– there are sections of periodicity interspersed with periods of chaos, and the trend is towards higher-period orbits as increases. For example, the period one orbit only appears for values of around 4 and is never found again in the bifurcation diagram. The same phenomenon is seen with period three; until , period three orbits can be found, but thereafter, they do not appear. A graphical illustration of the changing attractor over a range of values illustrates the general behavior seen for all of these parameter analyses – the frequent transitions between periodicity and aperiodicity. The above set of images illustrates the variations in the post-transient Rössler system as is varied over a range of values. These images were generated with . , period-1 orbit. , period-2 orbit. , period-4 orbit. , period-8 orbit. , sparse chaotic attractor. , period-3 orbit. , period-6 orbit. , sparse chaotic attractor. , filled-in chaotic attractor. Periodic orbits The attractor is filled densely with periodic orbits: solutions for which there exists a nonzero value of such that . These interesting solutions can be numerically derived using Newton's method. Periodic orbits are the roots of the function , where is the evolution by time and is the identity. As the majority of the dynamics occurs in the x-y plane, the periodic orbits can then be classified by their winding number around the central equilibrium after projection. It seems from numerical experimentation that there is a unique periodic orbit for all positive winding numbers. This lack of degeneracy likely stems from the problem's lack of symmetry. The attractor can be dissected into easier to digest invariant manifolds: 1D periodic orbits and the 2D stable and unstable manifolds of periodic orbits. These invariant manifolds are a natural skeleton of the attractor, just as rational numbers are to the real numbers. For the purposes of dynamical systems theory, one might be interested in topological invariants of these manifolds. Periodic orbits are copies of embedded in , so their topological properties can be understood with knot theory. The periodic orbits with winding numbers 1 and 2 form a Hopf link, showing that no diffeomorphism can separate these orbits. Links to other topics The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the half-twist that occurs in the Rössler attractor only affects a part of the attractor. Rössler showed that his