For values of 1<r<13.926, the origin is a 3D unstable saddle node, stable in 2 directions, but unstable in the third (2 eigenvalues with negative real parts, one with a positive real part).
Two stable fixed points, C+ and C- appear. Trajectories will approach either one, depending on the initial conditions.
For 13.926<r<24.06, a pair of homoclinic orbit connect the origin with itself.
An unstable limit surrounds the fixed points in the planes of the homoclinic orbits.
A very complicated, non-attracting invariant set appears in the phase plane, causing trajectories to be sensitive to initial conditions as they pass near the invariant set.
Trajectories starting inside the homoclinic orbit, but outside the unstable limit cycle, have no future in that plane. They will wander around around in the plane before jumping out of it and landing in the opposing plane. If the trajectory lands between the homoclinic orbit and the limit cycle of the opposite plane, it will have the same experience of wandering in a plane where it has no future. Eventually it will land inside the unstable limit cycle of one plane, where it will spend the rest of its existence spiraling into the stable fixed point.
As r increases, the pathways that lead from one plane into the unstable limit cycle of the opposite plane shrink, and it takes trajectories longer and longer to find a final home. This is called Transient Chaos because trajectories are extremely sensitive to initial conditions, but they do not have an infinitely complex trajectory – they eventually collapse into a stable fixed point. Therefore it is not Chaos.
To show sensitivity to initial conditions, the following animation shows two trajectories with initial conditions differing by less than one one-hundredth. The two trajectories initially move together very closely, but their differences compounds exponentially, and the quickly diverge forever. To make the point, the two even end their journeys at opposite fixed points — though the paths they took to them are completely different.
When r=24.06, there are no possible pathways from one plane into the unstable limit cycle of the opposite plane. Although the stable fixed points still exist, there is no way to get them other than to start near them. All trajectories starting anywhere other than inside the unstable limit cycles will never reach a fixed point, and thus they will travel along the Strange Attractor along an infinitely complex, non-repeating trajectory. This is Chaos. When r=24.74, a subcritical Hopf bifurcation occurs, the unstable limit cycles strangle the stable fixed points and cause them to become unstable. The only effective difference is that now all trajectories are chaotic, as there is no possibility of crawling into a stable fixed point.
The second of the two videos above shows two trajectories with very similar initial conditions. One can see that their trajectories do not remain similar for long.
The system remains chaotic as values of r increase until a periodic window appears, from about 99.5<r<100.8 ! In this window, trajectories are drawn into a Stable Limit Cycle, a closed orbit! With changes in r, the complexity of this closed orbit increases, it experiences Period Doubling. More periodic windows appear between 126.4<r<126.55, 145<r<167, and 215<r<313. For r>313, the system has a limit cycle that does not change topologically.