  We introduce a one-dimensional model of Bose-Einstein condensates (BECs),
combining the double-well potential, which is a well-known setting for the
onset of spontaneous-symmetry-breaking (SSB) effects, and time-periodic
modulation of the nonlinearity, which may be implemented by means of the
Feshbach-resonance-management (FRM) technique. Both cases of the nonlinearity
which is repulsive or attractive on the average are considered. In the former
case, the main effect produced by the application of the FRM is spontaneous
self-trapping of the condensate in either of the two potential wells in
parameter regimes where it would remain untrapped in the absence of the
management. In the weakly nonlinear regime, the frequency of intrinsic
oscillations in the FRM-induced trapped state is very close to half the FRM
frequency, suggesting that the effect is accounted for by a parametric
resonance. In the case of the attractive nonlinearity, the FRM-induced effect
is the opposite, i.e., enforced detrapping of a state which is self-trapped in
its unmanaged form. In the latter case, the frequency of oscillations of the
untrapped mode is close to a quarter of the driving frequency, suggesting that
a higher-order parametric resonance may account for this effect.
