It is often discussed in 3 spatial dimensions and the need for third dimension to prevent self intersection is mentioned. But shouldn't the phase space of the Lorenz system be 6 dimensional, i.e., the 3 momenta also?


The dimensions mentionned here are phase space dimensions. The dynamics of the Lorenz system are defined by its 3 coordinates X(t),Y(y),Z(t) solving a system of 3 non linear ODEs. This has of course nothing to do with ordinary space because the Lorenz variables X(t) etc are actually functions of temperatures. The Lorenz attractor (butterfly) shown everywhere is not a structure in an ordinary space but the trace of orbits that the system follows in the phase space. As the phase space of the Lorenz system is 3D, it is possible to draw an easy visualisation of the orbits in an ordinary 3D space too.

Of course the "visualisation" trick doesn't work anymore for Lorenz systems with more than 3 dimensions and more generally for high dimensional phase spaces which are necessary to study higher dimensional chaos.

  • $\begingroup$ "X is proportional to the intensity of the convective motion, while Y is proportional to the temperature difference between the ascending and descending currents [...] Z is proportional to the distortion of the vertical temperature profile from linearity" [Lorenz, Edward N., 1963: Deterministic Nonperiodic Flow. J. Atmos. Sci., 20, 130–141.] $\endgroup$
    – user9886
    Jun 26 '13 at 14:54
  • $\begingroup$ Yes user 9886, this is the correct description of the relation between temperature and the 3 dimensions of the Lorenz system. I suspect that the original poster confused the specific phase space of Hamiltonian 1 body mechanics with the general phase space of any dynamical system like the one of Lorenz which can be any number between 1 and infinity. $\endgroup$
    – Stan Won
    Jun 27 '13 at 10:14

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