Twisted Rossler Attractors by Bob Gilmore.

This page presents chaotic attractors that are locally diffeomorphic to the Rossler attractor, but not globally diffeomorphic. The Rossler attractor lives in a solid torus phase space. These image attractors can be formed visually by cutting open the torus, twisting a number n of times, and then closing it back up. If the twisting number is a fraction q/p in lowest terms, we must insert p copies of the attractor before matching the ends to ensure that the boundary conditions are satisfied. The attractors below are indexed by the numbers (p,q)=(n1,n2), where p is the number of copies and q the number of full twists.

Note that q can be either positive or negative, indicating the direction of the twist. Also, the indices (1,0) correspond to the original Rossler attractor. Finally, since these attractors are locally diffeomorphic they cannot be distinguished by their Lyapunov spectrum.

In each plot the coordinates are (u,du/d(phi),phi). phi is horizontal, as the axis indicates. The color code is black for du/d(phi)>0 and green for du/d(phi)<0. u is defined by the relation (x+iy) = u exp(i phi).