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No chaos parameter values
Some of the transverse sections found for the Lorenz system lie in a region of phase space that is crossed by the homoclinic orbit when the homoclinic bifurcation occurs.
We can see that in fact the transverse sections separate the unstable and stable manifolds of the origin, so that there cannot be any trajectory linking these two. This homoclinic bifurcation gives rise to the chaotic attractor, at least to a nonattracting invariant set, which is responsible for the transient chaos.
The conditions for the existence of the aforementioned
transverse sections are then conditions of non-existence
of the homoclinic orbit. Aside you can see the range of
parameters for which our method states that there is
no chaotic motion for the Lorenz system.
It would be very interesting to uncover conditions for the r parameter (which in the Lorenz system is related to the temperature gradient in the Rayleigh-Benard experiment) for which the homoclinic trajectory does not exist. This would lead to the discovery of an analytic lower bound for the values r=r_{ho} of the homoclinic bifurcation and hence for the chaotic behaviour.