Nonlinear Stochastic Dynamics Research Articles

Digital screener of socio-motor agency balancing motor autonomy and motor control.

Dyadic social interactions evoke complex dynamics between two agents that, while exchanging unequal levels of body autonomy and motor control, may find a fine balance to synergize, take turns, and gradually build social rapport. To study the evolution of such complex interactions, we currently rely exclusively on subjective pencil and paper means. Here, we complement this approach with objective biometrics of socio-motor behaviors conducive to socio-motor agency. Using a common clinical test as the backdrop of our study to probe social interactions between a child and a clinician, we demonstrate new ways to streamline the detection of social readiness potential in both typically developing and autistic children by uncovering a handful of tasks that enable quantification of levels of motor autonomy and levels of motor control. Using these biometrics of autonomy and control, we further highlight differences between males and females and uncover a new data type amenable to generalizing our results to any social setting. The new methods convert continuous dyadic bodily biorhythmic activity into spike trains and demonstrate that in the context of dyadic behavioral analyses, they are well characterized by a continuous Gamma process that can classify individual levels of our thus defined socio-motor agency during a dyadic exchange. Finally, we apply signal detection processing tools in a machine learning approach to show the validity of the streamlined version of the digitized ADOS test. We offer a new framework that combines stochastic analyses, non-linear dynamics, and information theory to streamline and facilitate scaling the screening and tracking of social interactions with applications to autism.

Spectral incremental dynamic methodology for nonlinear structural systems endowed with fractional derivative elements subjected to fully non-stationary stochastic excitation

A novel spectral incremental dynamic analysis methodology for analysing structural response in nonlinear systems with fractional derivative elements is presented, aligning with modern seismic design codes, like Eurocode 8. Drawing inspiration from the concept of fully non-stationary stochastic processes, the vector of the imposed seismic excitations is characterised by time and frequency evolving power spectra stochastically compatible with elastic response spectra of specified damping ratio and ground acceleration. The proposed method efficiently determines the nonlinear system time-dependent probability density functions for the non-stationary system response amplitude by employing potent nonlinear stochastic dynamics concepts, such as stochastic averaging and statistical linearisation. Unlike traditional incremental dynamic analysis curves found in the literature, the herein proposed method introduces a three-dimensional alternative counterpart, that of stochastic engineering demand parameter surfaces, providing with higher-order statistics of the system response. An additional noteworthy aspect involves the derivation of response evolutionary power spectra as function of spectral acceleration, offering a deeper insight into the underlying system dynamics. Besides its capabilities, the method maintains the coveted element of a particularly low associated computational cost, increasing its attractiveness and practicality among diverse applications of engineering interest. Numerical examples comprising the bilinear hysteretic model endowed with fractional derivative elements subject to an Eurocode 8 elastic design spectrum demonstrate the capabilities and reliability of the proposed methodology. Its accuracy is assessed by juxtaposing the derived results with germane Monte Carlo Simulation data.

Uncovering the stochastic dynamics of solitons of the Chaffee–Infante equation

In this paper, we apply stochastic differential equations with the Wiener process to investigate the soliton solutions of the Chaffee–Infante (CI) equation. The CI equation, a fundamental model in mathematical physics, explains concepts such as wave propagation and diffusion processes. Exact soliton solutions are obtained through the application of the modified extended tanh (MET) method. The obtained wave figures in 3D, 2D, and contour are highly localized and determine an individual frequency shift under the behavior of sharp peak, periodic wave, and singular soliton. The MET method shows to be a valuable analytical tool for obtaining soliton solutions, essential for understanding the dynamics of nonlinear wave phenomena. Numerical simulations enable us to explore soliton solutions in two and three dimensions, shedding light on their properties over time. Our results have wide applications in various domains, including stochastic processes and nonlinear dynamics, impacting advancements in physics, engineering, finance, biology, and beyond.

Permanent loss of barrier island resilience due to a critical transition in dune ecosystems

Barrier islands cover a large fraction of US coasts and support unique ecosystems and coastal infrastructure. The ‘barrier’ function of a barrier island depends on coastal dunes that can prevent storm flooding and widespread ecosystem loss. Furthermore, dune-less barriers are more susceptible to breaching and potential drowning under sea level rise. Here we study the transition from richly-vegetated barriers with mature dunes (‘high’ state) to dune-less barren barriers (‘barren’ state) using data from a representative set of barrier islands in Virginia, US. We find that these two states are possible stable solutions of a non-linear stochastic dynamics characterized by a tipping point at which barriers with elevation around beach berms experience a critical transition into a permanently barren state. Our results suggest that frequently-flooded dune-less barren islands are a natural endpoint of barrier’s evolution under sea level rise.

Updating Nonlinear Stochastic Dynamics of an Uncertain Nozzle Model Using Probabilistic Learning With Partial Observability and Incomplete Dataset

Abstract This article introduces a methodology for updating the nonlinear stochastic dynamics of a nozzle with uncertain computational model. The approach focuses on a high-dimensional nonlinear computational model constrained by a small target dataset. Challenges include the large number of degrees-of-freedom, geometric nonlinearities, material uncertainties, stochastic external loads, underobservability, and high computational costs. A detailed dynamic analysis of the nozzle is presented. An updated statistical surrogate model relating the observations of interest to the control parameters is constructed. Despite small training and target datasets and partial observability, the study successfully applies probabilistic learning on manifolds (PLoM) to address these challenges. PLoM captures geometric nonlinear effects and uncertainty propagation, improving conditional mean statistics compared to training data. The conditional confidence region demonstrates the ability of the methodology to accurately represent both observed and unobserved output variables, contributing to advancements in modeling complex systems.

Wait time to stochastic self-focusing

The first passage time extended to stochastic nonlinear beam dynamics emerges as a natural time scale and a meaningful estimator for the expected wait time to the first self-focusing event within a large sample of stochastic laser pulses. We show that the ratio P/Pcr of the laser peak power P to the critical power of self-focusing Pcr, which plays a central role in deterministic self-focusing, keeps its status as a key governing parameter in stochastic self-focusing. However, in contrast to its deterministic counterpart, the Pcr/P ratio of a stochastic laser beam is no longer an indicator of whether self-focusing will occur, but is, rather, a predictor of when the self-focusing is expected, in the sense of the first passage time, given the statistics of the laser field.

Multiplicative Renormalization of Stochastic Differential Equations for the Abelian Sandpile Model

The long-term, large-scale behavior in a problem of stochastic nonlinear dynamics corresponding to the Abelian sandpile model is studied with the use of the quantum-field theory renormalization group approach. We prove the multiplicative renormalization of the model including an infinite number of coupling parameters, calculate an infinite number of renormalization constants, identify a plane of fixed points in the infinite dimensional space of coupling parameters, discuss their stability and critical scaling in the model, and formulate a simple law relating the asymptotic size of an avalanche to a model exponent quantifying the time-scale separation between the slow energy injection and fast avalanche relaxation processes.

Non-equilibrium Onsager–Machlup theory

This paper proposes a simple mathematical model of non-stationary and non-linear stochastic dynamics, which approximates a (globally) non-stationary and non-linear stochastic process by its locally (or ‘piecewise’) stationary version. Profiting from the elegance and simplicity of both, the exact mathematical model referred to as the Ornstein–Uhlenbeck stochastic process (which is globally stationary, Markov and Gaussian) and of the Lyapunov criterion associated with the stability of stationarity, we show that the proposed non-linear non-stationary model provides a natural extension of the Onsager–Machlup theory of equilibrium thermal fluctuations, to the realm of non-stationary, non-linear and non-equilibrium processes. As an illustrative application, we then apply the extended non-equilibrium Onsager–Machlup theory, to the description of thermal fluctuations and irreversible relaxation processes in liquids, leading to the main exact equations employed to construct the non-equilibrium self-consistent generalised Langevin equation (NE-SCGLE) theory of irreversible processes in liquids. This generic theory has demonstrated that the most intriguing and long-unsolved questions of the glass and gel transitions are understood as a natural consequence of the second law of thermodynamics, enunciated in terms of the proposed piecewise stationary stochastic mathematical model.

Non-linear stochastic dynamics analysis of mechanical systems using non-intrusive polynomial chaos method: application to pole vaulting

Non-linear stochastic dynamics analysis of mechanical systems using non-intrusive polynomial chaos method: application to pole vaulting

Analytical equivalent transformation method for nonlinear stochastic dynamics with multiple noises in high dimensions

In many practical applications, stochastic problems are high dimensionality and multiple noises. The study focuses on developing an analytical approach for processing multi-noise systems in high dimensions, which overcomes the challenge solving stochastic dynamics problems. Applying an analytical equivalent transformation and introducing a new partial differentiation, the author demonstrate a convenient method for obtaining the Fokker–Planck equation corresponding to Langevin dynamics equations for high-dimensional and multi-noise systems. This methodology is applied to analyze the cubic laser model subjected to white noises and the saturation laser model subjected to bounded sine Wiener noise. The results verify the correction, efficiency, and superiority of the method of processing multiple noises in high dimensions. This approach provides a novel framework for solving other practical dynamics problems for high-dimensional and multi-noise systems.

Adaptive and robust fractional gain based interpolatory cubature Kalman filter

In this study, we put forward the robust fractional gain based interpolatory cubature Kalman filter (FGBICKF) and the adaptive FGBICKF (AFGBICKF) for the development of the state estimators for stochastic nonlinear dynamics system. FGBICKF introduces a fractional gain to interpolatory cubature Kalman filter to increase the robustness of state estimation. AFGBICKF is developed to enhance the state estimation adaptive to stochastic nonlinear dynamics system with unknown process noise covariance through recursive estimation. The simulations on re-entry target tracking system have shown that the performance of FGBICKF is superior to that of cubature Kalman filter and interpolatory cubature Kalman filter, and standard deviation of FGBICKF is closer to posterior Cramér-Rao lower bound. Moreover, our simulations have also demonstrated that AFGBICKF remains stable even when the initial process noise covariance increase, proving its adaptiveness, robustness, and effectiveness on state estimation.

Probabilistic-learning-based stochastic surrogate model from small incomplete datasets for nonlinear dynamical systems

We consider a high-dimensional nonlinear computational model of a dynamical system, parameterized by a vector-valued control parameter, in the presence of uncertainties represented by an uncontrolled parameter modeled by a vector-valued random variable, and possibly with stochastic excitation. The objective is to construct a statistical surrogate model where the input is any deterministic value of the control parameter, and the output is a vector-valued observation of the computational model, which is a random vector whose probability measure is updated using a target dataset. To construct this statistical surrogate model, the stochastic response of the computational model must be built, which is a vector-valued time-discretized stochastic process in high dimension, depending on the control parameter. It is assumed that the computational cost of a single evaluation of the deterministic model is high. For the probabilistic updating, we consider a subset of the components of the observation of the computational model, defined as the “identification observation” of the computational model, for which a small target dataset is available. Therefore, the target dataset is associated with partial observability, corresponding to an incomplete data case. Given a prior probability model of the random control and uncontrolled parameters, a training dataset is constructed, consisting of realizations of the random triplet composed of the stochastic response, the random identification observation, and the random control parameter. Since the computational cost of a single evaluation of the deterministic model is assumed to be large, the training dataset is also of small size. The main challenges in this problem are the high dimensionality, partial observability leading to incomplete data in the target dataset for the identification observation of the computational model (which is not sufficient to identify the computational stochastic responses), and the availability of a small training dataset. To address these challenges, we propose a methodology based on statistical methods for constructing necessary reduced representations, direct probabilistic learning under constraints using probabilistic learning on manifolds (PLoM) constrained by the target dataset, and the use of a weak formulation of the Fourier transform of probability measures. Statistical conditioning is also employed to explore the learned dataset. The constructed predictive statistical surrogate model can be implemented in the context of online computation. We apply this approach to a problem of nonlinear stochastic dynamics in high dimensions within the framework of deformable solids mechanics.

Patch formation driven by stochastic effects of interaction between viruses and defective interfering particles.

Defective interfering particles (DIPs) are virus-like particles that occur naturally during virus infections. These particles are defective, lacking essential genetic materials for replication, but they can interact with the wild-type virus and potentially be used as therapeutic agents. However, the effect of DIPs on infection spread is still unclear due to complicated stochastic effects and nonlinear spatial dynamics. In this work, we develop a model with a new hybrid method to study the spatial-temporal dynamics of viruses and DIPs co-infections within hosts. We present two different scenarios of virus production and compare the results from deterministic and stochastic models to demonstrate how the stochastic effect is involved in the spatial dynamics of virus transmission. We compare the spread features of the virus in simulations and experiments, including the formation and the speed of virus spread and the emergence of stochastic patchy patterns of virus distribution. Our simulations simultaneously capture observed spatial spread features in the experimental data, including the spread rate of the virus and its patchiness. The results demonstrate that DIPs can slow down the growth of virus particles and make the spread of the virus more patchy.

Spectral Representations of Iterated Stochastic Integrals and Their Application for Modeling Nonlinear Stochastic Dynamics

Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multiplicity, including integrals from Taylor–Itô and Taylor–Stratonovich expansions, are obtained by the spectral method. They are required for the implementation of numerical methods for solving Itô and Stratonovich stochastic differential equations with high orders of mean-square and strong convergence. The purpose of such numerical methods is the modeling of nonlinear stochastic dynamics in many fields. This paper contains necessary theoretical results, as well as the results of numerical experiments.

Effect of non-Gaussian coloured noise on ambient energy harvesting

Using nonlinear stochastic dynamics, the impact of non-Gaussian noise is investigated for vibration to electric energy conversion schemes. For the purpose of utilizing noise in energy harvesting, which offers helpful tips for the development of micropower generators of practical importance, we consider that noise can either be additive or multiplicative in nature. We found that when the property of the noise deviates from the Gaussian characteristics, maximum power generation due to autonomous stochastic resonances disappears. Furthermore, in terms of power generation, non-Gaussian noise outperforms the Gaussian noise.

Regeneration in calcareous sponge relies on 'purse-string' mechanism and the rearrangements of actin cytoskeleton.

The crucial step in any regeneration process is epithelization, i.e. the restoration of anepithelium structural and functional integrity. Epithelization requires cytoskeletal rearrangements, primarily of actin filaments and microtubules. Sponges (phylum Porifera) are early branching metazoans with pronounced regenerative abilities. Calcareous sponges have a unique step during regeneration: theformation of a temporary structure, calledregenerative membrane which initially covers a wound. It forms due to the morphallactic rearrangements of exopinaco- and choanoderm epithelial-like layers. The current study quantitatively evaluates morphological changes and characterises underlying actin cytoskeleton rearrangements during regenerative membrane formation in asconoid calcareous sponge Leucosolenia variabilis through a combination of time-lapse imaging, immunocytochemistry, and confocal laser scanning microscopy. Regenerative membrane formation has non-linear stochastic dynamics with numerous fluctuations. The pinacocytes at the leading edge of regenerative membrane form a contractile actomyosin cable. Regenerative membrane formation either depends on its contraction or being coordinated through it. The cell morphology changes significantly during regenerative membrane formation. Exopinacocytes flatten, their area increases, while circularity decreases. Choanocytes transdifferentiate into endopinacocytes, losing microvillar collar and flagellum. Their area increases and circularity decreases. Subsequent redifferentiation of endopinacocytes into choanocytes is accompanied by inverse changes in cell morphology. All transformations rely on actin filament rearrangements similar to those characteristic of bilaterian animals. Altogether, we provide here a qualitative and quantitative description of cell transformations during reparative epithelial morphogenesis in a calcareous sponge.

On the Nonlinear Stochastic Dynamics of an Atomic Force Microscope Cantilever

AbstractAtomic force microscopy (AFM) serves characterization and actuation in nanoscale applications. We study the stochastic dynamics of an AFM cantilever under tip-sample interactions represented by the Lennard–Jones and Morse potential energy functions. In both cases, we also study the contrasting dynamic effects of additive (external) and multiplicative (internal) noise. Moreover, for multiplicative noise, we study the two sub-cases arising from the Itô and Stratonovich interpretations of stochastic integrals. In each case, we also investigate the stochastic stability of the system by tracing the time evolution of the maximal Lyapunov exponent. Additionally, we obtain stationary probability densities for the unforced dynamics using stochastic averaging.

Cooperative output feedback tracking control of stochastic nonlinear heterogeneous multi‐agent systems

SummaryWe study cooperative output feedback tracking control of stochastic nonlinear heterogeneous leader‐following multi‐agent systems. Each agent has a continuous‐time stochastic nonlinear heterogeneous dynamics with an unmeasurable state, and there are additive and multiplicative noises along with information exchange among agents. We propose admissible distributed observation strategies for estimating the leader's and the followers' states, and admissible cooperative output feedback control strategies based on the certainty equivalence principle. By output regulation theory and stochastic analysis, we show if the dynamics of each agent satisfies the Lipschitz condition, and the product of the leader's Lipschitz coefficient, the intensity of multiplicative measurement noises, and the constant related to the leader's Lipschitz coefficient and dimension is less than the minimum nonzero eigenvalue of graph Laplacian, then there exist admissible distributed observation and cooperative control strategies to ensure mean square bounded output tracking. Finally, the effectiveness of our control strategies is demonstrated by a numerical simulation.

Stochastic trajectory optimization for 6-DOF spacecraft autonomous rendezvous and docking with nonlinear chance constraints

This paper addresses the problem of stochastic trajectory planning for 6-DOF spacecraft close proximity in the presence of external disturbances and initial state uncertainties. Nonlinear chance constraints on collision avoidance, docking corridor, sensor field-of-view, and control magnitude are considered in the corresponding optimal control problem. Nominal trajectory along with affine feedback controller that restricts the dispersion caused by disturbances are optimized simultaneously. The nonlinear stochastic dynamics are transformed into equality constraints on the first two statistical moments of the random states by linearization and discretization, and further, the non-convex covariance update equation is relaxed to a semidefinite cone constraint. The nonlinear path chance constraints are first approximated by affine chance formations, and then are convexified by introducing auxiliary variables instead of direct linearization of the covariance matrix. Combined with the tail bound of sub-exponential random variables, the constraints on control input are replaced with conservative deterministic inequalities. The sub-optimal solution to the original problem is obtained by iteratively solving a series of semidefinite cone programs. The effectiveness of the proposed framework is verified using numerical simulations.

A purely data-driven framework for prediction, optimization, and control of networked processes

Networks are landmarks of many complex phenomena where interweaving interactions between different agents transform simple local rule-sets into nonlinear emergent behaviors. While some recent studies unveil associations between the network structure and the underlying dynamical process, identifying stochastic nonlinear dynamical processes continues to be an outstanding problem. Here, we develop a simple data-driven framework based on operator-theoretic techniques to identify and control stochastic nonlinear dynamics taking place over large-scale networks. The proposed approach requires no prior knowledge of the network structure and identifies the underlying dynamics solely using a collection of two-step snapshots of the states. This data-driven system identification is achieved by using the Koopman operator to find a low-dimensional representation of the dynamical patterns that evolve linearly. Further, we use the global linear Koopman model to solve critical control problems by applying to model predictive control (MPC)–typically, a challenging proposition when applied to large networks. We show that our proposed approach tackles this by converting the original nonlinear programming into a more tractable optimization problem that is both convex (quadratic programming) and with far fewer variables.