this section we present numerical examples of nonadiabatic jump behavior and the cycle-to-cycle changes in emin, emax, PK etc. that they cause.

We show three examples, isolating one SRF at a time.

For each example we integrate equations (2) and (3) up to τ = t/Psec = 400.

We perform integrations employing either the DA potential (12) or the SA potential (6), and in each case we run both with and without SRFs.

Each example below represents an astrophysically realistic system, with definite masses, semimajor axes and eccentricities.

The dynamics, however, depend only on the nine dimensionless numbers collected in Eq. (A1), built on the dimensionless framework of §2, so these astrophysical choices can be rescaled to apply to other systems (see Appendix A for details).

In our code, we specify each system directly using the dimensionless set (A1), reconstruct the initial j and e from those, and integrate in this dimensionless space.

Thus, when we report below the initial conditions in terms of orbital elements, they are really just consequences of our choice of the numbers (A1).

For reproducibility, the exact initial conditions are reported in Appendix A.1 and Table 1.

Pout
(17)
Psec ≪jDA min, where jDA min ≡(1 −e2 max,DA)1/2 is computed in the DA approximation (e.g., from Liu et al. 2015 when SRFs are included).

Alternatively, one might have demanded that the typical SA fluctuation in the binary’s angular momentum at high eccentricity be sufficiently small, |δj| ≪ jDA min.

However, one can show that |δj| ∼Pout/Psec (Ivanov et al. 2005; Grishin et al. 2018; Hamilton & Rafikov 2024), so this requirement is equivalent to (17).4 When the heuristic criterion (17) is broken, i.e., when Pout
(18)
Psec ≳jDA min, one might still hope to salvage some part of the DA program (Ivanov et al. 2005; Grishin et al. 2018; Hamilton & Rafikov 2019).

The hope is that jz and CK remain approximate adiabatic invariants, exhibiting only small oscillations about their DA values, so that the secular period is essentially unchanged and SA fluctuations only affect the peak eccentricity, which can be calculated (semi-)analytically.

Thus we might expect
3.1.

Example 1: GR precession We first consider a binary black hole (BBH) with m0 = m1 = 30 M⊙orbiting a supermassive black hole (SMBH) of mass m2 = 107 M⊙, with GR precession as the only SRF.

For the outer orbit we take initial conditions a2 ≃0.5 pc, e2 = 0.8 (implying an outer period Pout ≃1.2 × 104 yr) and an initial mean anomaly M2 that we will specify below.

For the inner orbit we choose a ≃16 AU (implying a characteristic secular time Psec ≃0.6 Myr), e = 0, ω = 0.0◦, i = 89.4◦, Ω= 180◦.

The only SRF is GR precession (so εTide = εRot = 0).

These choices return the values
PK ≃P DA K , (19) emax ≃eDA max + δe, (20) where δe is the SA eccentricity fluctuation corresponding to δj.

In this naive semi-analytic picture, the only role of SRFs is to reduce eDA max.

Our central technical claim is that, for systems lying in the fourth quadrant of Figure 1, this semi-analytic approach can deliver results that are highly misleading.

The underlying reason is the one already identified in §2.1: nonadiabatic jumps can place these systems on entirely different DA cycles, with a generally different
jz = 0.010, (21) CK = 1.0 × 10−6, (22) Pout/Psec = 0.021, (23)
4 Strictly we should use the Kozai period PK in the criterion (17) rather than the characteristic timescale Psec.

However, we nearly always have PK ∼Psec, and the latter is much easier to evaluate quickly.

The exception is near the separatrix CK = 0 where we can have PK ≫Psec.