|
Feigenbaum Function
In the study of dynamical systems the term Feigenbaum function has been used to describe two different function (mathematics), functions introduced by the physicist Mitchell Feigenbaum: * the solution to the Feigenbaum-Cvitanović functional equation; and * the scaling function that described the covers of the attractor of the logistic map Idea Period-doubling route to chaos In the logistic map, we have a function f_r(x) = rx(1-x), and we want to study what happens when we iterate the map many times. The map might fall into a fixed point (mathematics), fixed point, a fixed cycle, or chaos. When the map falls into a stable fixed cycle of length n, we would find that the graph of f_r^n and the graph of x\mapsto x intersects at n points, and the slope of the graph of f_r^n is bounded in (-1, +1) at those intersections. For example, when r=3.0, we have a single intersection, with slope bounded in (-1, +1), indicating that it is a stable single fixed point. As r increases to be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Map In The Chaotic Regime
Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, the inverse of the logistic function ** Logistic distribution, the derivative of the logistic function, a continuous probability distribution, used in probability theory and statistics * Mathematical logic, subfield of mathematics exploring the applications of formal logic to mathematics Other uses * Logistics, the management of resources and their distributions ** Logistic engineering, the scientific study of logistics ** Military logistics, the study of logistics at the service of military units and operations See also *Logic (other) Logic is the study of the principles and criteria of valid inference and demonstration. Logic may also refer to: *Mathematical logic, a branch of mathematics that grew out of symbolic logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presentation Function
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presentations usually require preparation, organization, event planning, writing, use of visual aids, dealing with stress, and answering questions. "The key elements of a presentation consists of presenter, audience, message, reaction and method to deliver speech for organizational success in an effective manner." Presentations are widely used in tertiary work settings such as accountants giving a detailed report of a company's financials or an entrepreneur pitching their venture idea to investors. The term can also be used for a formal or ritualized introduction or offering, as with the presentation of a debutante. Presentations in certain formats are also known as keynote address. Interactive presentations, in which the audience is involved, are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Map
The logistic map is a discrete dynamical system defined by the quadratic difference equation: Equivalently it is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple nonlinear dynamical equations. The map was initially utilized by Edward Lorenz in the 1960s to showcase properties of irregular solutions in climate systems. It was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation written down by Pierre François Verhulst. Other researchers who have contributed to the study of the logistic map include Stanisław Ulam, John von Neumann, Pekka Myrberg, Oleksandr Sharkovsky, Nicholas Metropolis, and Mitchell Feigenbaum. Two introductory examples Dynamical Systems example In the logistic map, x is a variable, and r is a parameter. It is a map in the sense that it map ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product ^ of the discrete two-point space \underline 2 . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The association between numbers and point (geometry), points on the line links elementary arithmetic, arithmetical operations on numbers to geometry, geometric relations between points, and provides a conceptual framework for learning mathematics. In elementary mathematics, the number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the pi, circle constant : Every point of the number line corresponds to a unique real number, and every real number to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Renormalization Group
In theoretical physics, the renormalization group (RG) is a formal apparatus that allows systematic investigation of the changes of a physical system as viewed at different scales. In particle physics, it reflects the changes in the underlying physical laws (codified in a quantum field theory) as the energy (or mass) scale at which physical processes occur varies. A change in scale is called a scale transformation. The renormalization group is intimately related to ''scale invariance'' and ''conformal invariance'', symmetries in which a system appears the same at all scales ( self-similarity), where under the fixed point of the renormalization group flow the field theory is conformally invariant. As the scale varies, it is as if one is decreasing (as RG is a semi-group and doesn't have a well-defined inverse operation) the magnifying power of a notional microscope viewing the system. In so-called renormalizable theories, the system at one scale will generally consist of self- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feigenbaum Constant
In mathematics, specifically bifurcation theory, the Feigenbaum constants and are two mathematical constants which both express ratios in a bifurcation diagram for a non-linear map. They are named after the physicist Mitchell J. Feigenbaum. History Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate. Feigenbaum made this discovery in 1975, and he officially published it in 1978. The first constant The first Feigenbaum constant or simply Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map :x_ = f(x_i), where is a function parameterized by the bifurcation parameter . It is given by the limit: :\delta = \lim_ \frac where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universality (dynamical Systems)
In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the Dynamics (mechanics), dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced by Leo Kadanoff in the 1960s, but a simpler version of the concept was already implicit in the van der Waals equation and in the earlier Landau theory of phase transitions, which did not incorporate scaling correctly. The term is slowly gaining a broader usage in several fields of mathematics, including combinatorics and probability theory, whenever the quantitative features of a structure (such as asymptotic behaviour) can be deduced from a few global parameters appearing in the definition, without requiring knowledge of the details of the system. The renormalization group provides an intuitively appealing, albeit mathematically non-rigoro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Predrag Cvitanović
Predrag Cvitanović (; born April 1, 1946) is a theoretical physicist regarded for his work in nonlinear dynamics, particularly his contributions to periodic orbit theory. Life Cvitanović earned his B.S. from MIT in 1969 and his Ph.D. at Cornell University in 1973. Before joining the physics department at the Georgia Institute of Technology he was the director of the Center for Chaos and Turbulence Studies of the Niels Bohr Institute in Copenhagen. Cvitanović is a member of the Royal Danish Academy of Sciences and Letters, a corresponding member of Croatian Academy of Sciences and Arts, a recipient of the Research Prize of the Danish Physical Society, and a fellow of the American Physical Society. In 2009 Cvitanović was the recipient of the prestigious Alexander von Humboldt Prize for his work in turbulence theory. He currently holds the Glen P. Robinson Chair in Non-Linear Science at the Georgia Institute of Technology. Scientific work Perhaps his best-known work is h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logistic Map Approaching The Period-3 Scaling Limit
Logistic may refer to: Mathematics * Logistic function, a sigmoid function used in many fields ** Logistic map, a recurrence relation that sometimes exhibits chaos ** Logistic regression, a statistical model using the logistic function ** Logit, the inverse of the logistic function ** Logistic distribution, the derivative of the logistic function, a continuous probability distribution, used in probability theory and statistics * Mathematical logic, subfield of mathematics exploring the applications of formal logic to mathematics Other uses * Logistics, the management of resources and their distributions ** Logistic engineering, the scientific study of logistics ** Military logistics, the study of logistics at the service of military units and operations See also *Logic (other) Logic is the study of the principles and criteria of valid inference and demonstration. Logic may also refer to: *Mathematical logic, a branch of mathematics that grew out of symbolic logic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
