A linear matrix difference equation of the homogeneous form has closed form solution predicated on the vector of initial conditions on the individual variables that are stacked into the vector; 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 variableX; that behavior is stable or unstable based on the eigenvalues of the matrixA but not based on the initial conditions. Alternatively, a dynamic process in a single variable x having multiple time lags is 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 to obtain the latter's k solutions, which are the characteristic values for use in the solution equation Here the constants 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 Is known.
Continuous time
A differential equation system of the first order with n variables stacked in a vector X is Its behavior through time can be traced with a closed form solution conditional on an initial condition vector. 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 of the system. A single kth order linear equation in a single variable x is 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 whose solutions are the characteristic values these are used in the solution equation This equation and its first k – 1derivatives form a system of k equations that can be solved for the k parameters given the known initial conditions on x and its k – 1 derivatives' values at some time t.
Nonlinear systems
s 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 region of values that some dynamic paths approach but never leave, has a basin of attraction such that state variables with initial conditions in that basin will evolve toward that attractor. Even nearby initial conditions could be in basins of attraction of different attractors. 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.
