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Period Doubling Bifurcation
In dynamical systems theory, a period-doubling bifurcation occurs when a slight change in a system's parameters causes a new periodic trajectory to emerge from an existing periodic trajectory—the new one having double the period of the original. With the doubled period, it takes twice as long (or, in a Discrete time and continuous time, discrete dynamical system, twice as many iterations) for the numerical values visited by the system to repeat themselves. A period-halving bifurcation occurs when a system switches to a new behavior with half the period of the original system. A period-doubling cascade is an infinite sequence of period-doubling bifurcations. Such cascades are one route by which dynamical systems can develop chaos. In hydrodynamics, they are one of the possible routes to turbulence. Examples Logistic map The logistic map is :x_ = r x_n (1 - x_n) where x_n is a function of the (discrete) time n = 0, 1, 2, \ldots. The parameter r is assumed to lie in the interv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems Theory
Dynamical systems theory is an area of mathematics used to describe the behavior of complex systems, complex dynamical systems, usually by employing differential equations by nature of the ergodic theory, ergodicity of dynamic systems. When differential equations are employed, the theory is called continuous time, ''continuous dynamical systems''. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a Principle of least action, least action principle. When difference equations are employed, the theory is called discrete time, ''discrete dynamical systems''. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inflation
In economics, inflation is an increase in the average price of goods and services in terms of money. This increase is measured using a price index, typically a consumer price index (CPI). When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reduction in the purchasing power of money. The opposite of CPI inflation is deflation, a decrease in the general price level of goods and services. The common measure of inflation is the inflation rate, the annualized percentage change in a general price index. Changes in inflation are widely attributed to fluctuations in Real versus nominal value (economics), real demand for goods and services (also known as demand shocks, including changes in fiscal policy, fiscal or monetary policy), changes in available supplies such as during energy crisis, energy crises (also known as supply shocks), or changes in inflation expectations, which may be self-fulfilling. Moderat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sharkovskii's Theorem
In mathematics, Sharkovskii's theorem (also spelled Sharkovsky, Sharkovskiy, Šarkovskii or Sarkovskii), named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period. Statement For some interval I\subset \mathbb, suppose that f : I \to I is a continuous function. The number x is called a ''periodic point of period m'' if f^(x)=x, where f^ denotes the iterated function obtained by composition of m copies of f. The number x is said to have ''least period m'' if, in addition, f^(x)\ne x for all 0 |
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] |
Feigenbaum Constants
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] |
Complex Quadratic Map
A complex quadratic polynomial is a quadratic polynomial whose coefficients and variable are complex numbers. Properties Quadratic polynomials have the following properties, regardless of the form: *It is a unicritical polynomial, i.e. it has one finite critical point in the complex plane, Dynamical plane consist of maximally 2 basins: the basin of infinity and basin of the finite critical point (if the finite critical point does not escape) *It can be postcritically finite, i.e. the orbit of the critical point can be finite, because the critical point is periodic or preperiodic. * It is a unimodal function, * It is a rational function, * It is an entire function. Forms When the quadratic polynomial has only one variable (univariate), one can distinguish its four main forms: * The general form: f(x) = a_2 x^2 + a_1 x + a_0 where a_2 \ne 0 * The factored form used for the logistic map: f_r(x) = r x (1-x) * f_(x) = x^2 +\lambda x which has an indifferent fixed point with m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Chaotic Maps
In mathematics, a chaotic map is a map (mathematics), map (an Discrete-time dynamical system, evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems. Chaotic maps and Iterated function, iterated functions often generate fractals. Some fractals are studied as objects themselves, as set (mathematics), sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems). List of chaotic maps List of fractals * Cantor set * de Rham curve * Gravity set, or Mitchell-Green gravity set * Julia set - derived from complex quadratic map * Koch snowflake - special case of de Rham curve * Lyapunov fractal * Mandelbrot set - derived from complex quadratic ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronic Circuit
An electronic circuit is composed of individual electronic components, such as resistors, transistors, capacitors, inductors and diodes, connected by conductive wires or Conductive trace, traces through which electric current can flow. It is a type of electrical circuit. For a circuit to be referred to as ''electronic'', rather than ''electrical'', generally at least one active component must be present. The combination of components and wires allows various simple and complex operations to be performed: signals can be amplified, computations can be performed, and data can be moved from one place to another. Circuits can be constructed of discrete components connected by individual pieces of wire, but today it is much more common to create interconnections by photolithographic techniques on a laminated Substrate (semiconductor), substrate (a printed circuit board or PCB) and solder the components to these interconnections to create a finished circuit. In an integrated circuit or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mercury (element)
Mercury is a chemical element; it has Symbol (chemistry), symbol Hg and atomic number 80. It is commonly known as quicksilver. A Heavy metal element, heavy, silvery d-block element, mercury is the only metallic element that is known to be liquid at standard temperature and pressure; the only other element that is liquid under these conditions is the halogen bromine, though metals such as caesium, gallium, and rubidium melt just above room temperature. Mercury occurs in deposits throughout the world mostly as cinnabar (mercuric sulfide). The red pigment vermilion is obtained by Mill (grinding), grinding natural cinnabar or synthetic mercuric sulfide. Exposure to mercury and mercury-containing organic compounds is toxic to the nervous system, immune system and kidneys of humans and other animals; mercury poisoning can result from exposure to water-soluble forms of mercury (such as mercuric chloride or methylmercury) either directly or through mechanisms of biomagnification. Mercu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convection Rolls
Horizontal convective rolls, also known as horizontal roll vortices or cloud streets, are long rolls of counter-rotating air that are oriented approximately parallel to the ground in the planetary boundary layer. Although horizontal convective rolls, also known as cloud streets, have been clearly seen in satellite photographs for the last 30 years, their development is poorly understood, due to a lack of observational data. From the ground, they appear as rows of cumulus or cumulus-type clouds aligned parallel to the low-level wind. Research has shown these eddies to be significant to the vertical transport of momentum, heat, moisture, and air pollutants within the boundary layer. Cloud streets are usually more or less straight; rarely, cloud streets assume paisley patterns when the wind driving the clouds encounters an obstacle. Those cloud formations are known as von Kármán vortex streets. Characteristics Horizontal rolls are counter-rotating vortex rolls that are near ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Money Supply
In macroeconomics, money supply (or money stock) refers to the total volume of money held by the public at a particular point in time. There are several ways to define "money", but standard measures usually include currency in circulation (i.e. physical cash) and demand deposits (depositors' easily accessed assets on the books of financial institutions). Money supply data is recorded and published, usually by the national statistical agency or the central bank of the country. Empirical money supply measures are usually named M1, M2, M3, etc., according to how wide a definition of money they embrace. The precise definitions vary from country to country, in part depending on national financial institutional traditions. Even for narrow aggregates like M1, by far the largest part of the money supply consists of deposits in commercial banks, whereas currency (banknotes and coins) issued by central banks only makes up a small part of the total money supply in modern economies. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
