scholarly journals Controlling nonlinear dynamical systems into arbitrary states using machine learning

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Alexander Haluszczynski ◽  
Christoph Räth
Keyword(s):  
Machine Learning ◽  
Dynamical Systems ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Large Data ◽  
Data Sets ◽  
Initial State ◽  
Nonlinear Dynamical ◽  
Phase Space Methods

AbstractControlling nonlinear dynamical systems is a central task in many different areas of science and engineering. Chaotic systems can be stabilized (or chaotified) with small perturbations, yet existing approaches either require knowledge about the underlying system equations or large data sets as they rely on phase space methods. In this work we propose a novel and fully data driven scheme relying on machine learning (ML), which generalizes control techniques of chaotic systems without requiring a mathematical model for its dynamics. Exploiting recently developed ML-based prediction capabilities, we demonstrate that nonlinear systems can be forced to stay in arbitrary dynamical target states coming from any initial state. We outline and validate our approach using the examples of the Lorenz and the Rössler system and show how these systems can very accurately be brought not only to periodic, but even to intermittent and different chaotic behavior. Having this highly flexible control scheme with little demands on the amount of required data on hand, we briefly discuss possible applications ranging from engineering to medicine.

scholarly journals Multidimensional Approximation of Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (6) ◽  
Author(s):  
Patrick Gelß ◽  
Stefan Klus ◽  
Jens Eisert ◽  
Christof Schütte
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Data Driven ◽  
High Dimensional ◽  
Data Sets ◽  
Nonlinear Dynamical ◽  
Tensor Network ◽  
Wide Range ◽  
Multidimensional Approximation

A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems.

scholarly journals Artificial Intelligence, Chaos, Prediction and Understanding in Science

2021 ◽  
Vol 31 (11) ◽  
pp. 2150173
Author(s):  
Miguel A. F. Sanjuán
Keyword(s):  
Artificial Intelligence ◽  
Machine Learning ◽  
Big Data ◽  
Deep Learning ◽  
Dynamical Systems ◽  
Chaotic Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Constructive Dialogue ◽  
Learning Techniques

Machine learning and deep learning techniques are contributing much to the advancement of science. Their powerful predictive capabilities appear in numerous disciplines, including chaotic dynamics, but they miss understanding. The main thesis here is that prediction and understanding are two very different and important ideas that should guide us to follow the progress of science. Furthermore, the important role played by nonlinear dynamical systems is emphasized for the process of understanding. The path of the future of science will be marked by a constructive dialogue between big data and big theory, without which we cannot understand.

Nonlinear Dynamical Systems, Their Stability, and Chaos

2014 ◽  
Vol 66 (2) ◽  
Author(s):  
Amol Marathe ◽  
Rama Govindarajan
Keyword(s):  
Fluid Flow ◽  
Dynamical Systems ◽  
Nonlinear Systems ◽  
Fluid Mechanics ◽  
Fixed Points ◽  
Chaotic Systems ◽  
Nonlinear Dynamical Systems ◽  
Not Given ◽  
Nonlinear Dynamical ◽  
Detailed Explanation

This introduction to nonlinear systems is written for students of fluid mechanics, so connections are made throughout the text to familiar fluid flow systems. The aim is to present how nonlinear systems are qualitatively different from linear and to outline some simple procedures by which an understanding of nonlinear systems may be attempted. Considerable attention is paid to linear systems in the vicinity of fixed points, and it is discussed why this is relevant for nonlinear systems. A detailed explanation of chaos is not given, but a flavor of chaotic systems is presented. The focus is on physical understanding and not on mathematical rigor.

Uncertainty Quantification Using Generalized Polynomial Chaos Expansion for Nonlinear Dynamical Systems With Mixed State and Parameter Uncertainties

2019 ◽  
Vol 14 (2) ◽  
Author(s):  
Rajnish Bhusal ◽  
Kamesh Subbarao
Keyword(s):  
Dynamical Systems ◽  
Polynomial Chaos ◽  
Nonlinear Dynamical Systems ◽  
Distribution Functions ◽  
Parameter Uncertainties ◽  
Generalized Polynomial Chaos ◽  
Initial State ◽  
Aeroelastic System ◽  
Nonlinear Dynamical ◽  
Generalized Polynomial

This paper develops a framework for propagation of uncertainties, governed by different probability distribution functions in a stochastic dynamical system. More specifically, it deals with nonlinear dynamical systems, wherein both the initial state and parametric uncertainty have been taken into consideration and their effects studied in the model response. A sampling-based nonintrusive approach using pseudospectral stochastic collocation is employed to obtain the coefficients required for the generalized polynomial chaos (gPC) expansion in this framework. The samples are generated based on the distribution of the uncertainties, which are basically the cubature nodes to solve expectation integrals. A mixture of one-dimensional Gaussian quadrature techniques in a sparse grid framework is used to produce the required samples to obtain the integrals. The familiar problem of degeneracy with high-order gPC expansions is illustrated and insights into mitigation of such behavior are presented. To illustrate the efficacy of the proposed approach, numerical examples of dynamic systems with state and parametric uncertainties are considered which include the simple linear harmonic oscillator system and a two-degree-of-freedom nonlinear aeroelastic system.

scholarly journals Optimal Initial State for Fast Parameter Estimation in Nonlinear Dynamical Systems

2015 ◽  
Vol 48 (20) ◽  
pp. 557-562
Author(s):  
Qiaochu Li ◽  
Carine Jauberthie ◽  
Lilianne Denis-vidal ◽  
Zohra Cherfi
Keyword(s):  
Parameter Estimation ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Initial State ◽  
Nonlinear Dynamical

scholarly journals Machine Learning in Nonlinear Dynamical Systems

Resonance ◽  
2021 ◽  
Vol 26 (7) ◽  
pp. 953-970
Author(s):  
Sayan Roy ◽  
Debanjan Rana
Keyword(s):  
Machine Learning ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical

Synergetics and Acoustic Emission Approach for Crazing Nonlinear Dynamical Systems

2020 ◽  
Vol 30 (03) ◽  
pp. 2050043
Author(s):  
Guodong Gao ◽  
Yongming Xing
Keyword(s):  
Acoustic Emission ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
The Self ◽  
Self Organization ◽  
Mathematical Explanation ◽  
Theoretical Derivation ◽  
Nonlinear Dynamical ◽  
Loading Mode

This paper reports that synergetics are used to analyze the crazing evolution. On this basis, chaotic effect is explored. The chaos equation is established and verified. The theoretical derivation are consistent with the experimental results. We design a special specimen with a special loading mode, the transient monitoring function of acoustic emission (AE) technology is used to track and detect the crazing inside the PMMA in real time, and the experiments show that synergetics can explain the crazing properties of polymer. Importantly, the mathematical explanation is also given. The AE analysis, synergetics, and craze photo reached a conclusion that the crazing has chaotic behavior. After analyzing the AE events and crazing at different stress levels, the accuracy of synergetic approach for crazing is verified. By studying the course of AE events and crazing, the self-organization effect is proposed. The research results will provide data support for the application of PMMA in ship, aircraft, and precision instruments.

Probabilistic Analysis of Bifurcations in Stochastic Nonlinear Dynamical Systems

2019 ◽  
Vol 14 (8) ◽  
Author(s):  
Ehsan Mirzakhalili ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical Systems ◽  
Probability Density Function ◽  
Probability Density ◽  
Lorenz System ◽  
Nonlinear Dynamical Systems ◽  
Chaotic Behavior ◽  
Bifurcation Diagrams ◽  
Nonlinear Dynamical ◽  
Pitchfork Bifurcations ◽  
The Lorenz System

Bifurcation diagrams are limited most often to deterministic dynamical systems. However, stochastic dynamics can substantially affect the interpretation of such diagrams because the deterministic diagram often is not simply the mean of the probabilistic diagram. We present an approach based on the Fokker-Planck equation (FPE) to obtain probabilistic bifurcation diagrams for stochastic nonlinear dynamical systems. We propose a systematic approach to expand the analysis of nonlinear and linear dynamical systems from deterministic to stochastic when the states or the parameters of the system are noisy. We find stationary solutions of the FPE numerically. Then, marginal probability density function (MPDF) is used to track changes in the shape of probability distributions as well as determining the probability of finding the system at each point on the bifurcation diagram. Using MPDFs is necessary for multidimensional dynamical systems and allows direct visual comparison of deterministic bifurcation diagrams with the proposed probabilistic bifurcation diagrams. Hence, we explore how the deterministic bifurcation diagrams of different dynamical systems of different dimensions are affected by noise. For example, we show that additive noise can lead to an earlier bifurcation in one-dimensional (1D) subcritical pitchfork bifurcation. We further show that multiplicative noise can have dramatic changes such as changing 1D subcritical pitchfork bifurcations into supercritical pitchfork bifurcations or annihilating the bifurcation altogether. We demonstrate how the joint probability density function (PDF) can show the presence of limit cycles in the FitzHugh–Nagumo (FHN) neuron model or chaotic behavior in the Lorenz system. Moreover, we reveal that the Lorenz system has chaotic behavior earlier in the presence of noise. We study coupled Brusselators to show how our approach can be used to construct bifurcation diagrams for higher dimensional systems.

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