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Planetary debris is observed in the atmospheres of over 1,000 white dwarfs,and two white dwarfs are now observed to contain orbiting minor planets.Exoasteroids and planetary core fragments achieve orbits close to the whitedwarf through scattering with major planets. However, the architectures thatallow for this scattering to take place are time-consuming to explore with$N$-body simulations lasting $\sim 10^{10}$ yr; these long-running simulationsrestrict the amount of phase space that can be investigated. Here we use planarand three-dimensional (spatial) elliptic periodic orbits, as well as chaoticindicators through dynamical stability maps, as quick scale-free analyticalternatives to $N$-body simulations in order to locate and predict instabilityin white dwarf planetary systems that consist of one major and one minor planeton very long timescales. We then classify the instability according to ejectionversus collisional events. We generalized our previous work by allowingeccentricity and inclination of the periodic orbits to increase, thereby addingmore realism but also significantly more degrees of freedom to ourarchitectures. We also carried out a suite of computationally expensive 10 Gyr$N$-body simulations to provide comparisons with chaotic indicators in alimited region of phase space. We compute dynamical stability maps that arespecific to white dwarf planetary systems and that can be used as tools infuture studies to quickly estimate pollution prospects and timescales forone-planet architectures. We find that these maps also agree well with theoutcomes of our $N$-body simulations. As observations of metal-polluted whitedwarfs mount exponentially, particularly in the era of Gaia, tools such asperiodic orbits can help infer dynamical histories for ensembles of systems.

Driving white dwarf metal pollution through unstable eccentric periodic orbits