An initial condition of a vibrating string

Evolution from the initial condition

Inmathematics and particularly indynamic systems, aninitial condition, in some contexts called aseed value,^{[1]: pp. 160} is a value of an evolvingvariable at some point in time designated as the initial time (typically denotedt = 0). For a system oforderk (the number of time lags indiscrete time, or the order of the largest derivative incontinuous time) anddimensionn (that is, withn different evolving variables, which together can be denoted by ann-dimensionalcoordinate vector), generallynk initial conditions are needed in order to trace the system's variables forward through time.

In bothdifferential equations in continuous time anddifference 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 aclosed form solution for the state variables as a function of time and of the initial conditions is called theinitial 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.

A linearmatrix difference equation of the homogeneous (having no constant term) form${\displaystyle X_{t+1}=A{X}_t}$ has closed form solution${\displaystyle X_t=A^t{X}_0}$ predicated on the vector${\displaystyle X_0}$ of initial conditions on the individual variables that are stacked into the vector;${\displaystyle X_0}$ is called the vector of initial conditions or simply the initial condition, and containsnk pieces of information,n being the dimension of the vectorX andk = 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 variableX; that behavior isstable or unstable based on theeigenvalues of the matrixA but not based on the initial conditions.

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

${\displaystyle x_t=a_1{x}_{t-1}+a_2{x}_{t-2}+\cdots +a_k{x}_{t-k}.}$

Here the dimension isn = 1 and the order isk, so the necessary number of initial conditions to trace the system through time, either iteratively or via closed form solution, isnk = 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 itscharacteristic equation${\displaystyle {\lambda }^k-a_1{\lambda }^{k-1}-a_2{\lambda }^{k-2}-\cdots -a_{k-1}\lambda -a_k=0}$ to obtain the latter'sk solutions, which are thecharacteristic values${\displaystyle {\lambda }_1,\dots ,{\lambda }_k,}$ for use in the solution equation

${\displaystyle x_t=c_1{\lambda }_1^t+\cdots +c_k{\lambda }_k^t.}$

Here the constants${\displaystyle c_1,\dots ,c_k}$ are found by solving a system ofk different equations based on this equation, each using one ofk different values oft for which the specific initial condition${\displaystyle x_t}$ Is known.

A differential equation system of the first order withn variables stacked in a vectorX is

${\displaystyle \frac{dX}{dt}=AX.}$

Its behavior through time can be traced with a closed form solution conditional on an initial condition vector${\displaystyle X_0}$. The number of required initial pieces of information is the dimensionn of the system times the orderk = 1 of the system, orn. The initial conditions do not affect the qualitative behavior (stable or unstable) of the system.

A singlek^th order linear equation in a single variablex is

${\displaystyle \frac{d^{k}x}{d{t}^k}+a_{k-1}\frac{d^{k-1}x}{d{t}^{k-1}}+\cdots +a_1\frac{dx}{dt}+a_{0}x=0.}$

Here the number of initial conditions necessary for obtaining a closed form solution is the dimensionn = 1 times the orderk, or simplyk. In this case thek initial pieces of information will typically not be different values of the variablex at different points in time, but rather the values ofx and its firstk – 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. Thecharacteristic equation of this dynamic equation is${\displaystyle {\lambda }^k+a_{k-1}{\lambda }^{k-1}+\cdots +a_1\lambda +a_0=0,}$ whose solutions are thecharacteristic values${\displaystyle {\lambda }_1,\dots ,{\lambda }_k;}$ these are used in the solution equation

${\displaystyle x(t)=c_1{e}^{{\lambda }_{1}t}+\cdots +c_k{e}^{{\lambda }_{k}t}.}$

This equation and its firstk – 1 derivatives form a system ofk equations that can be solved for thek parameters${\displaystyle c_1,\dots ,c_k,}$ given the known initial conditions onx and itsk – 1 derivatives' values at some timet.

Another initial condition

Evolution of this initial condition for an example PDE

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 itconverges to one or anotherattractor 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 exampleNewton's method#Basins of attraction).

Moreover, in those nonlinear systems showingchaotic behavior, the evolution of the variables exhibitssensitive dependence on initial conditions: the iterated values of any two very nearby points on the samestrange 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 accuratesimulation 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.

Every empirical law has the disquieting quality that one does not know its limitations. We have seen that there are regularities in the events in the world around us which can be formulated in terms of mathematical concepts with an uncanny accuracy. There are, on the other hand, aspects of the world concerning which we do not believe in the existence of any accurate regularities. We call these initial conditions.^[2]

• Boundary condition
• Initialization vector, in cryptography

• Quotations related toInitial condition at Wikiquote

• Recurrence relations
• Differential equations

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