The effects of the energy spread of secondary electrons are theoretically investigated for a dc-biased single-surface multipactor. In our previous publication ͓S. G. Jeon et al., Phys. Plasmas 16, 073101 ͑2009͔͒, we obtained the conditions for the phase lock of an electron bunch, assuming zero velocity spread of the secondary electrons. In this work, we extended our previous theory to derive a quadratic map, by which the stability and bifurcation of the electron bunch can be systematically investigated. For the study of the energy spread of the secondary electrons, a randomized term was added to this map. The modified map then showed significant smearing-out of the bifurcated branches. The theoretical results were verified by particle-in-cell simulations, which showed good agreement in wide parameter ranges for both cases of monoenergetic and energy-spread secondary electrons.