Modeling Nonlinear Oscillators via Variable-Order Fractional Operators

2019 ◽  
Author(s):  
Sansit Patnaik ◽  
Fabio Semperlotti
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Models ◽  
Contact Problems ◽  
Variable Order ◽  
Mathematical Tool ◽  
Fractional Operators ◽  
State Variables ◽  
Governing Equations ◽  
Seamless Transition ◽  
Physical Phenomena

Abstract Fractional derivatives and integrals are intrinsically multiscale operators that can act on both space and time dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or variable order (VO) where, in the latter case, the order can also be function of either independent or state variables. When using VO differential governing equations to describe the response of dynamical systems, the order can evolve as a function of the response itself therefore allowing a natural and seamless transition between largely dissimilar dynamics (e.g. linear, nonlinear, and even contact problems). Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. In this study, we present the possible application of VO operators to a class of nonlinear lumped parameter models that has great practical relevance in mechanics and dynamics. Specific examples include hysteresis and contact problems for discrete oscillators. Within this context, we present a methodology to define VO operators capable of capturing such complex physical phenomena. Despite using simplified lumped parameters nonlinear models to present the application of VO operators to mechanics and dynamics, we provide a more qualitative discussion of the possible applications of this mathematical tool in the broader context of continuous multiscale systems.

scholarly journals Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators

2020 ◽  
Vol 15 (9) ◽  
Author(s):  
Sansit Patnaik ◽  
Fabio Semperlotti
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Contact Problems ◽  
Discrete Systems ◽  
Hysteretic Behavior ◽  
Variable Order ◽  
Fractional Operators ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Integer Order

Abstract The modeling of nonlinear dynamical systems subject to strong and evolving nonsmooth nonlinearities is typically approached via integer-order differential equations. In this study, we present the possible application of variable-order (VO) fractional operators to a class of nonlinear lumped parameter models that have great practical relevance in mechanics and dynamics. Fractional operators are intrinsically multiscale operators that can act on both space- and time-dependent variables. Contrarily to their integer-order counterpart, fractional operators can have either fixed or VO. In the latter case, the order can be function of either independent or state variables. We show that when using VO equations to describe the response of dynamical systems, the order can evolve as a function of the response itself; therefore, allowing a natural and seamless transition between widely dissimilar dynamics. Such an intriguing characteristic allows defining governing equations for dynamical systems that are evolutionary in nature. Within this context, we present a physics-driven strategy to define VO operators capable of capturing complex and evolutionary phenomena. Specific examples include hysteresis in discrete oscillators and contact problems. Despite using simplified models to illustrate the applications of VO operators, we show numerical evidence of their unique modeling capabilities as well as their connection to more complex dynamical systems.

scholarly journals Applications of Distributed-Order Fractional Operators: A Review

Entropy ◽  
2021 ◽  
Vol 23 (1) ◽  
pp. 110
Author(s):  
Wei Ding ◽  
Sansit Patnaik ◽  
Sai Sidhardh ◽  
Fabio Semperlotti
Keyword(s):  
Fractional Calculus ◽  
Real World ◽  
Transport Processes ◽  
Model Systems ◽  
Variable Order ◽  
Fractional Operators ◽  
Research Activity ◽  
Road Map ◽  
Real World Applications ◽  
Distributed Order

Distributed-order fractional calculus (DOFC) is a rapidly emerging branch of the broader area of fractional calculus that has important and far-reaching applications for the modeling of complex systems. DOFC generalizes the intrinsic multiscale nature of constant and variable-order fractional operators opening significant opportunities to model systems whose behavior stems from the complex interplay and superposition of nonlocal and memory effects occurring over a multitude of scales. In recent years, a significant amount of studies focusing on mathematical aspects and real-world applications of DOFC have been produced. However, a systematic review of the available literature and of the state-of-the-art of DOFC as it pertains, specifically, to real-world applications is still lacking. This review article is intended to provide the reader a road map to understand the early development of DOFC and the progressive evolution and application to the modeling of complex real-world problems. The review starts by offering a brief introduction to the mathematics of DOFC, including analytical and numerical methods, and it continues providing an extensive overview of the applications of DOFC to fields like viscoelasticity, transport processes, and control theory that have seen most of the research activity to date.

scholarly journals Discovering governing equations from data by sparse identification of nonlinear dynamical systems

2016 ◽  
Vol 113 (15) ◽  
pp. 3932-3937 ◽  
Author(s):  
Steven L. Brunton ◽  
Joshua L. Proctor ◽  
J. Nathan Kutz
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Climate Science ◽  
Model Complexity ◽  
Governing Equations ◽  
Canonical Systems ◽  
Nonlinear Dynamical ◽  
Balance Accuracy ◽  
Wide Range

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

Physical Realization of Generic-Element Parameters in Model Updating

2002 ◽  
Vol 124 (4) ◽  
pp. 628-633 ◽  
Author(s):  
H. Ahmadian ◽  
J. E. Mottershead ◽  
M. I. Friswell
Keyword(s):  
Model Updating ◽  
Finite Element Models ◽  
Physical Realization ◽  
Governing Equations ◽  
Stiffness Matrices ◽  
Model Predictions ◽  
Physical Effects ◽  
Realization Process ◽  
Physical Phenomena ◽  
Selection Of

The selection of parameters is most important to successful updating of finite element models. When the parameters are chosen on the basis of engineering understanding the model predictions are brought into agreement with experimental observations, and the behavior of the structure, even when differently configured, can be determined with confidence. Physical phenomena may be misrepresented in the original model, or may be absent altogether. In any case the updated model should represent an improved physical understanding of the structure and not simply consist of unrepresentative numbers which happen to cause the results of the model to agree with particular test data. The present paper introduces a systematic approach for the selection and physical realization of updated terms. In the realization process, the discrete equilibrium equation formed by mass, and stiffness matrices is converted to a continuous form at each node. By comparing the resulting differential equation with governing equations known to represent physical phenomena, the updated terms and their physical effects can be recognized. The approach is demonstrated by an experimental example.

scholarly journals Limitations of demand- and pressure-driven modeling for large deficient networks

2017 ◽  
Vol 10 (2) ◽  
pp. 93-98 ◽  
Author(s):  
Mathias Braun ◽  
Olivier Piller ◽  
Jochen Deuerlein ◽  
Iraj Mortazavi
Keyword(s):  
Water Distribution ◽  
Network Models ◽  
Distribution Networks ◽  
Computer Applications ◽  
State Variables ◽  
Modeling Process ◽  
Uniqueness Of The Solution ◽  
Basic Equations ◽  
Physical Phenomena ◽  
Pressure Driven

Abstract. The calculation of hydraulic state variables for a network is an important task in managing the distribution of potable water. Over the years the mathematical modeling process has been improved by numerous researchers for utilization in new computer applications and the more realistic modeling of water distribution networks. But, in spite of these continuous advances, there are still a number of physical phenomena that may not be tackled correctly by current models. This paper will take a closer look at the two modeling paradigms given by demand- and pressure-driven modeling. The basic equations are introduced and parallels are drawn with the optimization formulations from electrical engineering. These formulations guarantee the existence and uniqueness of the solution. One of the central questions of the French and German research project ResiWater is the investigation of the network resilience in the case of extreme events or disasters. Under such extraordinary conditions where models are pushed beyond their limits, we talk about deficient network models. Examples of deficient networks are given by highly regulated flow, leakage or pipe bursts and cases where pressure falls below the vapor pressure of water. These examples will be presented and analyzed on the solvability and physical correctness of the solution with respect to demand- and pressure-driven models.

scholarly journals A new hybridization filtering-based linear-nonlinear models for time series forecasting

2021 ◽  
Author(s):  
Mehrnaz Ahmadi ◽  
◽  
Mehdi Khashei ◽  
Keyword(s):  
Hybrid Methods ◽  
Nonlinear Models ◽  
Prediction Methods ◽  
Short Term ◽  
Forecasting Methods ◽  
Primary Relationships ◽  
Time Dependencies ◽  
Physical Laws ◽  
Face Time ◽  
Physical Phenomena

In recent years, the idea of using a mathematical model to describe the behavior of physical phenomena has been very much considered. Specifically, a definitive model, based on physical laws, enables researchers to calculate the number of time dependencies precisely at any moment in time. However, in the real world, we often face time-dependent phenomena with many unknown or unavailable factors (Lindley, 2010; Roulston et al., 2003). In this case, when it is not possible to achieve a definite - model, the prediction methods are wide used, especially when the past observations of a variable and primary relationships between specific observations are available. Forecasting methods that are used in different fields of science can be categorized based on various aspects. For example, the prediction methods used in the field of wind energy can be divided into four categories of 1) ultra short term (several seconds to four hours), 2) short term (4 to 24 hours), 3) medium-term (1 to 7 days), and 4) long term (more than 7 days) (Zack, 2003; Soman et al., 2010). Also, the structure of forecasting methods can be divided into two types of 1) single methods and 2) hybrid methods. Each of these categories can also be subdivided into smaller subgroups.

PARTICLE FILTERS IN A MULTISCALE ENVIRONMENT: WITH APPLICATION TO THE LORENZ-96 ATMOSPHERIC MODEL

2011 ◽  
Vol 11 (02n03) ◽  
pp. 569-591 ◽  
Author(s):  
HOONG CHIEH YEONG ◽  
JUN HYUN PARK ◽  
N. SRI NAMACHCHIVAYA
Keyword(s):  
Dynamical Systems ◽  
Particle Filter ◽  
Observational Data ◽  
Particle Filtering ◽  
Atmospheric Model ◽  
Random Dynamical Systems ◽  
Coarse Grained ◽  
Conditional Density ◽  
State Variables ◽  
Lorenz 96

The study of random dynamical systems involves understanding the evolution of state variables that contain uncertainties and that are usually hidden, or not directly observable. Therefore, state variables have to be estimated and updated based on system models using information from observational data, which themselves are noisy, in the sense that they contain uncertainties and disturbances due to imperfections in observational devices and disturbances in the environment within which data are being collected. The development of efficient data assimilation methods for integrating observational data in predicting the evolution of random state variables is thus an important aspect in the study of random dynamical systems. In this paper, we consider a particle filtering approach to nonlinear filtering in multiscale dynamical systems. Particle filtering methods [1–3] utilizes ensembles of particles to represent the conditional density of state variables using particle positions, distributed over a sample space. The distribution of an ensemble of particles is updated using observational data to obtain the best representation of the conditional density of the state variables of interest. On the other hand, homogenization theory [4, 5], allows us to estimate the coarse-grained (slow) dynamics of a multiscale system on a larger timescale without having to explicitly study the fast variable evolution on a small timescale. The results of filter convergence presented in [6] shows the convergence of the filter of the actual state variable to a homogenized solution to the original multiscale system, and thus we develop a particle filtering scheme for multiscale random dynamical systems that utilizes this convergence result. This particle filtering method is called the Homogenized Hybird Particle Filter, and it incorporates a multiscale computation scheme, the Heterogeneous Multiscale Method developed in [7], with the novel branching particle filter described in [8–10]. By incorporating a multiscale scheme based on homogenization of the original system, estimation of the coarse-grained dynamics using observational data is performed over a larger timescale, thus resulting in computational time and cost reduction in terms of the evolution of the state variables as well as functional evaluations for the filtering aspect. We describe the theory behind this combined scheme and its general algorithm, concluded with an application to the Lorenz-96 [11] atmospheric model that mimics midlatitude geophysical dynamics with microscopic convective processes.

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