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Brusselator Oscillations HQ Render
The Brusselator is a theoretical model for a type of autocatalytic reaction. The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles. It is a portmanteau of Brussels and oscillator. It is characterized by the reactions : A \rightarrow X : 2X + Y \rightarrow 3X : B + X \rightarrow Y + D : X \rightarrow E Under conditions where A and B are in vast excess and can thus be modeled at constant concentration, the rate equations become :\left\ = \left\ + \left\^2 \left\ - \left\ \left\ - \left\ \, :\left\ = \left\ \left\ - \left\^2 \left\ \, where, for convenience, the rate constants have been set to 1. The Brusselator has a fixed point at :\left\ = A \, :\left\ = \,. The fixed point becomes unstable when : B>1+A^2 \, leading to an oscillation of the system. Unlike the Lotka–Volterra equation, the oscillations of the Brusselator do not depend on the amount of reactant present initial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such behavior is exhibited in some nonlinear systems. Limit cycles have been used to model the behavior of a great many real-world oscillatory systems. The study of limit cycles was initiated by Henri Poincaré (1854–1912). Definition We consider a two-dimensional dynamical system of the form x'(t)=V(x(t)) where V : \R^2 \to \R^2 is a smooth function. A ''trajectory'' of this system is some smooth function x(t) with values in \mathbb^2 which satisfies this differential equation. Such a trajectory is called ''closed'' (or ''periodic'') if it is not constant but returns to its starting point, i.e. if there exists some t_0>0 such that x(t + t_0) = x(t) for all t \in \R. An orbit is the image of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaotic Maps
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, Cartoon Network and Disney XD. It was brought over to the United States from Denmark by Bryan C. Gannon and Chaotic USA Entertainment Group, and produced by Chaotic USA Entertainment Group, 4Kids Productions and Bardel Entertainment. The trading card game came out 6 September 2006 in the U.S. and Canada. Each card comes with a unique code which the owner can upload onto the Chaotic website. This allows the owner to trade and play online using their own card collection. The game was well known to be the only game with a TV show, an online game, and a TCG that were all integrated. However, the online game is currently closed. History Chaotic started out as a trading card game known as "Grolls and Gorks" and an idea for a cartoon se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-equilibrium Thermodynamics
Non-equilibrium thermodynamics is a branch of thermodynamics that deals with physical systems that are not in thermodynamic equilibrium but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with transport processes and with the rates of chemical reactions. Almost all systems found in nature are not in thermodynamic equilibrium, for they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems and to chemical reactions. Some systems and processes are, however, in a useful sense, near enough to thermodynamic equilibrium to allow description with useful accuracy by currently known non-equilibrium thermodynamics. Nevertheless, many natural systems and processes will always remain far beyond the scope o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oregonator
The Oregonator is a theoretical model for a type of autocatalytic reaction. The Oregonator is the simplest realistic model of the chemical dynamics of the oscillatory Belousov–Zhabotinsky reaction. It was created by Richard Field and Richard M. Noyes at the University of Oregon. It is a portmanteau of Oregon and oscillator. Earlier, the Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université libre de Bruxelles, Free University of Brussels as a portmanteau of Brussels and oscillator. The Oregonator is a reduced model of the FKN mechanism (developed by Richard Field, Endre Kőrös, and Richard M. Noyes) which still involved 11 reactions and 12 species (21 intermediate species and 18 elementary steps). The Oregonator is characterized by the reactions : \begin \text \quad & A+Y \longrightarrow X+P \\[5pt] \text \quad & X+Y \longrightarrow 2P \\[5pt] \text \quad & A+X \longrightarrow 2X+2Z \\[5pt] \text \quad & 2X \longrightarrow A+P \\[5pt] \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Belousov–Zhabotinsky Reaction
A Belousov–Zhabotinsky reaction, or BZ reaction, is one of a class of reactions that serve as a classical example of non-equilibrium thermodynamics, resulting in the establishment of a nonlinear chemical oscillator. The only common element in these oscillators is the inclusion of bromine and an acid. The reactions are important to theoretical chemistry in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior. These reactions are far from equilibrium and remain so for a significant length of time and evolve chaotically. In this sense, they provide an interesting chemical model of nonequilibrium biological phenomena; as such, mathematical models and simulations of the BZ reactions themselves are of theoretical interest, showing phenomenon as noise-induced order. An essential aspect of the BZ reaction is its so called "excitability"; under the influence of stimuli, patterns develop in what would otherwise be a perfectly quies ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clock Reaction
A chemical clock (or clock reaction) is a complex mixture of reacting chemical compounds in which the onset of an observable property (discoloration or coloration) occurs after a predictable induction time due to the presence of clock species at a detectable amount. In cases where one of the reagents has a visible color, crossing a concentration threshold can lead to an abrupt color change after a reproducible time lapse. Types Clock reactions may be classified into three or four types: Substrate-depletive clock reaction The simplest clock reaction featuring two reactions: :A → C (rate k1) :B + C → products (rate k2, fast) When substrate (B) is present, the clock species (C) is quickly consumed in the second reaction. Only when substrate B is all used up or depleted, species C can build up in amount causing the color to change. An example for this clock reaction is the sulfite/iodate reaction or iodine clock reaction, also known as Landolt's reaction. Sometimes, a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brusselator Space
The Brusselator is a theoretical model for a type of autocatalytic reaction. The Brusselator model was proposed by Ilya Prigogine and his collaborators at the Université Libre de Bruxelles. It is a portmanteau of Brussels and oscillator. It is characterized by the reactions : A \rightarrow X : 2X + Y \rightarrow 3X : B + X \rightarrow Y + D : X \rightarrow E Under conditions where A and B are in vast excess and can thus be modeled at constant concentration, the rate equations become :\left\ = \left\ + \left\^2 \left\ - \left\ \left\ - \left\ \, :\left\ = \left\ \left\ - \left\^2 \left\ \, where, for convenience, the rate constants have been set to 1. The Brusselator has a fixed point at :\left\ = A \, :\left\ = \,. The fixed point becomes unstable when : B>1+A^2 \, leading to an oscillation of the system. Unlike the Lotka–Volterra equation, the oscillations of the Brusselator do not depend on the amount of reactant present initial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscillator
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy. The term ''vibration'' is precisely used to describe a mechanical oscillation. Oscillation, especially rapid oscillation, may be an undesirable phenomenon in pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brussels
Brussels (french: Bruxelles or ; nl, Brussel ), officially the Brussels-Capital Region (All text and all but one graphic show the English name as Brussels-Capital Region.) (french: link=no, Région de Bruxelles-Capitale; nl, link=no, Brussels Hoofdstedelijk Gewest), is a region of Belgium comprising 19 municipalities, including the City of Brussels, which is the capital of Belgium. The Brussels-Capital Region is located in the central portion of the country and is a part of both the French Community of Belgium and the Flemish Community, but is separate from the Flemish Region (within which it forms an enclave) and the Walloon Region. Brussels is the most densely populated region in Belgium, and although it has the highest GDP per capita, it has the lowest available income per household. The Brussels Region covers , a relatively small area compared to the two other regions, and has a population of over 1.2 million. The five times larger metropolitan area of Brusse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
