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# Topic: Dynamical system

###### In the News (Tue 23 Apr 19)

 Dynamical system - Wikipedia, the free encyclopedia Linear dynamical systems and systems that have two numbers describing a state are examples of dynamical systems where the possible classes of orbits are understood. The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms. Hyperbolic systems are precisely defined dynamical systems that exhibit the properties ascribed to chaotic systems. en.wikipedia.org /wiki/Dynamical_system   (3731 words)

 Measure-preserving dynamical system - Wikipedia, the free encyclopedia In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of ergodic theory. A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. The set of symbolic names with respect to a partition is called the symbolic dynamics of the dynamical system. en.wikipedia.org /wiki/Kolmogorov-Sinai_entropy   (540 words)

 A CELL DYNAMICAL SYSTEM MODEL FOR THUNDERCLOUD ELECTRIFICATION To begin with, a brief summary of latest developments in the modelling of dynamical systems, in particular, the concept of deterministic chaos is presented in the context of modelling the cloud electrification processes. systems which evolve with time are formulated by tradition using Newtonian continuum dynamics where it is assumed that all change is continuous and the evolution equations of dynamical systems are given by a system of partial differential equations representing continuous rates of change. The cell dynamical system model for atmospheric flows enables prediction of the logarithmic spiral profile for wind in the entire ABL and further the value of the Von Karman's constant k is obtained as equal to 0.382 as a natural consequence of environmental mixing during large eddy growth and is in agreement with observations. www.geocities.com /CapeCanaveral/Lab/5833/thunder/THUNDER.html   (4598 words)

 HYLE 6-2 (2000): Biogeochemical Models in the Environmental Sciences Dynamical systems have become the formal paradigm in the ‘discovery of complexity’ across a range of disciplines: Dynamical systems as universal paradigm propelled the diffusion of complexity concepts in the empirical sciences and have become the leading paradigm for both conceptual and numerical models of complex phenomena. In an exo-perspective on the dynamical system, the system collapses to a closed system (Kampis 1994): The system and its boundaries are defined externally and analytically, closing the system towards its environment except for the vector of environmental input (external variables). The dynamical system paradigm remains within the limits of an exo-physically conceived systems theory which is based on conceptually closed systems and which claims that essential, systemic properties arise from the particular configuration of system components. www.hyle.org /journal/issues/6/haag.htm   (9201 words)

 CBofN - Glossary - S Systems that are continuous in time will form a smooth trajectory through this volume, while discrete systems may jump to different locations on subsequent time steps. Strange Attractor An attractor of a dynamical system that is usually fractal in dimension and is indicative of chaos. Systems are generally understood to have an internal state, inputs from an environment, and methods for manipulating the environment or themselves. mitpress.mit.edu /books/FLAOH/cbnhtml/glossary-S.html   (1060 words)

 Dynamics   (Site not responding. Last check: 2007-10-22) A dynamical system is a set of related phenomena that change over time in a deterministic way. This property of dynamical systems is important for a theory of action, and for dynamical approaches to psychology and linguistics, more generally. The behavior of a dynamical system can be qualitatively (as well as quantitatively) different as a function of the value of its parameters. sapir.ling.yale.edu:16080 /ling165/Action/Dynamics/Da.html   (364 words)

 [No title]   (Site not responding. Last check: 2007-10-22) Mathematical dynamical system (MDS): abstract mathematical structure that can be used to describe the change of a real system as an evolution through a series of states (a model of change in a real system). Dynamical Systems Theory: "to understand cognition we must first of all understand the state space evolution of a certain system" (p.556). COGNITIVE SYSTEMS AND THEIR MODELS (18.4) He talks about different interpretations of the "instantiation relation" because his culminating proposal is based on an interpretation different from the "simulation" interpretation that underlies present dynamical systems theory for cognition. www.cnbc.cmu.edu /~noelle/researcher/mind-as-motion/discussion/06.04.97.1453.mmelliss.txt   (1765 words)

 Introduction to Learning Dynamical Systems Dynamical systems are mathematical objects used to model physical phenomena whose state (or instantaneous description) changes over time. Some phenomena appear to be highly stochastic in the sense that the evolution of the system state appears to be governed by influences similar to those governing the role of dice or the decay of radioactive material. In some control problems, we are given a model for the dynamical system so we can predict the future of the system if we are able to observe the current state or accurately estimate the current state from a sequence of observations. www.cs.brown.edu /research/ai/dynamics/tutorial/Documents/DynamicalSystems.html   (1917 words)

 Dynamical Systems A dynamical system is any process that moves or changes in time. While, in the same system, there can also be an exponential chain of errors, for example the butterfly effect, indicating that the causality principle breaks down (12). To simplify the situation, let’s begin by discussing mathematical dynamical systems, a very simple abstraction of the kinds of dynamical systems that arise in nature. www.emayzine.com /infoage/math/math3.htm   (2567 words)

 Dynamical Systems and Action   (Site not responding. Last check: 2007-10-22) However the motion can be described as a virtual point attractor spring system, in which the system state being governed by the dynamical system is the distance between the hand and the target. System state modeled by dynamical system is abstract and defined in terms of the spatial/perceptual properties of real-world tasks, not the properties of individual muscles and joints. Dynamical systems at the level of the individual moving body parts (individual articulators and muscles) which are sapir.ling.yale.edu /ling165/Action/Actb.html   (1054 words)

 Introduction - dynamical systems, vector fields Dynamical systems on the other hand are usually defined analytically, for example, by a set of differential equations. Dynamical systems are a description of the evolution of some (usually inter-dependent) entities within a common system. Basically, it consists of two numbers representing the amount of both species present in the system at a certain time, and a description of the temporal change of these numbers due to the given setting of the system. www.cg.tuwien.ac.at /research/theses/helwig/node6.htm   (681 words)

 Nonlinear Science FAQ   (Site not responding. Last check: 2007-10-22) Dynamical systems are "deterministic" if there is a unique consequent to every state, and "stochastic" or "random" if there is more than one consequent chosen from some probability distribution (the "perfect" coin toss has two consequents with equal probability for each initial state). The new aspects raised by dynamical systems theory are (i) the implied geometric view of temporal behavior and (ii) the existence of "geometric invariants", such as dimension and Lyapunov exponents. System: Requires Windows 95, at least 256 colours Available : http://www.fssc.demon.co.uk/rdiffusion/ilya.htm INSITE (It's a Nonlinear Systems Investigative Toolkit for Everyone) is a collection for the simulation and characterization of dynamical systems, with an emphasis on chaotic systems. www.faqs.org /faqs/sci/nonlinear-faq   (11547 words)

 NASA - Prognostic Tools for Complex Dynamical Systems Modeling systems as networks of coupled nonlinear dynamical systems allows the structure of the coupling between the systems to be used as a diagnostic tool. On line detection of changes in the system behavior and prognostic of trends allows system maintenance to proceed as required– either by on-line controllers which modify the system to counter the effects of the dynamical instabilities, or by replacement of defective subsystems. These changes in couplings can cause the system to switch to a different mode of operation, one that may be at reduced efficiency (e.g., solar panel based power systems have an undesirable mode where only about 30% of rated power is delivered) or may result in a catastrophic failure. www.nasa.gov /lb/centers/ames/research/technology-onepagers/prognostic-tools.html   (918 words)

 [No title] On the other hand the system with a non-chaotic attractor, which occurs at the limit of the period doubling bifurcations (at the edge of chaos), is not of this class. The simplest systems with chaotic limits are the shift map $\sigma$ on $2^{N}$ defined by $\sigma(u)_{i}=u_{i+1}$, which itself is chaotic, and its inverse $\sigma^{-1}_{0}$ defined by $\sigma^{-1}_{0}(u)_{i+1} = u_{i}$,\ $\sigma^{-1}_{0}(u)_{0} = 0$, whose limit is the fixed point 0. This works again in the quadratic family: all systems at the period doubling and band merging bifurcations are factors of finite automata, while the system at their common limit is not. www.ma.utexas.edu /mp_arc/papers/93-78   (3516 words)

 PlanetMath: dynamical system A smooth dynamical system is the same definition as above but with the differentiable map being smooth. A planar dynamical system is the same definition as above but with This is version 3 of dynamical system, born on 2004-01-10, modified 2004-01-12. planetmath.org /encyclopedia/DynamicalSystem.html   (115 words)

 NASA - Critical State Detection for Safe and Affordable Mission Design Recent advances in dynamical system modeling have provided the ability to reduce the state space near a critical region so that reinforcement learning can be effective. In large systems designed to sustain long-term space missions, accurately modeling the dynamics of the system to pinpoint state transitions leading to catastrophic failures is a particularly challenging problem. In other words, whereas describing the normal operation of the system may require thousands of variables, once the system enters a critical state, the system’s evolution can be described by a handful of variables: a reduction in state space. www.nasa.gov /centers/ames/research/technology-onepagers/critical-state-detection.html   (810 words)

 The Math Forum - Math Library - Dynamical Systems Dynamical Systems preprints, from the U.C. Davis front end for the xxx.lanl.gov e-Print archive, a major site for mathematics preprints that has incorporated many formerly independent specialist archives. A database of around 50,000 planar dynamical systems of the type described by the formula: x = f(x,y), y = g(x,y), where f and g are algebraic functions containing constants and the four operations. The system is defined by a set of two differential equations, which are evaluated within the chosen region to form a two-dimensional vector field. mathforum.org /library/topics/dynamical_systems   (2054 words)

 Lesson 4: LINEAR DYNAMICAL SYSTEMS OF ORDER TWO Another example of a second-order, homogeneous, discrete dynamical system occurs in the situation where the number of seeds generated by a plant is one seed in the first year and two seeds in the second and successive years. Recall that a dynamical system has an equilibrium point, or value, e if and only if when the initi al values for the system are set equal to e, all remaining values for the system are also identically equal to e also. Experimentation with other systems will show such convergence and the both slow and rapid divergence for other systems when the roots of their characteristic equations exceed 1 in absolute value. mtl.math.uiuc.edu /modules/discrete/4/part4.html   (3042 words)

 Taylor & Francis Journals: Welcome The primary goal of Dynamical Systems: An International Journal (founded as Dynamics and Stability of Systems) is to act as a forum for communication across all branches of modern dynamical systems, and especially to facilitate interaction between theory and applications. As the remit of the journal is fairly wide, authors are requested to present their work in a way that enables a wide audience to understand the context and motivation of the results in their article. High quality papers describing the application of the modern theory of dynamics to practical problems in other disciplines and reports of experiments or numerical simulations are also welcome, as long as they clearly illustrate important theoretical issues or highlight deficiencies in the theoretical development of dynamical systems. www.tandf.co.uk /journals/titles/14689367.asp   (282 words)

 Glossary of Dynamical Systems Terms A limit cycle is orbitally asymptotically stable if the Floquet multipliers of the linearized system lie inside the unit circle with the exception of a multiplier with value 1. Attractor An attractor is a trajectory of a dynamical system such that initial conditions nearby it will tend toward it in forward time. When referring to a dynamical system, it means that all initial conditions tend to one of the attractors. mrb.niddk.nih.gov /glossary/glossary.html   (1119 words)

 Lexicon From its beginnings ca 1880 with Poincare' until the computer revolution and the advent of computer graphics in the 1970s, dynamical systems theory consisted in the analysis of a given dynamical system: determining its attractors and basins, and their transformations (called bifurcations) as the dynamical rule is gradually changed. After the computer simulation of dynamical systems along with the computer graphic representation of their trajectories was developed by John von Neumann and associates in the 1940s, scientists became interested in the inverse process: given an attractor-basin portrait (that is, experimental data) find a dynamical rule fitting the data. A family of dynamical systems is chosen, for example polynomials of degree three. www.visual-chaos.org /lex.html   (754 words)

 Lesson 2: Linear DDS's To identify discrete dynamical systems that are linear, homogeneous, or affine. are actually general solutions for the respective dynamical systems because we have not specified the initial value c of the systems. The more non-linear a dynamical system is, the more equilibrium values it may have. mtl.math.uiuc.edu /modules/discrete/2/part2.html   (1898 words)

 Introduction: So what's a dynamical system anyway?   (Site not responding. Last check: 2007-10-22) Dynamics is the study of change, and a Dynamical System is just a recipe for saying how a system of variables interacts and changes with time. The different systems may seem to be distinct, but they can often be investigated using the same powerful tools. When we speak of dynamical systems mathematically, we are talking about a system of equations that describe how each variable (e.g. www.ldeo.columbia.edu /~mspieg/Complexity/Problems/node1.html   (303 words)

 Dynamical systems: Simulation and visualization   (Site not responding. Last check: 2007-10-22) The restricted three body problem is a model for the motion of a spacecraft moving under the gravitational influence of two larger bodies which are prescribing circular orbits around each other. Finite dimensional dynamical systems are often used to approximate PDE problems for the evolution of some sort of surface or interface. Here we demonstrate in an mpeg movie (385K) a simulation of the motion of such a surface whose motion is specified by a 2-dimensional dynamical system. www.geom.uiuc.edu /software/dstool   (338 words)

 Dynamical Systems in Mathematical Modeling II   (Site not responding. Last check: 2007-10-22) Thus, the solar system is a dynamical system; the United States economy is a dynamical system; the weather is a dynamical system; the human heart is a dynamical system. Mathematically speaking, a dynamical system is a system whose behavior at a given time depends, in some sense, on its behavior at one or more previous times. Furthermore, it is the task of the mathematical modeler to come up with a mathematical construct, a model, that will describe this relationship between current and past states of the system so that predictions about the future course of events for the system may be made with some degree of accuracy. home.earthlink.net /~srrobin/dynsysb.html   (770 words)

 CBofN - Glossary - D Dissipative System A dynamical system that contains internal friction that deforms the structure of its attractor, thus making motion such as fixed points, limit cycles, quasiperiodicity, and chaos possible. Dissipative systems often have internal structure despite being far from equilibrium, like a whirlpool that preserves its basic form despite being in the midst of constant change. Dynamical System A system that changes over time according to a set of fixed rules that determine how one state of the system moves to another state. mitpress.mit.edu /books/FLAOH/cbnhtml/glossary-D.html   (486 words)

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