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fayled
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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 :)
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 :)