Brusselator

Last updated • 1 min readFrom Wikipedia, The Free Encyclopedia
Top: The Brusselator in the unstable regime (A=1, B=3): The system approaches a limit cycle Bottom: The Brusselator in a stable regime with A=1 and B=1.7: For B<1+A the system is stable and approaches a fixed point. Bruesselator.svg
Top: The Brusselator in the unstable regime (A=1, B=3): The system approaches a limit cycle Bottom: The Brusselator in a stable regime with A=1 and B=1.7: For B<1+A the system is stable and approaches a fixed point.
Simulation of the Brusselator as reaction diffusion system in two spatial dimensions Brusselator space.gif
Simulation of the Brusselator as reaction diffusion system in two spatial dimensions
Simulation of the reaction-diffusion system of the Brusselator with reflective border conditions Brusselator Oscillations HQ Render.gif
Simulation of the reaction-diffusion system of the Brusselator with reflective border conditions

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. [2] [3]

It is a portmanteau of Brussels and oscillator.

It is characterized by the reactions

Under conditions where A and B are in vast excess and can thus be modeled at constant concentration, the rate equations become

where, for convenience, the rate constants have been set to 1.

The Brusselator has a fixed point at

.

The fixed point becomes unstable when

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 initially. Instead, after sufficient time, the oscillations approach a limit cycle. [4]

The best-known example is the clock reaction, the Belousov–Zhabotinsky reaction (BZ reaction). It can be created with a mixture of potassium bromate (KBrO3), malonic acid (CH2(COOH)2), and manganese sulfate (MnSO4) prepared in a heated solution of sulfuric acid (H2SO4). [5]

See also

Related Research Articles

<span class="mw-page-title-main">Fick's laws of diffusion</span> Mathematical descriptions of molecular diffusion

Fick's laws of diffusion describe diffusion and were first posited by Adolf Fick in 1855 on the basis of largely experimental results. They can be used to solve for the diffusion coefficient, D. Fick's first law can be used to derive his second law which in turn is identical to the diffusion equation.

<span class="mw-page-title-main">Oscillation</span> Repetitive variation of some measure about a central value

Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value 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.

<span class="mw-page-title-main">Aircraft flight dynamics</span> Science of air vehicle orientation and control in three dimensions

Flight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as pitch, roll and yaw. These are collectively known as aircraft attitude, often principally relative to the atmospheric frame in normal flight, but also relative to terrain during takeoff or landing, or when operating at low elevation. The concept of attitude is not specific to fixed-wing aircraft, but also extends to rotary aircraft such as helicopters, and dirigibles, where the flight dynamics involved in establishing and controlling attitude are entirely different.

A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dissipative systems stand in contrast to conservative systems.

In chemistry, a chemical reaction is said to be autocatalytic if one of the reaction products is also a catalyst for the same reaction. Many forms of autocatalysis are recognized.

In chemistry, the law of mass action is the proposition that the rate of a chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dynamic equilibrium. Specifically, it implies that for a chemical reaction mixture that is in equilibrium, the ratio between the concentration of reactants and products is constant.

The Lotka–Volterra equations, also known as the Lotka–Volterra predator–prey model, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems.

In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. Unlike the quasispecies equation, the replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies.

<span class="mw-page-title-main">Metapopulation</span> Group of separated yet interacting ecological populations

A metapopulation consists of a group of spatially separated populations of the same species which interact at some level. The term metapopulation was coined by Richard Levins in 1969 to describe a model of population dynamics of insect pests in agricultural fields, but the idea has been most broadly applied to species in naturally or artificially fragmented habitats. In Levins' own words, it consists of "a population of populations".

The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to the generalized Lotka–Volterra equation to include trophic interactions.

In chemistry, the rate equation is an empirical differential mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters only. For many reactions, the initial rate is given by a power law such as

In the physical sciences, relaxation usually means the return of a perturbed system into equilibrium. Each relaxation process can be categorized by a relaxation time τ. The simplest theoretical description of relaxation as function of time t is an exponential law exp(−t/τ).

In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind.

In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc. (any pair of variables). It is a two-dimensional case of the general n-dimensional phase space.

A multi-compartment model is a type of mathematical model used for describing the way materials or energies are transmitted among the compartments of a system. Sometimes, the physical system that we try to model in equations is too complex, so it is much easier to discretize the problem and reduce the number of parameters. Each compartment is assumed to be a homogeneous entity within which the entities being modeled are equivalent. A multi-compartment model is classified as a lumped parameters model. Similar to more general mathematical models, multi-compartment models can treat variables as continuous, such as a differential equation, or as discrete, such as a Markov chain. Depending on the system being modeled, they can be treated as stochastic or deterministic.

In mathematical biology, the community matrix is the linearization of a generalized Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point.

The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator–prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their dynamics can be analysed analytically to some extent. This makes them useful as a theoretical tool for modeling food webs. However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth.

<span class="mw-page-title-main">Chemical oscillator</span> Reacting chemical mixture in which the concentrations change periodically

In chemistry, a chemical oscillator is a complex mixture of reacting chemical compounds in which the concentration of one or more components exhibits periodic changes. They are a class of reactions that serve as an example of non-equilibrium thermodynamics with far-from-equilibrium behavior. The reactions are theoretically important in that they show that chemical reactions do not have to be dominated by equilibrium thermodynamic behavior.

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.

References

  1. Lukas Wittmann. "BACHELOR'S THESIS - DEVELOPMENT OF A PYTHON PROGRAM FOR INVESTIGATION OF REACTION-DIFFUSION COUPLING IN OSCILLATING REACTIONS".
  2. "IDEA - Internet Differential Equations Activities". Washington State University. Archived from the original on 2017-09-09. Retrieved 2010-05-16.
  3. Prigogine, I.; Lefever, R. (1968-02-15). "Symmetry Breaking Instabilities in Dissipative Systems. II". The Journal of Chemical Physics. 48 (4): 1695–1700. doi:10.1063/1.1668896. ISSN   0021-9606.
  4. http://www.bibliotecapleyades.net/archivos_pdf/brusselator.pdf Dynamics of the Brusselator
  5. BZ reaction Archived December 31, 2012, at the Wayback Machine