初始条件

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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.

在数学中,特别是在动态系统中,初始条件,在某些情况下称为种子值:第160页,是指在某个指定为初始时间(通常表示为t=0)的时间点上的一个演化变量的值。对于一个阶数为k(离散时间中的时间滞后数,或连续时间中最大导数的阶数)、维数为n(即有n个不同的演化变量,这些变量可以用n维坐标矢量表示)的系统,一般需要nk个初始条件,以便通过时间追踪系统的变量。


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.

在连续时间的微分方程和离散时间的差分方程中,初始条件会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,为状态变量寻找一个与时间和初始条件有关的封闭形式的解决方案的问题被称为初始值问题。对于离散时间的情况,也存在相应的问题。虽然闭合形式的解决方案并不总是可以得到的,但离散时间系统的未来值可以通过每次迭代向前推进一个时间段来找到,尽管四舍五入的误差可能使这种做法在很长的时间范围内不切实际。


Linear system 线性系统

Discrete time 离散时间

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.

同质(没有常数项)形式的线性矩阵差分方程 [math]\displaystyle{ X_{t+1}=AX_t }[/math] 具有封闭形式的解[math]\displaystyle{ X_t=A^tX_0 }[/math],其前提是堆积到该向量的各个变量的初始条件向量 [math]\displaystyle{ X_0 }[/math][math]\displaystyle{ X_0 }[/math]被称为初始条件向量或简单的初始条件,它包含nk个信息,n是向量X的维度,k=1是系统中的时间滞后数。这个线性系统中的初始条件并不影响状态变量X未来行为的定性;该行为是基于矩阵A的特征值的稳定或不稳定,但不是基于初始条件。


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

另外,一个具有多个时间滞后的单一变量x的动态过程是[math]\displaystyle{ 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

这里的维度是n=1,阶数是k,所以通过时间追踪系统的必要的初始条件数量,无论是迭代还是通过闭合形式解决,都是nk=k。这个方程的解是通过使用其特征方程 [math]\displaystyle{ \lambda^k-a_1\lambda^{k-1} -a_2\lambda^{k-2}-\cdots -a_{k-1}\lambda-a_k=0 }[/math]来获得后者的k解。这是特征值 [math]\displaystyle{ \lambda_1, \dots , \lambda_k, }[/math],用于解方程中[math]\displaystyle{ 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.

这里的常数[math]\displaystyle{ c_1, \dots , c_k }[/math]是通过解决基于此方程的k个不同的方程组而找到的,每个方程组使用k个不同的t值中的一个,对于这个t值,特定的初始条件 [math]\displaystyle{ x_t }[/math]是已知的。


Continuous time 连续时间

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]


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.

它通过时间的行为可以用一个封闭形式的解决方案来追踪,条件是初始条件向量 [math]\displaystyle{ X_0 }[/math]。所需的初始信息的数量是系统的维数n乘以系统的阶数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

单一变量x中的第k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]


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,获得闭合形式解所需的初始条件的数量是维数n=1乘以阶数k,或者简单说是k。在这种情况下,k个初始信息通常不是变量x在不同时间点的不同值,而是x及其前k-1个导数的值,都在某个时间点上,如时间零。初始条件并不影响系统行为的质量性质。这个动态方程的特征方程是[math]\displaystyle{ \lambda^k+a_{k-1}\lambda^{k-1}+\cdots +a_1\lambda +a_0=0, }[/math] 其解是特征值 [math]\displaystyle{ \lambda_1,\dots , \lambda_k; }[/math]这些都用于解方程式中[math]\displaystyle{ x(t)=c_1e^{\lambda_1t}+\cdots + 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.

这个方程和它的第一个k-1导数形成了一个k方程组,可以解决k个参数[math]\displaystyle{ c_1, \dots , c_k, }[/math],给定x的已知初始条件和它在某个时间t的k-1导数值。



Nonlinear systems 非线性系统

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).

非线性系统可以表现出比线性系统更丰富的行为。特别是,初始条件可以影响系统是否发散到无限大,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(可能是不相连的)数值区域,一些动态路径接近但从未离开,有一个(可能是不相连的)吸引盆地,这样初始条件在该盆地(而不是其他地方)的状态变量将向该吸引子演变。甚至附近的初始条件也可能处于不同吸引子的吸引盆地中(例如,见牛顿方法#吸引盆地)。



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,在那些表现出混沌行为的非线性系统中,变量的演化表现出对初始条件的敏感依赖:在同一个奇怪的吸引子上,任何两个非常接近的点的迭代值,虽然都留在吸引子上,但会随着时间的推移而相互背离。因此,即使在一个吸引子上,初始条件的精确值也会对迭代的未来位置产生重大影响。这一特点使得对未来值的精确模拟变得很困难,而且在很长的时间内也不可能,因为很少有可能精确地说明初始条件,而且从一个精确的初始条件开始,即使只有几次迭代,也不可避免地会出现舍入误差。



See also 另见


References 参考文献

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