DOCSLIB.ORG

Towards a Stability Theory of General Hybrid Dynamical Systems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Towards a Stability Theory of General Hybrid Dynamical Systems Automatica 35 (1999) 371—384 Towards a stability theory of general hybrid dynamical systems Anthony N. Michel* ,BoHu Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA Received 29 September 1997; revised 18 May 1998; received in final form 27 August 1998 A general model for hybrid dynamical systems is presented which incorporates a concept of generalized time. For such systems, a stability theory is developed. Both Comparison ¹heorems and the Principal ¸yapunov Stability Results are included. Abstract In recent work we proposed a general model for hybrid dynamical systems whose states are defined on arbitrary metric space and evolve along some notion of generalized abstract time. For such systems we introduced the usual concepts of Lyapunov and Lagrange stability. We showed that it is always possible to transform this class of hybrid dynamical systems into another class of dynamical systems with equivalent qualitative properties, but defined on real time R>"[0, R). The motions of this class of systems are in general discontinuous. This class of systems may be finite or infinite dimensional. For the above discontinuous dynamical systems (and hence, for the above hybrid dynamical systems), we established the Principal Lyapunov Stability Theorems as well as Lagrange Stability Theorems. For some of these, we also established converse theorems. We demonstrated the applicability of these results by means of specific classes of hybrid dynamical systems. In the present paper we continue the work described above. In doing so, we first develop a general comparison theory for the class of hybrid dynamical systems (resp., discontinuous dynamical systems) considered herein, making use of stability preserving mappings. We then show how these results can be applied to establish some of the Principal Lyaponov Stability Theorems. For the latter, we also state and prove a converse theorem not considered previously. Finally, to demonstrate the applicability of our results, we consider specific examples throughout the paper. ( 1999 Elsevier Science Ltd. All rights reserved. Keywords: Hybrid dynamical systems; stability analysis; Lyapunov stability; Lagrange stability 1. Introduction of the type encountered in science and applied mathe- matics (such as, e.g., ordinary differential equations, The demands of modern technology have resulted in ordinary difference equations, partial differential equa- the synthesis and implementation of systems of increas- tions, functional differential equations, Volterra integ- ing complexity. Such contemporary systems frequently rodifferential equations, and the like). Perhaps the most defy simple and tidy descriptions by classical equations important class of such contemporary systems are hybrid systems, whose descriptions involve mixtures of equa- tions of the type enumerated above; subsystems endowed with uncertainties which can best be represented by dif- ferential inclusions (e.g., differential inequalities rather * Corresponding author. Tel.: #001 219 6314395; fax: #001 219 than differential equations); ‘‘equation free’’ subsystems, 6314393; e-mail: [email protected]. such as discrete event systems (along with the necessary This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Guest Editors J.M. interface elements) for which there is no description by Schumacher, A.S. Morse, C. C. Pantelides, and S. Sastry. means of classical equations possible (e.g., subsystems Supported in part by an Alexander von Hamboldt Foundation representable by Petri nets, temporal logic units, etc.); Senior Research Award. Supported in part by a Center of Applied Mathematics Fellowship, and the like (see, e.g., Grossman et al., 1993; Antsaklis University of Notre Dame. et al., 1993; Go¨ llu¨ and Varaiya, 1989; Brockett, 1993; 0005-1098/99/$—see front matter ( 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 1 6 5 - 4 372 A.N. Michel, B. Hu/Automatica 35 (1999) 371—384 Branicky et al., 1994, Branicky, 1995; Pettersson, 1996 ¹"N"+0, 1, 2, 2, we speak of a discrete-time dynam- and Ye, 1996). ical system. Later on, when discussing hybrid dynamical In recent papers (Ye, 1996; Ye et al., 1995a; Ye et al., systems, we will generalize ¹. Every motion of a dynam- 3¹ 1995b; Ye et al., 1996a; Michel and Hou, 1997; Hou and ical system depends on initial data (t, a), where t is Michel, 1997; Ye et al., 1998), we proposed a general called initial time and a3ALX is called initial point, model for hybrid dynamical systems whose states are where X, the state space,isametric space with metric defined on arbitrary metric space and evolve along some d (i.e., (X, d) is a metric space), and A is an appropriate notion of generalized abstract time. For such systems we subset of X. For a given (t, a), we denote a motion,ifit introduced qualitative characterizations which include exists, by p(t, t , a), t3¹ , where ¹ "[t , t )5¹, and R ? R ? the usual notions of Lyapunov and Lagrange stability, where t may be finite or infinite. Thus, a motion is where convergence is defined relative to generalized time. a mapping p( ) , t , a):¹ PX with p(t , t , a)"a, and R ? Next, we showed that it is always possible to transform the family of motions which makes up the dynamical this class of hybrid dynamical systems into another class system is obtained by varying the initial point a over the ¹ of dynamical systems with equivalent qualitative proper- set A and the initial time t over , an appropriate ties, but is defined on real time. The motions of this class subset of ¹ (the set of initial times). If we denote such of systems are in general discontinuous. Furthermore, a family by S, then the dynamical system is signified by +¹ ¹ , ¹"¹ this class of systems may be finite or infinite dimensional, the quintuple , X, A, S, . When , we simply and the motions may be generated by the solutions of write +¹, X, A, S,, and when all is clear from context, we classical equations of the type enumerated above, or by will simply speak of a dynamical system S (rather than +¹ ¹ , ¹" > ‘‘equation free’’ characterizations (e.g., Petri nets, discrete a dynamical system , X, A, S, ). If R and all event systems, temporal logic elements, and the like). For p3S are continuous with respect to t, we speak of a the above discontinuous dynamical systems (and hence, continuous dynamical system. If the elements of S are for the above hybrid dynamical systems), we established not continuous, we speak of a discontinuous dynamical the Principal Lyapunov Stability Results (uniform sta- system (DDS). Most frequently, the system motions are bility, instability, uniform asymptotic stability (in the determined by means of the solutions of initial-value large), and exponential stability (in the large) of invariant problems. When such problems are defined on a finite- sets (resp., equilibria)) as well as Lagrange Stability Re- dimensional space (e.g., X"RL), they will generate sults (uniform boundedness and uniform ultimate boun- a finite-dimensional dynamical system. When X is not dedness of motions). For some of these results, we also finite-dimensional, the system on hand is said to be an established converse theorems. Finally, we demonstrated infinite-dimensional dynamical system. the applicability of these results by means of specific Examples of continuous-time, finite-dimensional classes of hybrid dynamical systems (including digital dynamical systems include systems determined by control systems and systems whose states are subjected ordinary differential equations and ordinary differential to impulses) (see Hou et al., 1997 and Ye et al., 1996b). inequalities; examples of discrete-time, finite-dimensional In the present paper we continue the work described dynamical systems include systems determined by ordi- above. In doing so, we first develop a general comparison nary difference equations and ordinary difference theory for the class of hybrid dynamical systems (resp., inequalities; and examples of infinite-dimensional dy- discontinuous dynamical systems) considered in these namical systems include systems determined by delay works, making use of stability preserving mappings (see differential equations, functional differential equations, Section 2). We then show how these results can be ap- Volterra integro-differential equations, partial differen- plied to establish some of the Principal Lyapunov Sta- tial equations, differential equations and inclusions on bility Theorems (see Section 3). For the latter, we state Banach space, and the like (Zubov 1964; Hahn 1967a; and prove a new converse theorem (see Section 4). To Yoshizawa 1996; Michel and Wang 1995 and Lakshmi- demonstrate the applicability of our results, we consider kantham and Leela 1969). specific examples throughout the paper. In the reminder of this section, we establish a proper 1.2. Hybrid dynamical systems context for what is to follow. Hybrid dynamical systems are capable of exhibiting 1.1. Dynamical systems simultaneously several kinds of dynamic behavior in different parts of the system (e.g., continuous-time dy- Dynamical systems are families of motions determined namics, discrete-time dynamics, logic commands, dis- by evolutionary processes (see, e.g., Zubov, 1964; Hahn, crete events, jump phenomena, and the like). Typical 1967a; Yoshizawa, 1996 and Michel and Wang, 1995). examples of such systems of varying degrees of com- The evolution of such processes takes place over time plexity include sampled-data control
Load more

Recommended publications
  • Solution Bounds, Stability and Attractors' Estimates of Some Nonlinear Time-Varying Systems
    1 Solution Bounds, Stability and Attractors’ Estimates of Some Nonlinear Time-Varying Systems Mark A. Pinsky and Steve Koblik Abstract. Estimation of solution norms and stability for time-dependent nonlinear systems is ubiquitous in numerous applied and control problems. Yet, practically valuable results are rare in this area. This paper develops a novel approach, which bounds the solution norms, derives the corresponding stability criteria, and estimates the trapping/stability regions for a broad class of the corresponding systems. Our inferences rest on deriving a scalar differential inequality for the norms of solutions to the initial systems. Utility of the Lipschitz inequality linearizes the associated auxiliary differential equation and yields both the upper bounds for the norms of solutions and the relevant stability criteria. To refine these inferences, we introduce a nonlinear extension of the Lipschitz inequality, which improves the developed bounds and estimates the stability basins and trapping regions for the corresponding systems. Finally, we conform the theoretical results in representative simulations. Key Words. Comparison principle, Lipschitz condition, stability criteria, trapping/stability regions, solution bounds AMS subject classification. 34A34, 34C11, 34C29, 34C41, 34D08, 34D10, 34D20, 34D45 1. INTRODUCTION. We are going to study a system defined by the equation n (1) xAtxftxFttt , , [0 , ), x , ft ,0 0 where matrix A nn and functions ft:[ , ) nn and F : n are continuous and bounded, and At is also continuously differentiable. It is assumed that the solution, x t, x0 , to the initial value problem, x t0, x 0 x 0 , is uniquely defined for tt0 . Note that the pertained conditions can be found, e.g., in [1] and [2].
    [Show full text]
  • Strange Attractors and Classical Stability Theory
    Nonlinear Dynamics and Systems Theory, 8 (1) (2008) 49–96 Strange Attractors and Classical Stability Theory G. Leonov ∗ St. Petersburg State University Universitetskaya av., 28, Petrodvorets, 198504 St.Petersburg, Russia Received: November 16, 2006; Revised: December 15, 2007 Abstract: Definitions of global attractor, B-attractor and weak attractor are introduced. Relationships between Lyapunov stability, Poincare stability and Zhukovsky stability are considered. Perron effects are discussed. Stability and instability criteria by the first approximation are obtained. Lyapunov direct method in dimension theory is introduced. For the Lorenz system necessary and sufficient conditions of existence of homoclinic trajectories are obtained. Keywords: Attractor, instability, Lyapunov exponent, stability, Poincar´esection, Hausdorff dimension, Lorenz system, homoclinic bifurcation. Mathematics Subject Classification (2000): 34C28, 34D45, 34D20. 1 Introduction In almost any solid survey or book on chaotic dynamics, one encounters notions from classical stability theory such as Lyapunov exponent and characteristic exponent. But the effect of sign inversion in the characteristic exponent during linearization is seldom mentioned. This effect was discovered by Oscar Perron [1], an outstanding German math- ematician. The present survey sets forth Perron’s results and their further development, see [2]–[4]. It is shown that Perron effects may occur on the boundaries of a flow of solutions that is stable by the first approximation. Inside the flow, stability is completely determined by the negativeness of the characteristic exponents of linearized systems. It is often said that the defining property of strange attractors is the sensitivity of their trajectories with respect to the initial data. But how is this property connected with the classical notions of instability? For continuous systems, it was necessary to remember the almost forgotten notion of Zhukovsky instability.
    [Show full text]
  • Chapter 8 Stability Theory
    Chapter 8 Stability theory We discuss properties of solutions of a first order two dimensional system, and stability theory for a special class of linear systems. We denote the independent variable by ‘t’ in place of ‘x’, and x,y denote dependent variables. Let I ⊆ R be an interval, and Ω ⊆ R2 be a domain. Let us consider the system dx = F (t, x, y), dt (8.1) dy = G(t, x, y), dt where the functions are defined on I × Ω, and are locally Lipschitz w.r.t. variable (x, y) ∈ Ω. Definition 8.1 (Autonomous system) A system of ODE having the form (8.1) is called an autonomous system if the functions F (t, x, y) and G(t, x, y) are constant w.r.t. variable t. That is, dx = F (x, y), dt (8.2) dy = G(x, y), dt Definition 8.2 A point (x0, y0) ∈ Ω is said to be a critical point of the autonomous system (8.2) if F (x0, y0) = G(x0, y0) = 0. (8.3) A critical point is also called an equilibrium point, a rest point. Definition 8.3 Let (x(t), y(t)) be a solution of a two-dimensional (planar) autonomous system (8.2). The trace of (x(t), y(t)) as t varies is a curve in the plane. This curve is called trajectory. Remark 8.4 (On solutions of autonomous systems) (i) Two different solutions may represent the same trajectory. For, (1) If (x1(t), y1(t)) defined on an interval J is a solution of the autonomous system (8.2), then the pair of functions (x2(t), y2(t)) defined by (x2(t), y2(t)) := (x1(t − s), y1(t − s)), for t ∈ s + J (8.4) is a solution on interval s + J, for every arbitrary but fixed s ∈ R.
    [Show full text]
  • Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory
    MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 – p.1/17 Stability in the sense of Lyapunov A dynamic system x˙ = f(x) is Lyapunov stable or internally stable about an equilibrium point xeq if state trajectories are confined to a bounded region whenever the initial condition x0 is chosen sufficiently close to xeq. Mathematically, given R> 0 there always exists r > 0 so that if ||x0 − xeq|| <r, then ||x(t) − xeq|| <R for all t> 0. As seen in the figure R defines a desired confinement region, while r defines the neighborhood of xeq where x0 must belong so that x(t) does not exit the confinement region. R r xeq x0 x(t) MCE503 – p.2/17 Stability in the sense of Lyapunov... Note: Lyapunov stability does not require ||x(t)|| to converge to ||xeq||. The stronger definition of asymptotic stability requires that ||x(t)|| → ||xeq|| as t →∞. Input-Output stability (BIBO) does not imply Lyapunov stability. The system can be BIBO stable but have unbounded states that do not cause the output to be unbounded (for example take x1(t) →∞, with y = Cx = [01]x). The definition is difficult to use to test the stability of a given system. Instead, we use Lyapunov’s stability theorem, also called Lyapunov’s direct method. This theorem is only a sufficient condition, however. When the test fails, the results are inconclusive. It’s still the best tool available to evaluate and ensure the stability of nonlinear systems.
    [Show full text]
  • Analysis of ODE Models
    Analysis of ODE Models P. Howard Fall 2009 Contents 1 Overview 1 2 Stability 2 2.1 ThePhasePlane ................................. 5 2.2 Linearization ................................... 10 2.3 LinearizationandtheJacobianMatrix . ...... 15 2.4 BriefReviewofLinearAlgebra . ... 20 2.5 Solving Linear First-Order Systems . ...... 23 2.6 StabilityandEigenvalues . ... 28 2.7 LinearStabilitywithMATLAB . .. 35 2.8 Application to Maximum Sustainable Yield . ....... 37 2.9 Nonlinear Stability Theorems . .... 38 2.10 Solving ODE Systems with matrix exponentiation . ......... 40 2.11 Taylor’s Formula for Functions of Multiple Variables . ............ 44 2.12 ProofofPart1ofPoincare–Perron . ..... 45 3 Uniqueness Theory 47 4 Existence Theory 50 1 Overview In these notes we consider the three main aspects of ODE analysis: (1) existence theory, (2) uniqueness theory, and (3) stability theory. In our study of existence theory we ask the most basic question: are the ODE we write down guaranteed to have solutions? Since in practice we solve most ODE numerically, it’s possible that if there is no theoretical solution the numerical values given will have no genuine relation with the physical system we want to analyze. In our study of uniqueness theory we assume a solution exists and ask if it is the only possible solution. If we are solving an ODE numerically we will only get one solution, and if solutions are not unique it may not be the physically interesting solution. While it’s standard in advanced ODE courses to study existence and uniqueness first and stability 1 later, we will start with stability in these note, because it’s the one most directly linked with the modeling process.
    [Show full text]
  • Combinatorial and Analytical Problems for Fractals and Their Graph Approximations
    UPPSALA DISSERTATIONS IN MATHEMATICS 112 Combinatorial and analytical problems for fractals and their graph approximations Konstantinos Tsougkas Department of Mathematics Uppsala University UPPSALA 2019 Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 15 February 2019 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Professor Ben Hambly (Mathematical Institute, University of Oxford). Abstract Tsougkas, K. 2019. Combinatorial and analytical problems for fractals and their graph approximations. Uppsala Dissertations in Mathematics 112. 37 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2739-8. The recent field of analysis on fractals has been studied under a probabilistic and analytic point of view. In this present work, we will focus on the analytic part developed by Kigami. The fractals we will be studying are finitely ramified self-similar sets, with emphasis on the post-critically finite ones. A prototype of the theory is the Sierpinski gasket. We can approximate the finitely ramified self-similar sets via a sequence of approximating graphs which allows us to use notions from discrete mathematics such as the combinatorial and probabilistic graph Laplacian on finite graphs. Through that approach or via Dirichlet forms, we can define the Laplace operator on the continuous fractal object itself via either a weak definition or as a renormalized limit of the discrete graph Laplacians on the graphs. The aim of this present work is to study the graphs approximating the fractal and determine connections between the Laplace operator on the discrete graphs and the continuous object, the fractal itself.
    [Show full text]
  • Stability Theory for Ordinary Differential Equations 1. Introduction and Preliminaries
    JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 1, June 1988 Stability Theory for Ordinary Differential Equations Sung Kyu Choi, Keon-Hee Lee and Hi Jun Oh ABSTRACT. For a given autonomous system xf = /(a?), we obtain some properties on the location of positive limit sets and investigate some stability concepts which are based upon the existence of Liapunov functions. 1. Introduction and Preliminaries The classical theorem of Liapunov on stability of the origin 鉛 = 0 for a given autonomous system xf = makes use of an auxiliary function V(.x) which has to be positive definite. Also, the time deriva­ tive V'(⑦) of this function, as computed along the solution, has to be negative definite. It is our aim to investigate some stability concepts of solutions for a given autonomous system which are based upon the existence of suitable Liapunov functions. Let U C Rn be an open subset containing 0 G Rn. Consider the autonomous system x' = /(⑦), where xf is the time derivative of the function :r(t), defined by the continuous function f : U — Rn. When / is C1, to every point t()in R and Xq in 17, there corresponds the unique solution xq) of xf = f(x) such that ^(to) = ⑦⑴ For such a solution, let us denote I(xq) = (aq) the interval where it is defined (a, ⑵ may be infinity). Let f : (a, &) -나 R be a continuous function. Then f is decreasing on (a, b) if and only if the upper right-sided Dini derivative D+f(t) = lim sup f〔t + h(-f(t) < o 九 一>o+ h Received by the editors on March 25, 1988.
    [Show full text]
  • Stability and Bifurcation of Dynamical Systems
    STABILITY AND BIFURCATION OF DYNAMICAL SYSTEMS Scope: • To remind basic notions of Dynamical Systems and Stability Theory; • To introduce fundaments of Bifurcation Theory, and establish a link with Stability Theory; • To give an outline of the Center Manifold Method and Normal Form theory. 1 Outline: 1. General definitions 2. Fundaments of Stability Theory 3. Fundaments of Bifurcation Theory 4. Multiple bifurcations from a known path 5. The Center Manifold Method (CMM) 6. The Normal Form Theory (NFT) 2 1. GENERAL DEFINITIONS We give general definitions for a N-dimensional autonomous systems. ••• Equations of motion: x(t )= Fx ( ( t )), x ∈ »N where x are state-variables , { x} the state-space , and F the vector field . ••• Orbits: Let xS (t ) be the solution to equations which satisfies prescribed initial conditions: x S(t )= F ( x S ( t )) 0 xS (0) = x The set of all the values assumed by xS (t ) for t > 0is called an orbit of the dynamical system. Geometrically, an orbit is a curve in the phase-space, originating from x0. The set of all orbits is the phase-portrait or phase-flow . 3 ••• Classifications of orbits: Orbits are classified according to their time-behavior. Equilibrium (or fixed -) point : it is an orbit xS (t ) =: xE independent of time (represented by a point in the phase-space); Periodic orbit : it is an orbit xS (t ) =:xP (t ) such that xP(t+ T ) = x P () t , with T the period (it is a closed curve, called cycle ); Quasi-periodic orbit : it is an orbit xS (t ) =:xQ (t ) such that, given an arbitrary small ε > 0 , there exists a time τ for which xQ(t+τ ) − x Q () t ≤ ε holds for any t; (it is a curve that densely fills a ‘tubular’ space); Non-periodic orbit : orbit xS (t ) with no special properties.
    [Show full text]
  • Arnold: Swimming Against the Tide / Boris Khesin, Serge Tabachnikov, Editors
    ARNOLD: Real Analysis A Comprehensive Course in Analysis, Part 1 Barry Simon Boris A. Khesin Serge L. Tabachnikov Editors http://dx.doi.org/10.1090/mbk/086 ARNOLD: AMERICAN MATHEMATICAL SOCIETY Photograph courtesy of Svetlana Tretyakova Photograph courtesy of Svetlana Vladimir Igorevich Arnold June 12, 1937–June 3, 2010 ARNOLD: Boris A. Khesin Serge L. Tabachnikov Editors AMERICAN MATHEMATICAL SOCIETY Providence, Rhode Island Translation of Chapter 7 “About Vladimir Abramovich Rokhlin” and Chapter 21 “Several Thoughts About Arnold” provided by Valentina Altman. 2010 Mathematics Subject Classification. Primary 01A65; Secondary 01A70, 01A75. For additional information and updates on this book, visit www.ams.org/bookpages/mbk-86 Library of Congress Cataloging-in-Publication Data Arnold: swimming against the tide / Boris Khesin, Serge Tabachnikov, editors. pages cm. ISBN 978-1-4704-1699-7 (alk. paper) 1. Arnold, V. I. (Vladimir Igorevich), 1937–2010. 2. Mathematicians–Russia–Biography. 3. Mathematicians–Soviet Union–Biography. 4. Mathematical analysis. 5. Differential equations. I. Khesin, Boris A. II. Tabachnikov, Serge. QA8.6.A76 2014 510.92–dc23 2014021165 [B] Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are now being handled by Copyright Clearance Center’s RightsLink service.
    [Show full text]
  • An Introduction to Dynamical Systems and Chaos G.C
    An Introduction to Dynamical Systems and Chaos G.C. Layek An Introduction to Dynamical Systems and Chaos 123 G.C. Layek Department of Mathematics The University of Burdwan Burdwan, West Bengal India ISBN 978-81-322-2555-3 ISBN 978-81-322-2556-0 (eBook) DOI 10.1007/978-81-322-2556-0 Library of Congress Control Number: 2015955882 Springer New Delhi Heidelberg New York Dordrecht London © Springer India 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer (India) Pvt. Ltd. is part of Springer Science+Business Media (www.springer.com) Dedicated to my father Late Bijoychand Layek for his great interest in my education.
    [Show full text]
  • The Fractional View of Complexity
    entropy Editorial The Fractional View of Complexity António M. Lopes 1,∗ and J.A. Tenreiro Machado 2 1 UISPA–LAETA/INEGI, Faculty of Engineering, University of Porto, R. Dr. Roberto Frias, 4200-465 Porto, Portugal 2 Institute of Engineering, Polytechnic of Porto, Deptartment of Electrical Engineering, R. Dr. António Bernardino de Almeida, 431, 4249-015 Porto, Portugal; [email protected] * Correspondence: [email protected]; Tel.: +351-220413486 Received: 27 November 2019; Accepted: 11 December 2019; Published: 13 December 2019 Keywords: fractals; fractional calculus; complex systems; dynamics; entropy; information theory Fractal analysis and fractional differential equations have been proven as useful tools for describing the dynamics of complex phenomena characterized by long memory and spatial heterogeneity. There is a general agreement about the relation between the two perspectives, but the formal mathematical arguments supporting their relation are still being developed. The fractional derivative of real order appears as the degree of structural heterogeneity between homogeneous and inhomogeneous domains. A purely real derivative order would imply a system with no characteristic scale, where a given property would hold regardless of the scale of the observations. However, in real-world systems, physical cut-offs may prevent the invariance spreading over all scales and, therefore, complex-order derivatives could yield more realistic models. Information theory addresses the quantification and communication of information. Entropy and complexity are concepts that often emerge in relation to systems composed of many elements that interact with each other, which appear intrinsically difficult to model. This Special Issue focuses on the synergies of fractals or fractional calculus and information theory tools, such as entropy, when modeling complex phenomena in engineering, physics, life, and social sciences.
    [Show full text]
  • Linear Hydrodynamic Stability
    COMMUNICATION Linear Hydrodynamic Stability Y. Charles Li Communicated by Christina Sormani Introduction Hydrodynamic instability and turbulence are ubiquitous in nature, such as the air flow around an airplane of Figure 1. Turbulence was also the subject of paintings by Leonardo da Vinci and Vincent van Gogh (Figure 2). Tur- bulence is the last unsolved problem of classical physics. Mathematically, turbulence is governed by the Navier– Stokes equations, for which even the basic question of well-posedness is not completely solved. Indeed, the global well-posedness of the 3D Navier–Stokes equations is one of the seven Clay millennium prize problems [1]. Hydrodynamic stability goes back to such luminaries as Helmholtz, Kelvin, Rayleigh, O. Reynolds, A. Sommerfeld, Heisenberg, and G. I. Taylor. Many treatises have been written on the subject [5]. Linear hydrodynamic stability theory applies the sim- ple idea of linear approximation of a function through its tangent to function spaces. Hydrodynamicists have been Figure 1. Hydrodynamic instability and turbulence trying to lay down a rigorous mathematical foundation are ubiquitous in nature, such as the turbulent vortex for linear hydrodynamic stability theory ever since its air flow around the tip of an airplane wing. beginning [5]. Recent results on the solution operators of Euler and Navier–Stokes equations [2] [3] [4] imply that the simple idea of tangent line approximation does not apply because the derivative does not exist in the inviscid Linear Approximation in the Simplest Setting case, thus the linear Euler equations fail to provide a linear Let us start with a real-valued function of one variable approximation to inviscid hydrodynamic stability.
    [Show full text]