  Mounting discoveries of debris discs orbiting newly-formed stars and white
dwarfs (WDs) showcase the importance of modeling the long-term evolution of
small bodies in exosystems. WD debris discs are in particular thought to form
from very long-term (0.1-5.0 Gyr) instability between planets and asteroids.
However, the time-consuming nature of $N$-body integrators which accurately
simulate motion over Gyrs necessitates a judicious choice of initial
conditions. The analytical tools known as \textit{periodic orbits} can
circumvent the guesswork. Here, we begin a comprehensive analysis directly
linking periodic orbits with $N$-body integration outcomes with an extensive
exploration of the planar circular restricted three-body problem (CRTBP) with
an outer planet and inner asteroid near or inside of the $2$:$1$ mean motion
resonance. We run nearly 1000 focused simulations for the entire age of the
Universe (14 Gyr) with initial conditions mapped to the phase space locations
surrounding the unstable and stable periodic orbits for that commensurability.
In none of our simulations did the planar CRTBP architecture yield a
long-timescale ($\gtrsim 0.25$% of the age of the Universe) asteroid-star
collision. The pericentre distance of asteroids which survived beyond this
timescale ($\approx 35$ Myr) varied by at most about 60%. These results help
affirm that collisions occur too quickly to explain WD pollution in the planar
CRTBP $2$:$1$ regime, and highlight the need for further periodic orbit studies
with the eccentric and inclined TBP architectures and other significant orbital
period commensurabilities.
