The $\omega$-limit set in a positively invariant compact region and a new description of the Lorenz attractor

The $\omega$-limit set in a compact positively invariant region $R\subset {\mathbb{R}}^n$ has been identified for $n=1$, 2, and 3, with examples in each case. It has been shown that the $\omega$-limit set becomes more complex as $n$ increases from 1 to 3, and we expect this to also be true for $n>3$. Our example for $n=3$ is the Lorenz equations, for which we have shown that its $\omega$-limit set is a {\em twisted torus}