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Isochron
In the mathematical theory of dynamical systems, an isochron is a set of initial conditions for the system that all lead to the same long-term behaviour. Mathematical isochron An introductory example Consider the ordinary differential equation for a solution y(t) evolving in time: : \frac + \frac = 1 This ordinary differential equation (ODE) needs two initial conditions at, say, time t=0. Denote the initial conditions by y(0)=y_0 and dy/dt(0)=y'_0 where y_0 and y'_0 are some parameters. The following argument shows that the isochrons for this system are here the straight lines y_0+y'_0=\mbox. The general solution of the above ODE is :y=t+A+B\exp(-t) Now, as time increases, t\to\infty, the exponential terms decays very quickly to zero (exponential decay). Thus ''all'' solutions of the ODE quickly approach y\to t+A. That is, ''all'' solutions with the same A have the same long term evolution. The exponential decay of the B\exp(-t) term brings together a host of soluti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
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] |
Ordinary Differential Equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematics), function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with partial differential equation, ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with stochastic differential equations, ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Conditions
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For a system of order ''k'' (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension ''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional coordinate vector), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exponential Decay
A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda Lambda (; uppercase , lowercase ; , ''lám(b)da'') is the eleventh letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoen ...) is a positive rate called the exponential decay constant, disintegration constant, rate constant, or transformation constant: :\frac = -\lambda N(t). The solution to this equation (see #Solution_of_the_differential_equation, derivation below) is: :N(t) = N_0 e^, where is the quantity at time , is the initial quantity, that is, the quantity at time . Measuring rates of decay Mean lifetime If the decaying quantity, ''N''(''t''), is the number of discrete elements in a certain set (mathematics), se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. The term "ordinary" is used in contrast with ''partial'' differential equations (PDEs) which may be with respect to one independent variable, and, less commonly, in contrast with ''stochastic'' differential equations (SDEs) where the progression is random. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where a_0(x),\ldots,a_n(x) and b(x) are arbitrary differentiable functions that do not need to be linear, and y',\ldots, y^ are the successive derivatives of the unknown function y of the variable x. Among ord ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Form (mathematics)
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an object and allows it to be identified in a unique way. The distinction between "canonical" and "normal" forms varies from subfield to subfield. In most fields, a canonical form specifies a ''unique'' representation for every object, while a normal form simply specifies its form, without the requirement of uniqueness. The canonical form of a positive integer in decimal representation is a finite sequence of digits that does not begin with zero. More generally, for a class of objects on which an equivalence relation is defined, a canonical form consists in the choice of a specific object in each class. For example: *Jordan normal form is a canonical form for matrix similarity. *The row echelon form is a canonical form, when one considers as eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
