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In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value[1]:pp. 160, 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 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.

在数学领域中(特别是在某些动态系统中),初始条件,指的是某个演化变量在规定初始时间上的某一点(通常表示为 [math]\displaystyle{ t = 0 }[/math])的值,在一些情境中也叫做种子值。对于 [math]\displaystyle{ k }[/math]阶系统(即离散时间中的时滞个数,或连续时间中最大导数的阶数)和 [math]\displaystyle{ n }[/math]维系统(即 [math]\displaystyle{ n }[/math]个不同的演化变量,可以用一个 [math]\displaystyle{ n }[/math]维坐标向量表示) 来说,通常需要 [math]\displaystyle{ nk }[/math]作为初始条件来跟踪系统的变量。


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 problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.

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 problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.


在连续时间的微分方程和离散时间的差分方程中,初始条件影响未来任何时间的动态变量(状态变量)的值。在连续时间中,将状态变量作为时间函数,求其与初始条件的闭合形式解,此问题称为初值问题。对于离散时间来说,也存在类似的问题。虽然这种封闭形式解并不能总是求得,但离散时间系统下,未来值还是可以通过不停迭代时间周期来求得,(但这样做,过程中可能会出现舍入误差,从而使求得的值与应求得的值相去甚远。)




Linear system

Discrete time

A linear matrix difference equation of the homogeneous (having no constant term) form [math]\displaystyle{ X_{t+1}=AX_t }[/math] has closed form solution [math]\displaystyle{ X_t=A^tX_0 }[/math] predicated on the vector [math]\displaystyle{ X_0 }[/math] of initial conditions on the individual variables that are stacked into the vector; [math]\displaystyle{ X_0 }[/math] is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable X; that behavior is stable or unstable based on the eigenvalues of the matrix A but not based on the initial conditions.

A linear matrix difference equation of the homogeneous (having no constant term) form [math]\displaystyle{ X_{t+1}=AX_t }[/math] has closed form solution [math]\displaystyle{ X_t=A^tX_0 }[/math] predicated on the vector [math]\displaystyle{ X_0 }[/math] of initial conditions on the individual variables that are stacked into the vector; [math]\displaystyle{ X_0 }[/math] is called the vector of initial conditions or simply the initial condition, and contains nk pieces of information, n being the dimension of the vector X and k = 1 being the number of time lags in the system. The initial conditions in this linear system do not affect the qualitative nature of the future behavior of the state variable X; that behavior is stable or unstable based on the eigenvalues of the matrix A but not based on the initial conditions.

一个齐次(无常数项)的线性矩阵差分方程式 < math > x _ t + 1} = AX _ t </math > ,其闭式解 < math > x _ t = a ^ tX _ 0 </math > 取向量 < math > x _ 0 </math > 叠加到向量中的个别变量的初始条件的向量; < math > x _ 0 </math > 被称为初始条件的向量或简单的初始条件,其中包含 nk 信息片段,n 是向量 x 的维数,k = 1是系统中时间滞后的数目。这个线性系统的初始条件并不影响状态变量 x 未来行为的定性本质; 这种行为是稳定的或不稳定的,基于矩阵 a 的特征值,但不是基于初始条件。


Alternatively, a dynamic process in a single variable x having multiple time lags is

Alternatively, a dynamic process in a single variable x having multiple time lags is

另外,一个动态过程在一个单一的变量 x 具有多个时间滞后是


[math]\displaystyle{ x_t=a_1x_{t-1} +a_2x_{t-2}+\cdots +a_kx_{t-k}. }[/math]

[math]\displaystyle{ x_t=a_1x_{t-1} +a_2x_{t-2}+\cdots +a_kx_{t-k}. }[/math]

< math > x _ t = a _ 1x _ { t-1} + a _ 2x _ { t-2} + cdots + a _ kx _ { t-k } . </math >


Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its characteristic equation [math]\displaystyle{ \lambda^k-a_1\lambda^{k-1} -a_2\lambda^{k-2}-\cdots -a_{k-1}\lambda-a_k=0 }[/math] to obtain the latter's k solutions, which are the characteristic values [math]\displaystyle{ \lambda_1, \dots , \lambda_k, }[/math] for use in the solution equation

Here the dimension is n = 1 and the order is k, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, is nk = k. Again the initial conditions do not affect the qualitative nature of the variable's long-term evolution. The solution of this equation is found by using its characteristic equation [math]\displaystyle{ \lambda^k-a_1\lambda^{k-1} -a_2\lambda^{k-2}-\cdots -a_{k-1}\lambda-a_k=0 }[/math] to obtain the latter's k solutions, which are the characteristic values [math]\displaystyle{ \lambda_1, \dots , \lambda_k, }[/math] for use in the solution equation

这里的维数是 n = 1,顺序是 k,所以必要数目的初始条件跟踪系统的时间,无论是迭代或通过封闭形式的解,是 nk = k。这个方程的解是用它的特征方程 < math > lambda ^ k-a _ 1 lambda ^ { k-1}-a _ 2 lambda ^ { k-2}-cdots-a _ { k-1} lambda-a _ k = 0 </math > 求出后者的 k 解,即用于解方程的特征值 < math > λ _ 1,dots,da _ k,</math >


[math]\displaystyle{ x_t=c_1\lambda _1^t+\cdots + c_k\lambda _k^t. }[/math]

[math]\displaystyle{ x_t=c_1\lambda _1^t+\cdots + c_k\lambda _k^t. }[/math]

< math > x _ t = c _ 1 lambda _ 1 ^ t + cdots + c _ k lambda _ k ^ t </math >


Here the constants [math]\displaystyle{ c_1, \dots , c_k }[/math] are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition [math]\displaystyle{ x_t }[/math] Is known.

Here the constants [math]\displaystyle{ c_1, \dots , c_k }[/math] are found by solving a system of k different equations based on this equation, each using one of k different values of t for which the specific initial condition [math]\displaystyle{ x_t }[/math] Is known.

在这里,常数 c _ 1,点,c _ k </math > 是通过基于这个方程的 k 个不同的方程组求得的,每个方程组使用 t 的 k 个不同的值中的一个,其特定的初始条件 < math > x _ t </math > 是已知的。


Continuous time

A differential equation system of the first order with n variables stacked in a vector X is

A differential equation system of the first order with n variables stacked in a vector X is

一个向量 x 中包含 n 个变量的一阶微分方程系统是


[math]\displaystyle{ \frac{dX}{dt}=AX. }[/math]

[math]\displaystyle{ \frac{dX}{dt}=AX. }[/math]

[ math > frac { dX }{ dt } = AX


Its behavior through time can be traced with a closed form solution conditional on an initial condition vector [math]\displaystyle{ X_0 }[/math]. The number of required initial pieces of information is the dimension n of the system times the order k = 1 of the system, or n. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.

Its behavior through time can be traced with a closed form solution conditional on an initial condition vector [math]\displaystyle{ X_0 }[/math]. The number of required initial pieces of information is the dimension n of the system times the order k = 1 of the system, or n. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.

它在一段时间内的行为可以用一个以初始条件向量 < math > x _ 0 </math > 为条件的封闭形式的解来追踪。所需的初始信息的数量是系统的维数 n 乘以系统的次序 k = 1,或 n。初始条件不影响系统的定性行为(稳定或不稳定)。


A single kth order linear equation in a single variable x is

A single kth order linear equation in a single variable x is

单变量 x 中的单个 k < sup > th 阶线性方程为


[math]\displaystyle{ \frac{d^{k}x}{dt^k}+a_{k-1}\frac{d^{k-1}x}{dt^{k-1}}+\cdots +a_1\frac{dx}{dt} +a_0x=0. }[/math]

[math]\displaystyle{ \frac{d^{k}x}{dt^k}+a_{k-1}\frac{d^{k-1}x}{dt^{k-1}}+\cdots +a_1\frac{dx}{dt} +a_0x=0. }[/math]

< math > frac { d ^ { k } x }{ dt ^ k } + a _ { k-1} frac { d ^ { k-1} x }{ dt ^ { k-1} + cdots + a _ 1 frac { dx }{ dt } + a _ 0x = 0. </math >


Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In this case the k initial pieces of information will typically not be different values of the variable x at different points in time, but rather the values of x and its first k – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The characteristic equation of this dynamic equation is [math]\displaystyle{ \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, }[/math] whose solutions are the characteristic values [math]\displaystyle{ \lambda_1,\dots , \lambda_k; }[/math] these are used in the solution equation

Here the number of initial conditions necessary for obtaining a closed form solution is the dimension n = 1 times the order k, or simply k. In this case the k initial pieces of information will typically not be different values of the variable x at different points in time, but rather the values of x and its first k – 1 derivatives, all at some point in time such as time zero. The initial conditions do not affect the qualitative nature of the system's behavior. The characteristic equation of this dynamic equation is [math]\displaystyle{ \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, }[/math] whose solutions are the characteristic values [math]\displaystyle{ \lambda_1,\dots , \lambda_k; }[/math] these are used in the solution equation

在这里,获得闭式解所需的初始条件的个数是维数 n = 1乘以顺序 k,或简单地 k。在这种情况下,k 的初始信息通常不是变量 x 在不同时间点的不同值,而是 x 及其第一个 k-1导数的值,所有这些都在时间的某一点,比如时间零点。初始条件不影响系统行为的定性本质。这个动力学方程的特征方程是 < math > lambda ^ k + a _ { k-1} lambda ^ { k-1} + cdots + a _ 1 lambda + a _ 0 = 0,</math > 其解是特征值 < math > lambda _ 1,dots,lambda _ k; </math > 这些用于解方程


[math]\displaystyle{ x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. }[/math]

[math]\displaystyle{ x(t)=c_1e^{\lambda_1t}+\cdots + c_ke^{\lambda_kt}. }[/math]

< math > x (t) = c _ 1e ^ { lambda _ 1t } + c _ ke ^ { lambda _ kt } </math >


This equation and its first k – 1derivatives form a system of k equations that can be solved for the k parameters [math]\displaystyle{ c_1, \dots , c_k, }[/math] given the known initial conditions on x and its k – 1 derivatives' values at some time t.

This equation and its first k – 1derivatives form a system of k equations that can be solved for the k parameters [math]\displaystyle{ c_1, \dots , c_k, }[/math] given the known initial conditions on x and its k – 1 derivatives' values at some time t.

这个方程和它的第一个 k-1导数组成了一个 k 方程组,可以用 k 参数来求解。


Nonlinear systems

Nonlinear systems can exhibit a substantially richer variety of behavior than linear systems can. In particular, the initial conditions can affect whether the system diverges to infinity or whether it converges to one or another attractor of the system. Each attractor, a (possibly disconnected) region of values that some dynamic paths approach but never leave, has a (possibly disconnected) basin of attraction such that state variables with initial conditions in that basin (and nowhere else) will evolve toward that attractor. Even nearby initial conditions could be in basins of attraction of different attractors (see for example Newton's method#Basins of attraction).

Nonlinear systems can exhibit a substantially richer variety of behavior than linear systems can. In particular, the initial conditions can affect whether the system diverges to infinity or whether it converges to one or another attractor of the system. Each attractor, a (possibly disconnected) region of values that some dynamic paths approach but never leave, has a (possibly disconnected) basin of attraction such that state variables with initial conditions in that basin (and nowhere else) will evolve toward that attractor. Even nearby initial conditions could be in basins of attraction of different attractors (see for example Newton's method#Basins of attraction).

与线性系统相比,非线性系统可以表现出更多种类的行为。特别地,初始条件可以影响系统是否发散到无穷远,或者是否收敛到系统的一个或另一个吸引子。每个吸引子,一个(可能不连续的)价值区域,一些动态路径接近但永远不会离开,有一个(可能不连续的)吸引盆,这样的状态变量的初始条件在该盆地(没有其他地方)将向该吸引子演化。甚至附近的初始条件也可能是不同吸引子的吸引盆(例如牛顿方法 # 吸引盆)。


Moreover, in those nonlinear systems showing chaotic behavior, the evolution of the variables exhibits sensitive dependence on initial conditions: the iterated values of any two very nearby points on the same strange attractor, while each remaining on the attractor, will diverge from each other over time. Thus even on a single attractor the precise values of the initial conditions make a substantial difference for the future positions of the iterates. This feature makes accurate simulation of future values difficult, and impossible over long horizons, because stating the initial conditions with exact precision is seldom possible and because rounding error is inevitable after even only a few iterations from an exact initial condition.

Moreover, in those nonlinear systems showing chaotic behavior, the evolution of the variables exhibits sensitive dependence on initial conditions: the iterated values of any two very nearby points on the same strange attractor, while each remaining on the attractor, will diverge from each other over time. Thus even on a single attractor the precise values of the initial conditions make a substantial difference for the future positions of the iterates. This feature makes accurate simulation of future values difficult, and impossible over long horizons, because stating the initial conditions with exact precision is seldom possible and because rounding error is inevitable after even only a few iterations from an exact initial condition.

此外,在具有混沌行为的非线性系统中,变量的演化对初始条件具有敏感的依赖性: 同一个奇异吸引子上任意两个相邻点的迭代值在吸引子上保持不变时,随着时间的推移会发生偏离。因此,即使在单个吸引子上,初始条件的精确值对迭代器的未来位置也有很大的影响。这一特性使得准确模拟未来值变得困难,而且在长时间范围内是不可能的,因为精确地描述初始条件几乎是不可能的,而且在从一个精确的初始条件开始的几次迭代之后,舍入误差是不可避免的。


See also


References

Category:Recurrence relations

类别: 循环关系

Category:Differential equations

类别: 微分方程


This page was moved from wikipedia:en:Initial condition. Its edit history can be viewed at 初始状态/edithistory