We present the results of a systematic study of cataclysmic variables ( CVs ) and related systems , combining detailed binary-population synthesis ( BPS ) models with a grid of 120 binary evolution sequences calculated with a Henyey-type stellar evolution code .
In these sequences , we used 3 masses for the white dwarf ( 0.6 , 0.8 , 1.0 M _ { \odot } ) and seven masses for the donor star in the range of 0.6 - 1.4 \mbox { M$ { } _ { \odot } $ } .
The shortest orbital periods were chosen to have initially unevolved secondaries , and the longest orbital period for each secondary mass was taken to be just longer than the bifurcation period ( 16 - 22 hr ) , beyond which systems evolve towards long orbital periods .
These calculations show that systems which start with evolved secondaries near the end or just after their main-sequence phase become ultra-compact systems with periods as short as \sim 7 min .
These systems are excellent candidates for AM CVn stars .
Using a standard BPS code , we show how the properties of CVs at the beginning of mass transfer depend on the efficiency for common-envelope ( CE ) ejection and the efficiency of magnetic braking .
In our standard model , where CE ejection is efficient , some 10 per cent of all CVs have initially evolved secondaries ( with a central hydrogen abundance X _ { c } < 0.4 ) and ultimately become ultra-compact systems ( implying a Galactic birthrate for AM CVn-like stars of \sim 10 ^ { -3 } yr ^ { -1 } ) .
While these systems do not experience a period gap between 2 and 3 hr , their presence in the gap does not destroy its distinct appearance .
Almost all CVs with orbital periods longer than \sim 5 hr are found to have initially evolved or relatively massive secondaries .
We show that their distribution of effective temperatures is in good agreement with the distribution of spectral types obtained by Beuermann et al .
( 1998 ) .