Chaos and the Lorenz Equations

[fayled]

Take the Lorenz equations
x'=σ(y-x)
y'=rx-y-xz
z'=xy-bz
with σ=10, b=8/3 and r=28 as a typical example of chaos (I am using primes to indicate total time derivatives in this post).

A basic property of a chaotic system (where the flow in phase space is a strange attractor) is that if you pick two initial points (corresponding to initial conditions) in phase space a tiny distance apart and follow their flows, their separation will grow exponentially.

If we consider the evolution of volumes in phase space, we find that
V'=V(∇.u)
where here ∇.u=-(σ+1+b) so that
V(t)=V(0)exp[-(σ+1+b)t]
and hence volumes in phase space exhibit exponetial decay. Now let's pick a volume containing loads of initial points - this tells us it is going to contract exponentially. This makes sense if we are considering the regime in which normal attractors exist in phase space because I would expect everything to collapse towards stable orbits or stable fixed points.

However in our chaotic regime where we have a strange attractor, I feel as though the above two paragraphs are in contradiction. We pick two initial points in phase space - the first argument tells us that these points give rise to very different flows in phase space, whilst the second says they collapse onto the same flow.

Clearly I am misunderstanding something so would anybody be able to point me in the correct direction? Thank you :)

 

[S.G. Janssens]

A basic property of a chaotic system (where the flow in phase space is a strange attractor) is that if you pick two initial points (corresponding to initial conditions) in phase space a tiny distance apart and follow their flows, their separation will grow exponentially.

Can you state the precise mathematical definition that you use for sensitive dependence on initial conditions?

(...) whilst the second says they collapse onto the same flow.

What do you mean by this? The fact that the strange attractor has zero volume does not imply that orbits passing through two different initial conditions on the attractor will coincide either after finite or infinite time. Why do you think there is such an implication?

In any case, sensitive dependence on initial conditions does not mean that if $p$ and $q$ are two initial conditions, then there exist constants $C > 0$ and $\mu > 0$ such that $\|\phi^t(p) - \phi^t(q)\| \ge C e^{\mu t}$ for all $t \ge 0$. (Here $\phi$ is the flow corresponding to the ODE.) If this were the case, attractors admiting sensitive dependence would always be unbounded.