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 (stable or unstable) 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 – 1 derivatives 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 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). Moreover, in those nonlinear systems showing