1. In several cases, the existence of a periodic orbit was known. 2. After the bifurcation there is no longer a periodic orbit . 3. This is usually accompanied by the birth or death of a periodic orbit . 4. For other values of \ rho, the system displays knotted periodic orbits . 5. The trace formula asserts that each periodic orbit contributes a sinusoidal term to the spectrum. 6. Periodic-orbit theory gives a recipe for computing spectra from the periodic orbits of a system. 7. The set of points that never leaves the neighborhood of the given periodic orbit form a fractal. 8. Thus, a heteroclinic orbit can be understood as the transition from one periodic orbit to another. 9. Using the trace formula to compute a spectrum requires summing over all of the periodic orbits of a system. 10. Without these restrictions, no continuous time system with fixed points or periodic orbits could have been structurally stable.
