Dynamic Analysis of Nonlinear Stochastic Systems by Polynomial Chaos Expansion (PCE)

Point group identification algorithm in dynamic response analysis of nonlinear stochastic systems

A copula-based Gaussian mixture closure method for stochastic response of nonlinear dynamic systems

Estimation and analysis of nonlinear stochastic systems (Ph.D. Thesis abstr.)

The adjoint simulation method is an efficient, computerised method for the performance analysis of linear time varying systems excited by deterministic and/or stochastic inputs. It is based on the impulse response function (or weighting function) of the so-called adjoint system---an associated linear system derived from the original system by clever utilisation of the mathematical principle of duality. We demonstrate the power of adjoint simulation in the context of guided missile homing loop studies and present a Matlab tool for its speedy implementation and execution. As an example, the tool analyses the miss distance performance of a generic guided missile against an evading target in two dimensional space. References Statistical performance analysis of nonlinear stochastic systems by the Monte Carlo method volume 23-1 of Mathematics and Computers in Simulation J. , pages 21--23, 1981. doi:10.1016/0378-4754(81)90004-5 Zarchan, P. Tactical and Strategic Missile Guidance , 5th Edition, AIAA Progress in Astronautics and Aeronautics, 2007. Weiss, M. Adjoint Method for Missile Performance Analysis on State-Space Models, volume 28-2 of Guidance, Control and Dynamics J. , pages 236--248, AIAA ,2005. Laning, J. H. and Battin, R. H. Random Processes in Automatic Control , McGraw-Hill Book Company, New York, NY, 1956. D. Bucco and M. Weiss. Development of a Matlab/Simulink Tool to Facilitate System Analysis and Simulation via the Adjoint and Covariance Methods, AIAA MST Conference , Aug 2007. http://www.aiaa.org/agenda.cfm?lumeetingid=1501&formatview=1&dateget=22-Aug-07 DeRusso, P. M., Roy, R. J., Close, C. M., and Desrochers, A. A. State Variables for Engineers, 2nd Edition , John Wiley and Sons, Inc., New York, NY, 1998. Moretti, G. Best Practices for Adopting Model-Based Design in Electronic System Development, White-paper, Gabe on EDA , 2007. http//www.mathworks.com/products/simulink/technicalliterature.html

Method of matrix continued fractions for the analysis of nonlinear stochastic systems

In this paper, stability analysis of deterministic and stochastic nonlinear discrete-time dynamical systems is considered. It is shown that some difficulties, such as independency or identical distribution of random variable sequence, that may arise in using Lyapunov's and LaSalle's approaches for stability analysis of discrete-time stochastic systems may be overcome by means of fixed point theory. Furthermore, suitable definitions of stability are provided. Ultimately, numerical examples are given in order to show the results.

In this study stochastic analysis of non-linear dynamical systems under α-stable,
\nmultiplicative white noise has been conducted. The analysis has dealt with a special class of α-stable
\nstochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the
\nprobability density function or of the characteristic function of the dynamical response may be obtained
\nconsidering the dynamical system forced by a Gaussian white noise with an uncertain factor with α/2-
\nstable distribution. This consideration yields the probability density function or the characteristic function
\nof the response by means of a simple integral involving the probability density function of the system
\nunder Gaussian white noise and the probability density function of the α/2-stable random parameter. Some
\nnumerical applications have been reported assessing the reliability of the proposed formulation. Moreover
\na proper way to perform digital simulation of the sub-Gaussian α-stable random process preventing
\ndynamical systems from numerical overflows has been reported and discussed in detail.

Analysis of Nonlinear Systems with Stochastic Input and Stochastic Parameters—II

The [formula omitted]-compound of a difference–algebraic system

Theory of nonlinear systems

Nonlinear [formula omitted]-gain verification for nonlinear systems

In this study, a general technique for the analysis of time-period nonlinear dynamical systems is presented. The method is based on the fact that all quasilinear periodic systems can be replaced by similar systems whose linear parts are time invariant via the well-known Liapunov-Floquet (L-F) transformation. A general procedure for the computation of L-F transformation in terms of Chebyshev polynomials is outlined. Once the transformation has been applied, a periodic orbit in original coordinates has a fixed point representation in the transformed coordinates. The stability and bifurcation analysis of the transformed equations are studied by employing thetime-dependent normal form theory and time-dependent centre manifold reduction. For the two examples considered, the three generic codimension-one bifurcations, viz, Hopf, flip and tangent, are analysed. The methodology is semi-analytic in nature and provides a quantitative measure of stability even under critical conditions. Unlike the perturbation or averaging techniques, this method is applicable even to those systems where the periodic term in the linear part does not contain a small parameter or a generating solution does not exist due to the absence of the time-invariant term in the linear part.

Analysis of fast-sampled non-linear systems: generalised frequency response functions for δ-operator models

There are many physical phenomena in engineering applications that are mostly modeled as stochastic differential equations. Sensor noise and environmental disturbance are two main sources of randomness. Determination of a closed-form analytical solution for the dynamical systems under random excitation is significantly useful for further investigations of these systems. Linear systems with stochastic excitation obey simple classified rules, leading to straightforward procedures for derivation of their analytical solutions. However, it is not the case for nonlinear systems with stochastic excitation, where no comprehensive method has been developed to be applicable to all such systems. This article brings in a novel method for statistical analysis of stochastic nonlinear systems, especially the ones with multiple equilibria, for which the traditional methods such as statistical linearization give incorrect approximations. The proposed method is mainly based on the moment closure method and assumes the joint probability distribution function of the state vector, as a linear combination of several normal distribution functions. The method represents a solution for essentially nonlinear systems, which are far from being linearizable. Duffing oscillator with negative linear stiffness is discussed as a case study to illustrate the advantage of the proposed method compared to the traditional ones. Such nonlinear systems especially arise in energy harvesting applications, when linear harvester designs do not fulfill performance requirements.

IntroductionThe dynamic behavior analysis of nonlinear physical systems plays a critical role in understanding complex processes across various domains, including education, where interactive simulations of such systems can enhance conceptual learning. Traditional modeling techniques for nonlinear systems often fail to capture their high-dimensional, multi-scale, and chaotic nature due to oversimplified assumptions or reliance on linear approximations.MethodsIn this study, we present a novel framework leveraging computer vision and advanced neural architectures to analyze the dynamic behaviors of nonlinear physical systems. The proposed Physics-Informed Nonlinear Dynamics Network (PNDN) integrates data-driven embeddings with physics-based constraints, offering a robust solution for capturing intricate dynamics and ensuring adherence to physical principles.ResultsExperimental results highlight the model’s superior performance in reconstructing and predicting nonlinear system behaviors under diverse conditions, establishing its utility for real-time educational simulations.DiscussionThis approach bridges the gap between computational modeling and educational innovation, providing learners with interactive tools to explore complex physical phenomena.