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2.10] What is sensitive dependence on initial conditions? (nonlinear science)




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This article is from the Nonlinear Science FAQ, by James D. Meiss jdm@boulder.colorado.edu with numerous contributions by others.

2.10] What is sensitive dependence on initial conditions? (nonlinear science)

Consider a boulder precariously perched on the top of an ideal hill. The
slightest push will cause the boulder to roll down one side of the hill or the
other: the subsequent behavior depends sensitively on the direction of the
push--and the push can be arbitrarily small. Of course, it is of great
importance to you which direction the boulder will go if you are standing at
the bottom of the hill on one side or the other!

Sensitive dependence is the equivalent behavior for every initial condition--
every point in the phase space is effectively perched on the top of a hill.

More precisely a set S exhibits sensitive dependence if there is an r such
that for any epsilon > 0 and for each x in S, there is a y such that |x - y| <
epsilon, and |x_n - y_n| > r for some n > 0. Then there is a fixed distance r
(say 1), such that no matter how precisely one specifies an initial state
there are nearby states that eventually get a distance r away.

Note: sensitive dependence does not require exponential growth of
perturbations (positive Lyapunov exponent), but this is typical (see [2.14])
for chaotic systems. Note also that we most definitely do not require ALL
nearby initial points diverge--generically [2.14] this does not happen--some
nearby points may converge. (We may modify our hilltop analogy slightly and
say that every point in phase space acts like a high mountain pass.) Finally,
the words "initial conditions" are a bit misleading: a typical small
disturbance introduced at any time will grow similarly. Think of "initial" as
meaning "a time when a disturbance or error is introduced," not necessarily
time zero.

 

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