Physics
Dynamic Systems
Dynamic systems in physics refer to systems that change over time due to the interaction of various components or forces. These systems are characterized by their ability to exhibit complex behaviors and patterns, often influenced by feedback loops and non-linear relationships. Understanding dynamic systems is crucial for analyzing and predicting the behavior of physical phenomena, such as oscillations, chaos, and emergent properties.
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7 Key excerpts on "Dynamic Systems"
- Javier Kypuros(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
Chapter 1 Introduction to System Dynamics System Dynamics is the study of physical mechanisms and devices that ex-hibit dynamic behavior that can be characterized through the use of mathe-matical models for the purposes of analysis, design, and automation. It is a discipline that draws on a variety of subjects in order to examine whole mechanisms, devices, and physical phenomena from a “systems” perspec-tive. A system is composed of more basic elements that function together to form the whole. The basic components of that system can be described by various mathematical models or constitutive relations. In order to understand the whole, we must understand how the more basic elements interface and interact. Thus far in your engineering studies, you have learned separately a variety of subjects that are brought to bear together and expanded upon in System Dynamics. Here are a series of questions to consider: B What aspects of the system must you consider? B What tools, models, or information will you need? B How do you design or optimize the system to ensure reasonable per-formance? B What metrics do you use to measure the system’s performance? B How do you automate or control a system? B Where do you get started? 1 2 CHAPTER 1. INTRODUCTION TO SYSTEM DYNAMICS You probably have more questions than answers at this stage. As this course and text proceed, you will evolve an understanding of tools and skills em-ployed in examining Dynamic Systems. Each chapter purposely builds upon prior chapters and looks ahead to preview connections with chapters to come. The book can be broken down into three major parts – Chapters 1-6, Chap-ters 7 and 8, and Chapters 9 and 10. The first part focuses on synthesis of system models and the derivation of differential and algebraic equations that represent the dynamic response. In the second part we explore methods for analyzing Dynamic Systems including metrics to quantify their characteristics.
eBook - PDFStructures in Dynamics
Finite Dimensional Deterministic Studies
- H.W. Broer, F. Dumortier, S.J. van Strien, F. Takens(Authors)
- 1991(Publication Date)
- North Holland(Publisher)
1 Introduction to dynamical systems I H. W. Broer Deterministic time evolutions, and signals or time series derived from these, are well known from physics, biology, chemistry, economics, etc. The theory of dynamical systems provides an encompassing framework for this. We here restrict to systems with a finite dimensional state space, or as it is said in classical mechanics: with 'finitely many degrees of freedom'. Below, with help of several examples, we first develop a general def-inition of 'dynamical system', where continuous- resp. discrete-time systems are just special cases, linked together by the Poincare map-ping and the suspension. Second we distinguish between various types of dynamical behaviour: steady state or equilibrium behaviour, peri-odic behaviour, etc. Also, as a prelude to later chapters, we briefly meet more complicated behaviour, like quasi-periodic and chaotic dynamics. While introducing the corresponding language, we present a problem setting. Roughly speaking our interest is with the asymptotic dynamics for infinite time. To fix thoughts, one may think of evolutions in sys-tems with dissipation, that in the state space have settled down on an attractor. One important feature is the geometry of these attractors. Another point is the dependence of the asymptotic dynamics on the initial value of the evolution. As we shall see, certain types of asymp-totic behaviour can be more or less typical for the system at hand, than others. Finally, it will be of importance how this whole structure depends on perturbations of the system. We'll call a dynamical prop-erty persistent if it is unchanged under small enough perturbations of the system. Here the notion of persistent can be seen as opposed to pathological. As said before, many of these subjects will be explored in later chap-ters of this volume. Also, for background reading we refer to e.g. Arnold [1], Broer and Takens [4] and Ruelle [8,9].
eBook - PDFSystems Science
Methodological Approaches
- Yi Lin, Xiaojun Duan, Chengli Zhao, Li Da Xu(Authors)
- 2012(Publication Date)
- CRC Press(Publisher)
In this process, plans, designs, simulations, analysis of propos-als, etc., all belong to the class of conceptual systems. 2.4.2.3 Static and Dynamic Systems Systems’ statics and dynamics are relative. Roughly speaking, all structures and systems that do not have any moving parts as well as those that are at rest, such as bridges, houses, roads, etc., are con-sidered as static systems. As for Dynamic Systems, it means such entities that contain both static and moving parts. For example, each school is a dynamic system, which contains not only buildings but also teachers and students. Before the middle ages, people once believed that the cosmos is eternal 43 Concepts, Characteristics, and Classifications of Systems and invariant and liked to treat matters as constant and static. Such a world point of view is ideal-istic or mechanic materialistic in philosophy. Along with the advances of science, man gradually recognized that the physical world is not a collection of invariant matters; instead it is a collection of dynamic processes, and only movement is eternal. The cosmos is a dynamic system, in which rest is only a relative concept. 2.4.2.4 Open and Closed Systems A closed system is one that does not have any connection with the outside. No matter how the outside world changes, the closed system maintains its characteristics of equilibrium and internal stability. The chemical reaction that is taking place in a well-sealed container is an example of a closed system; under certain initial conditions, the reaction of the chemicals within the container reaches its equilibrium. Each open system exchanges information, materials, and energies with its environment. For example, business systems, production systems, or ecological systems are all open systems. When the environment undergoes changes, the open system stays in a dynamically stable state through mutual interactions of its parts and the environment and through its own capability of adjustment.
eBook - PDFChaos: From Simple Models To Complex Systems
From Simple Models to Complex Systems
- Angelo Vulpiani, Fabio Cecconi, Massimo Cencini(Authors)
- 2009(Publication Date)
- World Scientific(Publisher)
Electric circuits are described by the currents and voltages of different components which, typically, nonlinearly depend on each other. Therefore, dynamical systems theory encompasses the study of systems from chemistry, socio-economical sciences, engineering, and Newtonian mechanics described by F = m a , i.e. by the ODEs d q d t = p d p d t = F , (2.3) where q and p denote the coordinates and momenta, respectively. If q , p ∈ IR N the phase space, usually denoted by Γ, has dimension d = 2 × N . Equation (2.3) can be rewritten in the form (2.1) identifying x i = q i ; x i + N = p i and f i = p i ; f i + N = F i , for i = 1 , . . . , N . Interesting ODEs may also originate from approximation of more complex systems such as, e.g., the Lorenz (1963) model: d x 1 d t = − σx 1 + σx 2 d x 2 d t = − x 2 − x 1 x 3 + r x 1 d x 3 d t = − bx 3 + x 1 x 2 , where σ, r, b are control parameters, and x i ’s are variables related to the state of fluid in an idealized Rayleigh-B´ enard cell (see Sec. 3.2). 2.1.1 Conservative and dissipative dynamical systems We can identify two general classes of dynamical systems. To introduce them, let’s imagine to have N pendulums as that in Fig. 1.1a and to choose a slightly different initial state for any of them. Now put all representative points in phase space Γ forming an ensemble , i.e. a spot of points, occupying a Γ-volume, whose distribution is described by a probability density function (pdf) ρ ( x , t = 0) normalized in such a way that Γ d x ρ ( x , 0) = 1. How does such a pdf evolve in time? The number of 14 Chaos: From Simple Models to Complex Systems pendulums cannot change so that d N/ d t = 0. The latter result can be expressed via the continuity equation ∂ρ ∂t + d i =1 ∂ ( f i ρ ) ∂x i = 0 , (2.4) where ρ f is the flux of representative points in a volume d x around x .
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 6 Dynamical System and Chaos Theory Dynamical system The Lorenz attractor is an example of a non-linear dynamical system. Studying this system helped give rise to Chaos theory. A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake. ________________________ WORLD TECHNOLOGIES ________________________ At any given time a dynamical system has a state given by a set of real numbers (a vector) which can be represented by a point in an appropriate state space (a geometrical manifold). Small changes in the state of the system correspond to small changes in the numbers. The evolution rule of the dynamical system is a fixed rule that describes what future states follow from the current state. The rule is deterministic; in other words, for a given time interval only one future state follows from the current state. Overview The concept of a dynamical system has its origins in Newtonian mechanics. There, as in other natural sciences and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the state of the system only a short time into the future. (The relation is either a differential equation, difference equation or other time scale.) To determine the state for all future times requires iterating the relation many times—each advancing time a small step. The iteration procedure is referred to as solving the system or integrating the system . Once the system can be solved, given an initial point it is possible to determine all its future points, a collection known as a trajectory or orbit .
eBook - PDF- Jean-marc Ginoux(Author)
- 2009(Publication Date)
- World Scientific(Publisher)
D. Birkhoff (1912, p. 306) of his memoir originally presented in 1909 at a meeting of the American Mathematical Society and entitled: Quelques Th´ eor` emes sur le mouvement des syst` emes dynamiques . “A dynamical system in a very large meaning may be considered as being defined by any system of differential equations of the first order: dx 1 X 1 = dx 2 X 2 = · · · = dx n X n = dt where X 1 , . . . , X n are given functions, real and uniform depending on x 1 , . . . , x n , analytical with respect of these variables, and where t is the in-dependent variable. Variables x 1 , . . . , x n are the coordinates of the motion and t indicates the time.” This is exactly the same definition as previously proposed by Henri Poincar´ e (1886, p. 168) in one of his famous memoirs entitled: Sur les courbes d´ efinies par une ´ equation diff´ erentielle . “. . . any differential equation can be written as: dx 1 dt = X 1 , dx 2 dt = X 2 , . . . , dx n dt = X n where X i are real polynomials. If t is considered as the time, these equations will define the motion of a variable point in a space of dimension n .” Chapter 2 Dynamical Systems “One would have to have completely forgotten the history of science so as to not remember that the desire to know nature has had the most constant and the happiest influence on the development of mathematics.” — H. Poincar´ e — The notion of dynamical system is the mathematical description of the dynamics of a given physical, mechanical, electronic, biological, ecologi-cal, economical system from the point of view of a deterministic process which is expressed in terms of state variables , making it possible to de-fine the instantaneous state of the system, and equations of evolution of these variables between an initial and final instant. According to whether these instants are separated by a finite or infinitesimal time, equations of evolution are mapping iterations (Collet and Eckman, 1980) or differential equations (Hirsch and Smale, 1974).
eBook - PDFFoundations Of Complex Systems: Emergence, Information And Prediction (2nd Edition)
Emergence, Information and Prediction
- Gregoire Nicolis, Catherine Nicolis(Authors)
- 2012(Publication Date)
- World Scientific(Publisher)
In the discussion above it was understood that the control parameters λ are time independent and that the system is not subjected to time-dependent external forcings. Such autonomous dynamical systems constitute the core of nonlinear dynamics. They serve as a reference for identifying the different types of complex behaviors and for developing the appropriate methodologies. Accordingly, in this chapter we will focus entirely on this class of systems. Non-autonomous systems, subjected to random perturbations of intrinsic or environmental origin will be considered in Chapters 3, 4 and onwards. The case of time-dependent control parameters will be briefly discussed in Sec. 6.4.3. 2.1.1 Conservative systems Consider a continuum of initial states, enclosed within a certain phase space region ΔΓ 0 . As the evolution is switched on, each of these states will be the point from which will emanate a phase trajectory. We collect the points reached on these trajectories at time t and focus on the region ΔΓ t that they constitute. We define a conservative system by the property that ΔΓ t will keep the same volume as ΔΓ 0 in the course of the evolution, | ΔΓ t | = | ΔΓ 0 | although it may end up having a quite different shape and location in Γ compared to ΔΓ 0 . It can be shown that this property entails that the phase trajectories are located on phase space manifolds which constitute a con-tinuum, the particular manifold enclosing a given trajectory being specified uniquely by the initial conditions imposed on x 1 , ..., x n . We refer to these manifolds as invariant sets . A simple example of conservative dynamical system is the frictionless pendulum. The corresponding phase space is two-dimensional and is spanned by the particle’s position and instantaneous velocity.
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