文档介绍：Chapter 31
Nonlinear Dynamics
and Chaos
A variety of techniques including the Frobenius method of infinite power series
could solve almost all linear DEs of physical interest. However, some very fun-
damental questions such as the stability of the solar system led to DEs that
were not linear, and for such DEs no analytic (including series representation)
solution existed. In the 1890s, Henri Poincar´e, the great French mathemati-
cian, took upon himself the task of gleaning as much information from the
DEs describing the whole solar system as was possible. The result was the
invention of one of the most powerful branches of mathematics (topology) and
the realization that the qualitative analysis of (nonlinear) DEs could be very
useful.
One of the discoveries made by Poincar´e, which much later became the
cornerstone of many developments, was that
Box . Unlike the linear DEs, nonlinear DEs may be very sensitive
to the initial conditions.
In other words, if a nonlinear system starts from some initial conditions and
develops into a certain final configuration, then starting it with slightly dif-
ferent initial conditions may cause the system to develop into a final config-
pletely different from the first one. This is plete contrast
to the linear DEs where two nearby initial conditions lead to nearby final
configurations.
In general, the initial conditions are not known with infinite accuracy.
Therefore, the final states of a nonlinear dynamical system may exhibit an
indeterministic behavior resulting from the initial (small) uncertainties. This
is what e to be known as chaos. The reader should note that the inde- chaos due to
terminism discussed here has nothing to do with the quantum indeterminism. uncertainty in
initial conditions
754 Nonlinear Dynamics and Chaos
All equations here pletely deterministic. It is the divergence of the
initially nearby—pletely deterministic—trajectories that results in
unpredictable final states.
Ther