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Of the Lorenz equations I understand that for certain values of the parameters (the ones that cause chaos):
Any volume of initial conditions in the phase space contracts with the flow of time
All trajectories eventually enter and remain within a finite ellipsoid about the origin
Because the system is 'chaotic' (not entirely certain about the definition of chaos) neighboring trajectories with even very slight perturbations separate exponentially fast
Doesn't this third statement contradict the first two? Why not? Is the fractal nature of the attractor somehow responsible?
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