scholarly journals On a class of integro-differential equations modeling complex systems with nonlinear interactions

2012 ◽  
Vol 25 (3) ◽  
pp. 490-495 ◽  
Author(s):  
L. Arlotti ◽  
E. De Angelis ◽  
L. Fermo ◽  
M. Lachowicz ◽  
N. Bellomo
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Nonlinear Interactions

scholarly journals On local fractional operators View of computational complexity: Diffusion and relaxation defined on cantor sets

2016 ◽  
Vol 20 (suppl. 3) ◽  
pp. 755-767 ◽  
Author(s):  
Xiao-Jun Yang ◽  
Zhi-Zhen Zhang ◽  
Tenreiro Machado ◽  
Dumitru Baleanu
Keyword(s):  
Computational Complexity ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Systems ◽  
Exact Solutions ◽  
Fractional Derivatives ◽  
Cantor Sets ◽  
Fractional Operators ◽  
Fractional Partial Differential Equations ◽  
Comparative Results

This paper treats the description of non-differentiable dynamics occurring in complex systems governed by local fractional partial differential equations. The exact solutions of diffusion and relaxation equations with Mittag-Leffler and exponential decay defined on Cantor sets are calculated. Comparative results with other versions of the local fractional derivatives are discussed.

scholarly journals Specific Mathematical Aspects of Dynamics Generated by Coherence Functions

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Ezzat G. Bakhoum ◽  
Cristian Toma
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Integral Equations ◽  
Quantum Physics ◽  
Initial Conditions ◽  
Coherence Function ◽  
Multiple Integral ◽  
The State ◽  
Integrodifferential Equations ◽  
State Variables

This study presents specific aspects of dynamics generated by the coherence function (acting in an integral manner). It is considered that an oscillating system starting to work from initial nonzero conditions is commanded by the coherence function between the output of the system and an alternating function of a certain frequency. For different initial conditions, the evolution of the system is analyzed. The equivalence between integrodifferential equations and integral equations implying the same number of state variables is investigated; it is shown that integro-differential equations of second order are far more restrictive regarding the initial conditions for the state variables. Then, the analysis is extended to equations of evolution where the coherence function is acting under the form of a multiple integral. It is shown that for the coherence function represented under the form of annth integral, some specific aspects as multiscale behaviour suitable for modelling transitions in complex systems (e.g., quantum physics) could be noticed whennequals 4, 5, or 6.

scholarly journals Nonparametric inference of stochastic differential equations based on the relative entropy rate

2021 ◽  
Author(s):  
Min Dai ◽  
Jinqiao Duan ◽  
jianyu Hu ◽  
Xiangjun Wang
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Stochastic Differential Equations ◽  
Relative Entropy ◽  
Nonparametric Inference ◽  
Entropy Rate ◽  
Governing Equations ◽  
Nonparametric Approach ◽  
Information Detection ◽  
Kernel Theory

The information detection of complex systems from data is currently undergoing a revolution,driven by the emergence of big data and machine learning methodology. Discovering governingequations and quantifying dynamical properties of complex systems are among central challenges. Inthis work, we devise a nonparametric approach to learn the relative entropy rate from observationsof stochastic differential equations with different drift functions. The estimator corresponding tothe relative entropy rate then is presented via the Gaussian process kernel theory. Meanwhile, thisapproach enables to extract the governing equations. We illustrate our approach in several examples.Numerical experiments show the proposed approach performs well for rational drift functions, notonly polynomial drift functions.

Chaos Theory and Complexity Theory

2013 ◽  
Author(s):  
Keith Warren
Keyword(s):  
Nonlinear Dynamics ◽  
Dynamical Systems ◽  
Complex Systems ◽  
Systems Theory ◽  
Complexity Theory ◽  
Chaos Theory ◽  
Mathematical Framework ◽  
Nonlinear Interactions ◽  
Simple Systems ◽  
Over Time

Chaos theory and complexity theory, collectively known as nonlinear dynamics or dynamical systems theory, provide a mathematical framework for thinking about change over time. Chaos theory seeks an understanding of simple systems that may change in a sudden, unexpected, or irregular way. Complexity theory focuses on complex systems involving numerous interacting parts, which often give rise to unexpected order. The framework that encompasses both theories is one of nonlinear interactions between variables that give rise to outcomes that are not easily predictable. This entry provides a nonmathematical introduction, discussion of current research, and references for further reading.

scholarly journals Bundled Causal History Interaction

Entropy ◽  
2020 ◽  
Vol 22 (3) ◽  
pp. 360
Author(s):  
Peishi Jiang ◽  
Praveen Kumar
Keyword(s):  
Complex Systems ◽  
Evolutionary Dynamics ◽  
Graphical Model ◽  
Partial Information ◽  
Synergistic Effects ◽  
Nonlinear Interactions ◽  
Stream Chemistry ◽  
Multivariate System ◽  
Complex Dependence ◽  
Temporal Interactions

Complex systems arise as a result of the nonlinear interactions between components. In particular, the evolutionary dynamics of a multivariate system encodes the ways in which different variables interact with each other individually or in groups. One fundamental question that remains unanswered is: How do two non-overlapping multivariate subsets of variables interact to causally determine the outcome of a specific variable? Here, we provide an information-based approach to address this problem. We delineate the temporal interactions between the bundles in a probabilistic graphical model. The strength of the interactions, captured by partial information decomposition, then exposes complex behavior of dependencies and memory within the system. The proposed approach successfully illustrated complex dependence between cations and anions as determinants of pH in an observed stream chemistry system. In the studied catchment, the dynamics of pH is a result of both cations and anions through mainly synergistic effects of the two and their individual influences as well. This example demonstrates the potentially broad applicability of the approach, establishing the foundation to study the interaction between groups of variables in a range of complex systems.

scholarly journals On the modeling of nonlinear interactions in large complex systems

2010 ◽  
Vol 23 (11) ◽  
pp. 1372-1377 ◽  
Author(s):  
N. Bellomo ◽  
C. Bianca ◽  
M.S. Mongiovì
Keyword(s):  
Complex Systems ◽  
Nonlinear Interactions ◽  
Large Complex

Reconstruction of Complex Dynamical Systems Using Stochastic Differential Equations

2020 ◽  
Author(s):  
Forough Hassanibesheli ◽  
Niklas Boers ◽  
Jürgen Kurths
Keyword(s):  
Time Series ◽  
Differential Equations ◽  
Complex Systems ◽  
Stochastic Differential Equations ◽  
Langevin Equation ◽  
Degrees Of Freedom ◽  
Evolution Equations ◽  
Ice Core ◽  
Dynamical Behavior ◽  
Generalized Langevin Equation

<p>A complex system is a system composed of highly interconnected components in which the collective property of an underlying system cannot be described by dynamical behavior of the individual parts. Typically, complex systems are governed by nonlinear interactions and intricate fluctuations, thus to retrieve dynamics of a system, it is required to characterize and asses interactions between deterministic tendencies and random fluctuations. </p><p>For systems with large numbers of degrees of freedom, interacting across various time scales, deriving time-evolution equations from data is computationally expensive. A possible way to circumvent this problem is to isolate a small number of relatively slow degrees of freedom that may suffice to characterize the underlying dynamics and solve the governing motion equation for the reduced-dimension system in the framework of stochastic differential equations(SDEs).  For some specific example settings, we have studied the performance of three stochastic dimension-reduction methods (Langevin equation(LE), generalized Langevin Equation(GLE) and Empirical Model Reduction(EMR)) to model various synthetic and real-world time series. In this study corresponding numerical simulations of all models have been examined by probability distribution function(PDF) and Autocorrelation function(ACF) of the average simulated time series as statistical benchmarks for assessing the differnt models' performance. </p><p>First we reconstruct the Niño-3 monthly sea surface temperature (SST) indices averages across (5°N–5°S, 150°–90°W) from 1891 to 2015 using the three aforementioned stochastic models. We demonstrate that all these considered models can reproduce the same skewed and heavy-tailed distributions of Niño-3 SST, comparing ACFs, GLE exhibits a tendency towards achieving a higher accuracy than LE and EMR. A particular challenge for deriving the underlying dynamics of complex systems from data is given by situations of abrupt transitions between alternative states. We show how the Kramers-Moyal approach to derive drift and diffusion terms for LEs can help in such situations. A prominent example of such 'Tipping Events' is given by the Dansgaard-Oeschger events during previous glacial intervals. We attempt to obtain the statistical properties of high-resolution, 20yr average, δ<sup>18</sup>O and Ca<sup>+</sup><sup>2</sup> collected from the same ice core from the NGRIP on the GICC05 time scale. Through extensive analyses of various systems, our results signify that stochastic differential equation models considering memory effects are comparatively better approaches for understanding  complex systems.</p><p> </p>

Out-of-Plane Vibration of a Curved Pipe due to Pulsating Flow (Nonlinear Interactions Between In-Plane and Out-of-Plane Vibrations)

2010 ◽  
Author(s):  
Kiyotaka Yamashita ◽  
Keita Nakamura ◽  
Hiroshi Yabuno
Keyword(s):  
Fluid Flow ◽  
Differential Equations ◽  
Harmonic Component ◽  
Small Time ◽  
Nonlinear Interactions ◽  
Curved Pipe ◽  
Plane Vibration ◽  
Out Of Plane ◽  
Pulsating Fluid ◽  
Conveying Fluid

A great deal of study has been done on the dynamics of straight pipes conveying fluid. In contrast, only a few studies have been devoted to the dynamics of curved pipe conveying fluid. In this paper, a theoretical and experimental investigation was conducted into out-of-plane vibration of a curved pipe for the case that the fluid flow contains a small time-dependent harmonic component. The nonlinear out-of plane vibrations of a curved pipe, which is hanging horizontally and is supported at both ends, are examined when the frequency of the pulsating fluid flow is near twice the fundamental natural frequency of out-of-plane vibration. The main purpose of this paper is to investigate the nonlinear interactions between the in-plane and the out-of-plane vibrations analytically and experimentally. The partial differential equations of out-of-plane motions are reduced into a set of ordinary differential equations, which govern the amplitude and phase of out-of-plane vibration, using the method of Lyapnov-Schmidt reduction. It is clarified that the excitation of the in-plane vibration produces significant responses in the out-of-plane vibrations. Finally, the experiments were conducted with a silicon rubber pipe conveying water. The typical features of out-of-plane vibration are confirmed qualitatively by experiment.

Sign in / Sign up

Export Citation Format

Share Document