Editing Initial condition (section)

==Nonlinear systems==
{{Multiple image
| image1 = LF-Initial.png
| caption1 = Another initial condition
| image2 = LF-Solution.png
| caption2 = Evolution of this initial condition for an example PDE
}}
[[Nonlinear system]]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 [[convergence (mathematics)|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 [[chaos theory|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#Computer simulation|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.