In mathematics and particularly in dynamical systems, an initial condition is the initial value (often at time t = 0 {\displaystyle t=0} ) of a differential equation, difference equation, or other "time"-dependent equation which evolves in time. The most fundamental case, an ordinary differential equation of order k (the number of derivatives in the equation), generally requires k initial conditions to trace the equation's evolution through time. In other contexts, the term may refer to an initial value of a recurrence relation, discrete dynamical system, hyperbolic partial differential equation, or even a seed value of a pseudorandom number generator, at "time zero", enough such that the overall system can be evolved in "time", which may be discrete or continuous. The problem of determining a system's evolution from initial conditions is referred to as an initial value problem.
Linear system
Discrete time A linear matrix difference equation of the homogeneous (having no constant term) form X t + 1 = A X t {\displaystyle X_{t+1}=AX_{t}} has closed form solution X t = A t X 0 {\displaystyle X_{t}=A^{t}X_{0}} predicated on the vector X 0 {\displaystyle X_{0}} of initial conditions on the individual variables that are stacked into the vector; X 0 {\displaystyle X_{0}} 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. Alternatively, a dynamic process in a single variable x having multiple time lags is
x t = a 1 x t − 1 + a 2 x t − 2 + ⋯ + a k x t − k . {\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}
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 λ k − a 1 λ k − 1 − a 2 λ k − 2 − ⋯ − a k − 1 λ − a k = 0 {\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's k solutions, which are the characteristic values λ 1 , … , λ k , {\displaystyle \lambda _{1},\dots ,\lambda _{k},} for use in the solution equation
x t = c 1 λ 1 t + ⋯ + c k λ k t . {\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}
Here the constants c 1 , … , c k {\displaystyle c_{1},\dots ,c_{k}} 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 x t {\displaystyle x_{t}} Is known.
Continuous time A differential equation system of the first order with n variables stacked in a vector X is
d X d t = A X . {\displaystyle {\frac {dX}{dt}}=AX.}
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