文档介绍：15 Lyapunov exponents
Whereas fractals quantify the geometry of strange attractors, Lyaponov ex­
ponents quantify the sensitivity to initial conditions that is, in effect, their
most salient feature.
In this lecture we point broadly sketch some of the mathematical issues con­
cerning Lyaponov exponents. We also briefly describe how they ­
puted. We then conclude with a description of a simple model that shows
how both fractals and Lyaponov exponents manifest themselves in a simple
model.
Diverging trajectories
Lyapunov exponents measure the rate of divergence of trajectories on an
attractor.
Consider a flow θψ(t) in phase space, given by
dθ
= Fψ(θψ)
dt
If instead of initiating the flow at θψ(0), it is initiated at θψ(0)+π(0), sensitivity
to initial conditions would produce a divergent trajectory:
ε(t)
ε(0)
φ(0) φ( t)
Here ψπ grows with time. To the first order,
| |
d(θψ+ ψπ)
Fψ(θψ) + M(t) ψπ
dt ◦
141
where
ωF
M (t) = i .
ij ωθ
j πτ(t)
We thus find that
dψπ
= M(t) ψπ. (31)
dt
Consider the example of the Lorenz model. The Jacobian M is given by
P P 0
−
M(t) =
Z(t) + r 1 X(t) .
⎭−−−⎣
Y (t)
X(t) b
−
We cannot solve for ψπ because of the unknown time ⎤dependence of M(t).
However one may numerically solve for θψ(t), and thus ψπ(t), to obtain (for­
mally)
ψπ(t) = L(t) ψπ