaneously separating the complex data structures (e.g., DLCA; Asparouhov et al. in Struct Equ Model 24:257–269, 2017; DSEM; Asparouhov et al. in Struct Equ Model 25:359–388, 2018; NDLC-SEM, Kelava and Brandt in Struct Equ Model 26:509–528, 2019). From a methodological perspective, forecasting in dynamic frameworks allowing for real-time inferences on latent or observed variables based on ongoing data collection has not been an extensive research topic. From a practical perspective, there has been no empirical study on student dropout in math that integrates ILD, dynamic frameworks, and forecasting of critical states of the individuals allowing for real-time interventions. In this paper, we show how Bayesian forecasting of multivariate intra-individual variables and time-dependent class membership of individuals (affective states) can be performed in these dynamic frameworks using a Forward Filtering Backward Sampling method. To illustrate our approach, we use an empirical example where we apply the proposed forecasting method to ILD from a large university student dropout study in math with multivariate observations collected over 50 measurement occasions from multiple students ($N=122$\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$N = 122$$\end{document}). More specifically, we forecast emotions and behavior related to dropout. This allows us to predict emerging critical dynamic states (e.g., critical stress levels or pre-decisional states) 8 weeks before the actual dropout occurs.

Drylands provide multiple essential services to human society, and dryland vegetation is one of the foundations of these services. There is a paradox, however, in the vegetation productivity–precipitation relationship in drylands. Although water is the most limiting resource in these systems, a strong relationship between precipitation and productivity does not always occur. Such a paradox affects our understanding of dryland vegetation dynamics and hinders our capacity to predict dryland vegetation responses under future climates. In this perspective, we examine the possible causes of the dryland precipitation–productivity paradox. We argue that the underlying reasons depend on the location and scale of the study. Sometimes multiple factors may interact, resulting in a less significant relationship between vegetation growth and water availability. This means that when we observe a poor correlation between vegetation growth and water availability, there are potentially missing sources of water input or a lack of consideration of other important processes. The paradox could also be related to the inaccurate measurement of vegetation productivity and water availability indicators. Incorporating these complexities into predictive models will help us better understand the complex relationship between water availability and dryland ecosystem processes and improve our ability to predict how these ecosystems will respond to the multiple facets of climate change.

The paper deals with boundedness of solutions and uniform asymptotic stability of the zero solution. In our current undertaking, we aim to solve two open problems that were proposed by the author in his book Qualitative theory of Volterra difference equations (2018, Springer, Cham). Our approach centers on finding the appropriate Lyapunov functional that satisfies specific conditions, incorporating the concept of wedges.

Establishing a precise electromagnetic scattering model of surfaces is of great significance for comprehending the underlying mechanics of synthetic aperture radar (SAR) imaging. To describe surface electromagnetic scattering more comprehensively, this paper established a nonlinear integral equation model with the Creamer model and bispectrum (IEM-C). Based on the IEM-C model, the effect of parameters, such as radar wave incidence angle, wind speed and direction of sea surfaces, and different polarization modes on the backscattering coefficients of C-band radar waves, was systematically evaluated. The results show that the IEM-C model can characterize both the vertical nonlinear features due to wave interactions and the horizontal nonlinear features due to the wind direction. The sensitivity of the sea surface backscattering coefficient in the IEM-C model to nonlinear effects varies with different incident angles. At the incident angle of 30°, the IEM-C model exhibits the most significant nonlinear effects. The nonlinear effects of the IEM-C model vary under different wind speeds. By comparing with the measured data, it is proved that the IEM-C model is closer to the real sea surface scattering situation than the IEM model.

In this work, PbS and PbS/CdS core–shell quantum dots (QDs) were synthesized by a new photochemical approach. Prepared QDs were characterized by means of x-ray diffraction (XRD), field emission scanning electron microscopy (FESEM), energy dispersive x-ray analysis (EDAX), UV–Vis, and Z-scan analyses. Synthesized QDs were in a cubic phase with a spherical morphology, and the crystallite sizes are estimated using the strain–size method. A uniform shift of Bragg diffraction peaks and quenching (200) Bragg plane are interpreted as the growth of the CdS shell. Linear optical properties were investigated using Urbach analysis and Tauc formula. It was found that the density of states of QD conduction and valence bands are three dimensional. The estimated sizes of PbS QDs and PbS/CdS using exciton absorption peaks at room temperature are 1.8 and 2.7 nm, respectively, which are in good agreement with the strain–size plot analysis. The growth of the CdS shell resulted in a considerable decrease in the nonlinearity refractive index and a significant increase in the nonlinear absorption.

A Legendre-collocation method is proposed to solve the nonlinear Volterra integral equations of the second kind. We provide a rigorous error analysis for the proposed method, which indicate that the numerical errors in L2-norm and L∞-norm will decay exponentially provided that the kernel function is sufficiently smooth. Numerical results are presented, which confirm the theoretical prediction of the exponential rate of convergence.

An iterative discontinuous Galerkin (DG) method is proposed to solve the nonlinear Poisson Boltzmann (PB) equation. We first identify a function space in which the solution of the nonlinear PB equation is iteratively approximated through a series of linear PB equations, while an appropriate initial guess and a suitable iterative parameter are selected so that the solutions of linear PB equations are monotone within the identified solution space. For the spatial discretization we apply the direct discontinuous Galerkin method to those linear PB equations. More precisely, we use one initial guess when the Debye parameter λ = (1), and a special initial guess for λ ≫1 to ensure convergence. The iterative parameter is carefully chosen to guarantee the existence, uniqueness, and convergence of the iteration. In particular, iteration steps can be reduced for a variable iterative parameter. Both one and two-dimensional numerical results are carried out to demonstrate both accuracy and capacity of the iterative DG method for both cases of λ = (1) and λ ≪ 1. The (m + 1)th order of accuracy for L2 and mth order of accuracy for H1 for Pm elements are numerically obtained.

In the paper, nonlinear dynamic analysis of a circular plate composed of functionally graded material (FGM) is presented. Considering a transverse shear deformation and geometric nonlinearity, the von Karman geometric relation of the FGM circular plate is established. Based on the theory of the first-order shear deformation, a new set of equilibrium equations is developed by the principle of minimum total energy. Applying the finite difference method and Newmark scheme, the nonlinear transient problem is solved by the iterative method. To validate the presented method, the transient problem of the FGM circular plate is compared with those of the existed literature, and good agreement is observed. The effects of the volume fraction index, boundary conditions, mechanical load and the ratio of thickness to radius on the nonlinear transient problem of the FGM circular plate are investigated.

We present a simple yet effective and applicable scheme, based on quadrature, for constructing optimal iterative methods. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on n + 1 evaluations could achieve a maximum convergence order of 2n. Through quadrature, we develop optimal iterative methods of orders four and eight. The scheme can further be applied to develop iterative methods of even higher orders. Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

This paper explores analytically the nonlinear dynamic behavior of rotors. Coupled nonlinear equations of motion are formulated using Hamilton’s principle. The rotor model is composed of a rigid disk and a flexible shaft which is characterized as a beam of circular cross section. Various influences are taken into account like the effect of higher order large deformations, rotary inertia, gyroscopic effect, rotor unbalance and the effect of a dynamic axial force. Forced response due to a mass unbalance is presented first for the linear analysis and then perturbation techniques are used to solve the complete equations of motion including nonlinear terms. Method of multiple scales is applied to examine the nonlinear behaviour of the rotor system. Resonant curves are plotted for different possible resonance conditions. It is concluded that the higher order large deformations and axial force acting dynamically on the rotor have a significant effect on its nonlinear response. This response varies for different parameters of the rotor like an unbalance mass and diameter of the shaft.

The single-mode Richtmyer–Meshkov instability is investigated using a first-order perturbation of the two-dimensional Navier–Stokes equations about a one-dimensional unsteady shock-resolved base flow. A feature-tracking local refinement scheme is used to fully resolve the viscous internal structure of the shock. This method captures perturbations on the shocks and their influence on the interface growth throughout the simulation, to accurately examine the start-up and early linear growth phases of the instability. Results are compared to analytic models of the instability, showing some agreement with predicted asymptotic growth rates towards the inviscid limit, but significant discrepancies are noted in the transient growth phase. Viscous effects are found to be inadequately predicted by existing models.

We have studied the responses of striate cortical neurons to stimuli whose contrast is modulated in time by either a single sinusoid or by the sum of eight sinusoids. The sum-of-sinusoids stimulus resembles white noise and has been used to study the linear and nonlinear dynamics of retinal ganglion cells (Victor et al., 1977). In cortical neurons, we have found different linear and second-order responses to single-sinusoid and sum-of-sinusoids inputs. Specifically, while the responsivity near the optimal temporal frequency is lower for the sum-of-sinusoids stimulus, the responsivity at higher temporal frequencies is relatively greater. Along with this change in the response amplitudes, there is a systematic change in the time course of responses. For complex cells, the integration time, the effective delay due to a combination of actual delays and low-pass filter stages, changes from a median of 85 ms with single sinusoids to 57 ms with a sum of sinusoids. For simple cells, the integration times for single sinusoids range from 44–100 ms, but cluster tightly around 40 ms for the sum-of-sinusoids stimulus. The change in time constant would argue that the increased sensitivity to high frequencies cannot be explained by a static threshold, but must be caused by a fundamental alteration in the response dynamics. These effects are not seen in the retina (Shapley & Victor, 1981) and are most likely cortical in origin.

This paper deals with an investigation of the relative importance of robotic characteristics typically associated with nonlinear manipulators. An IBM 7540 SCARA type of robot is used for simulation, and results are presented for decentralized proportional plus derivative control action applied to individual robot joints, and the use of an adaptive computed torque control strategy is illustrated. The influence of variations in payload and robot parameters on trajectory tracking is also shown.

The stability and evolution of very thin, single component, metallic-melt films isstudied by analysis of coupled strongly nonlinear equations for gas-melt (GM) and crystal-melt (CM) interfaces, derived using the lubrication approximation. The crystal-melt interface is deformable by freezing and melting, and there is a thermal gradient applied across thefilm. Linear analysis reveals that there is a maximum applied far-field temperature in thegas, beyond which there is no film instability. Instabilities observed in the absence of CMsurface energy are oscillatory for all marginally stable states. The effect of the CM surfaceenergy is to expand the parameter range over which a film is unstable. The new range ofinstabilities are of longer wavelength and are stationary, compared to the range found inthe absence of CM surface energy. Numerical analysis illustrates how perturbations grow torupture by standing waves. With CM surface energy, an initially longer (stationary) wavelength perturbation has a relatively slow growth rate, but it can trigger the appearance ofmuch faster growing shorter wavelength (oscillatory) instabilities, leading to an acceleratedfilm rupture process.