Chapter 8 Numerical analysis of dynamical systems

This work delves into the theory of dynamic systems, focusing on the analysis of entropy in both classical and topological contexts. Beginning with an exposition of fundamental concepts in dynamical systems theory, particular attention is given to topological dynamical systems (TDS). The discussion progresses to explore discrete topological entropy and its significance within dynamical systems, culminating in the introduction of topological entropy pressure as a nuanced form of this concept. The study then investigates various applications of topological entropy within dynamic systems, emphasizing its utility in understanding chaotic systems and its role in ergodic theory. A novel theory, termed Topological Ergodic Entropy Theory (TEET), is presented, offering a fresh perspective on the analysis of ergodic dynamical systems. Furthermore, the work introduces the Ergodic Theory of Turbulent Flow (ETTF), which probes the interplay between topological entropy and the ergodic properties of dynamic systems governed by the Navier-Stokes equations. Through these explorations, the findings contribute significantly to the comprehension of the intricate nature of dynamical systems and their diverse applications across mathematics and physics. By scrutinizing topological entropy and its implications in dynamical systems, this research offers novel insights into the chaotic and stochastic behaviors exhibited by these systems. Additionally, the introduction of pioneering theories like ETTF opens up new avenues for understanding and modeling turbulent phenomena, thereby enriching our understanding of complex dynamical processes.

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

Parallel implementation for the two-stage SDIRK methods via diagonalization

The design of aircraft for ground maneuvers is an essential part in satisfying the demanding requirements of the aircraft operators. Extensive analysis is done to ensure that a new civil aircraft type will adhere to these requirements, for which the nonlinear nature of the problem generally adds to the complexity of such calculations. Small perturbations in velocity, steering angle, or brake application may lead to significant differences in the final turn widths that can be achieved. Here, the U-turn maneuver is analyzed in detail, with a comparison between the two ways in which this maneuver is conducted. A comparison is also made between existing turn-width prediction methods that consist mainly of geometric methods and simulations and a proposed new method that uses dynamical systems theory. Some assumptions are made with regard to the transient behavior, for which it is shown that these assumptions are conservative when an upper bound is chosen for the transient distance. Furthermore, we demonstrate that the results from the dynamical systems analysis are sufficiently close to the results from simulations to be used as a valuable design tool. Overall, dynamical systems methods provide an order-of-magnitude increase in analysis speed and capability for the prediction of turn widths on the ground when compared with simulations.

Variable step-size techniques in continuous Runge-Kutta methods for isospectral dynamical systems

Introduction: This article is the result of the research “Analysis of the dynamic systems component of the engineering programs in control and automation and technology in industrial electronics,” developed at the Francisco José de Caldas District University in the years 2022-2023. Problem: The document addresses the lack of a clear and comprehensive structure in the analysis of linear dynamic systems, as well as the absence of an innovative approach to addressing these topics in traditional courses, which hinders their understanding and application in different disciplines. Objective: The objective is to present a novel approach to analyzing linear dynamic systems, providing a complete framework of concepts, relationships, and important tools in this analysis, as well as a literature review based on over seventeen years of experience in courses and studies of this kind. Methodology: A journey of concepts is proposed from signals and systems, through modeling with methods such as black box, white box, gray box, to linear analysis based on examples with calculation of differential equations, state representation, transfer function, and block diagrams. Results: The document aggregates and articulates all the concepts of dynamic systems, along with the relationships and tools used, offering a more practical and intuitive approach to understanding the material. Conclusion: The document provides a comprehensive and articulated view of key concepts in the analysis of linear dynamic systems, highlighting an innovative approach that facilitates their understanding. Its main contribution is to aggregate and articulate these concepts, along with the tools and relationships used, to offer a more practical and clear approach for their study. Originality: The originality lies in proposing a novel and structured approach to analyzing linear dynamic systems, addressing the lack of clarity and completeness in traditional approaches. Limitations: Although the document proposes a novel approach, it does not delve into specific aspects of some topics covered, which could limit the detailed understanding of certain concepts. Deepening: The document could delve into the practical application of the proposed concepts and tools in real cases of dynamic systems, as well as into the comparison with traditional approaches to highlight the differences and advantages of the proposed new approach.

The dynamics of a differential algebraic equation takes place on a lower dimensional manifold in phase space. Applying a numerical integration scheme, it is natural to ask if and how this geometric property is preserved by the discrete dynamical system. In the index-1 case answers to this question are obtained from the singularly perturbed case treated by Nipp and Stoffer, Numer. Math. 70 (1995), 245–257, for Runge-Kutta methods and in K. Nipp and D. Stoffer, Numer. Math. 74 (1996), 305–323, for linear multistep methods. As main result of this paper it is shown that also for Runge-Kutta methods and linear multistep methods applied to a index-2 problem of Hessenberg form there is a (attractive) invariant manifold for the discrete dynamical system and this manifold is close to the manifold of the differential algebraic equation.

Most dynamical systems in real-life applications are usually nonlinear and nonstationary, such as biomechanical systems, neuroscience, healthcare, economics, and engineering. Nowadays, the availability of data makes it possible to investigate and discover the hidden dynamics of target systems. However, numerous concerns arise in modeling and analysis of such systems, including complex system structure, data-inherent uncertainty in observations, poor interpretability, etc. To this end, we will investigate dynamical behaviors by proposing data-driven machine learning and dynamical system techniques. Specifically, we present various approaches for extracting characteristics or patterns that have good interpretability in various complex dynamical system applications.In the first study, we consider a pattern recognition problem in biomechanical system using multivariate human kinematic data and propose data-driven analytical methods to model human movement. We first perform recurrence quantification analysis on the phase space of the kinematic system to visualize and evaluate the recurrence behavior of the folded phase space trajectory. However, this method could miss out the relationship between variables. We therefore introduce a new modeling approach based on the self-expressive assumption to discover the interactions between variables and characterize the patterns of human motion. Our model measures correlations between variables by underlying linear subspaces, which can be further extended to nonlinear subspaces. Based on these correlations, our model improves the classification accuracy by 20% - 30% over other standard state-of-art correlation measurements and clustering methods. In the second study, we consider a specific neuroscience problem, seizure detection, to recognize abnormal brain states by learning from highly nonlinear electroencephalography. Based on dynamical systems theory, Koopman operator theory, and dynamic mode decomposition, we project the complex brain dynamics into an infinite-dimensional linear space, where the intrinsic linear dynamic shares the same dynamical behavior as the original nonlinear dynamical system. We propose a new system measurement to unify the projections in different Koopman spaces. Our measurement indicates the stability of the dynamical system, which represents the system state trajectory with respect to the corresponding attractor. We demonstrate the effectiveness of our approach with a seizure onset detection task. In this case study, our model gives high accuracy with a low false-positive rate. In the last study, we consider dynamical systems with multiple modalities. We design a Koopman-based deep learning network to identify and simplify the intrinsic nonlinear dynamics from different sources/observations of a multi-modal dynamical system. Since these different modalities describe the same system, they jointly share the same intrinsic dynamics. We develop an autoencoder framework with custom loss functions to find proper project functions/operators and to obtain the intrinsic Koopman coordinates. Our framework can reconstruct the original system states based on the joint dynamics. Once the intrinsic linear dynamics is learned, future system states can be predicted by the evolution in the Koopman space. This approach allows us to uncover and linearize the hidden nonlinear dynamics, which enables linear dynamical tools, for example, system control, spectrum analysis, to analyze complex dynamical systems.--Author's abstract

This chapter deals with the numerical solution of initial value problems of ordinary differential equations. First, different approaches to obtain integration methods for ordinary differential equations are presented. Then the important properties consistency, stability, and convergence of integration methods are introduced and studied. This allows for a fruitful application of the results obtained in Chapter 12. Subsequently, one-step methods are considered in detail and their important properties are shown. Moreover, Runge–Kutta methods are thoroughly investigated. Finally, we deal with linear multistep methods and study the asymptotic behavior of integration methods and of stiff differential equations. In particular, we deal with Dahlquist’s root condition and derive the first Dahlquist barrier and the second Dahlquist barrier.

This chapter defines dynamic systems, its relevance to sports performance analysis, and explains the inter-personal coordination and perturbations during sport competition. It is concerned with outlining a dominant motor control theory that can aid the explanation and interpretation of performance analysis data dynamic systems theory. The chapter has three central principles of dynamic systems theory: synergies, self-organisation and constraints. It is built upon the existing research using dynamic systems theory to study attacker-defender interactions in rugby union, it could be argued that the practical implications were limited. Future research using dynamic systems theory will be a theoretical framework for analysing inter-personal coordination, ensuring the analysis can be more effectively linked to the attacking/defending principles of play, and be useful and informative for sports coach and performance analysts. There are clear links between the theory and the principles of play of the sport, i.e. the attacking principles of soccer: width, mobility, support, facilitating the exploration of the available playing space.

In this article, the identification and analysis of spatio-temporal dynamical systems is presented. An approximated B-spline wavelet representation of spatio-temporal dynamical systems is identified using an orthogonal least-squares algorithm from measured data. Control variables are incorporated to represent controlled external inputs and/or some system parameters for the purpose of analysis. The identified system models can be used to evaluate how the external inputs or system parameters affect the evolution of the spatio-temporal dynamics and pattern formation. Two examples are used to illustrate the proposed approach.

In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. Although their algorithm is powerful, it taxes the capabilities of most modern computers when the dimension of a system is high or even when the dimension is moderate in size. In this paper we remove these difficulties to a certain extent by generalizing the work of Brayton and Tong to interconnected dynamical systems. In our approach, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.

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.

In the previous chapter, we introduced explicit Runge-Kutta (ERK) methods and demonstrated how they can be implemented as a hierarchy of Python classes. For most ODE systems, replacing the simple forward Euler method with a higher-order ERK method will significantly reduce the number of time steps needed to reach a specified accuracy. Furthermore, it often leads to reduced computation time, since the additional cost per time step is outweighed by the reduced number of steps. However, there exists a class of ODEs known as stiff systems, where all the ERK methods require very small time steps, and any attempt to increase the time step leads to spurious oscillations and possible divergence of the solution. Stiff ODE systems pose a challenge for explicit methods, and they are better addressed by implicit solvers such as implicit Runge-Kutta (IRK) methods. IRK methods are well-suited for stiff problems and can offer substantial reductions in computation time when tackling challenging problems.

Chaos theory can be used for quantitative dynamics of uncertainty and finding order in its perturbances. Chaos theory is largely a colloquial notion and is referred to as an analysis of non-linear dynamical systems. The studies devoted to linear dynamical systems and theories of complexity involve studies of turbulence, or, more precisely, the transfer from stability to turbulence. The dynamical system by its nature does not follow long-term forecasts. There are two reasons for the unpredictability. Dynamical systems involve both feedback and critical levels. To paraphrase, one can say that a dynamical system forms a system of non-linear feedback. The most important properties of the system include: sensitivity to the change of the initial conditions, occurrence of critical points and a fractal dimension. Non-linear dynamical systems tend to have more than one solution. The case often is that the number of solutions is huge or even infinite. A visual representation of the data forms a finite space called the phase space of the system. The number of dimensions in the space is determined by the number of variable presents in the system. If there are two or three variables, it is possible to visually examine the data. In case of a larger number of dimensions, the data are examined with the use of mathematical data. Another term used in non-linear dynamical systems is the attractor. This is an area of a equilibrium of non-linear system within a time series. A system which aims to achieve an equilibrium in the form of a single value has a point attractor. Besides, there are phase attractors, which form periodic cycles or orbits in space forming a limit cycle. An attractor, in which none of the points in the space overlaps and whose orbits do not cross but both originate in the same area of phase space, is called strange attractor. Such attractors in contrast to point attractors are non-periodic and predominantly have a fractal dimension [10,11].