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The nonlinear dynamics book by Hilborn gives the following argument about the famous Lorenz system:

Let $\vec f$ represent the set of time evolution functions for the system. If we consider a set of initial points distributed through a volume of the torus and if $\nabla{\vec f} < 0$ everywhere, then the volume occupied by the initial points inside the torus must shrink to $0$, and the torus must disappear. This argument tells us that the Lorenz model cannot have quasi-periodic solutions since it has $\nabla{\vec f} < 0$ everywhere.

I understand the first part of the argument that the volume occupied by the initial points inside the torus must shrink to $0$. However, how does that imply that "the torus must disappear"? Can't the trajectories simply reside on its surface (which has volume $0$) and give rise to the quasi-periodic motion anyway?

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I think the quoted text is aiming for the following (not that it was good in communicating this):

Let’s consider the torus and the phase-space volume enclosed by it. This volume must be invariant under time evolution:

  • No trajectories can move out of this volume. Trajectories on the surface stay there (due to it being a solution). Trajectories from the inside of the torus stay inside (because they cannot intersect the surface). The latter point is why this argument does not translate to limit cycles.

  • Every point inside the torus has an appropriate predecessor (just invert time and you’ll find it), i.e., there are no holes appearing under time evolution.

However, due to the ubiquitously negative divergence, this volume must shrink. This is a contradiction.


Now, what about phase-space volumes “shrinking to the surface” (for an attracting surface of the torus)? If you considered a sufficiently small neighbourhood of the surface, this would indeed shrink over time. However, this does not apply to the entire basin of attraction. As already mentioned above, every trajectory in the basin of attraction can be traced back backwards arbitrarily to another point in the basin of attraction. Thus the basin of attraction does not change its volume under time evolution.

Another way of looking at this is: If the surface of the torus is attracting points from its interior, it is a sink of phase-space flow. Thus there must also be a source of phase-space flow within the torus (e.g., an unstable limit cycle). This source must have a positive divergence (because that’s exactly what divergence measures).


Note that the ubiquitously negative divergence of the Lorenz system is a special property not found in many dynamical systems. Most dissipative dynamical systems are only dissipative on average or for specific regions of interest, but have source of phase-space flow somewhere.

Finally note that you can make an analogous argument for a two-dimensional system (with ubiquitously negative divergence) and a limit cycle, if it helps visualising.

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Quoting Strogatz, while he discusses volume contraction for the Lorenz system,

...Pick an arbitrary closed surface S(t) of volume V(t). Think of the points on S as initial conditions for trajectories, and let them evolve for an infinitesimal time dt...

This is the sense in which volume of phase space has been referred to, i.e. not the volume of the surface of phase space points themselves but that of the volume enclosed by the closed surface defined by the phase space points. So, if there is a quasiperiodic orbit as a solution of the Lorenz system, points on a certain torus will enclose a constant volume over time evolution. However, this contradicts the fact that the phase space volume enclosed must contract.

Reference: Nonlinear dynamics and Chaos by Steven H. Strogatz

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  • $\begingroup$ You don't have to. Strogatz has of course chosen initial conditions on the surface but in general you don't have to. Hilborn does take initial conditions throughout the "meat" of the torus. $\endgroup$
    – Peaceful
    Nov 11, 2016 at 6:30
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Intuitively, one can always say that as volume of torus approaches zero, one does not get a surface but only a circle(a cross-section of torus would be a circle which gradually becomes a "point" as the density of points -> 0 and there are infinite such points, which makes a circle). Now, quasi-periodicity can not happen on a circle !

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  • $\begingroup$ The distribution of points is inside the "meat" of the torus. For example, when we say that we start from a sphere of initial conditions which go to limit cycle, the final limit cycle has 3-volume zero. On the other hand if we had started from initial conditions on a circle and fixed point is the final attractor, then we say that points' 2-volume is zero and so on. $\endgroup$
    – Peaceful
    Sep 2, 2016 at 10:59

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