Systems with double-loop hysteresis are used increasingly in engineering, but few studies on their dynamics are reported. In this study, the bifurcation characteristics of the primary resonance of a system with double-loop bilinear hysteresis are investigated on the background of a shape memory alloy damper. First, the frequency-amplitude response equation is obtained by using the averaging methods. Then, the influences of the temperature and the amplitude of excitation on amplitude-frequency responses are analyzed by the constrained bifurcation singularity analysis method of non-smooth systems. The calculation results show that the parameter space composed of the temperature and the amplitude of excitation can be divided into 11 regions, which suggest that there are 11 qualitatively different kinds of amplitude-frequency responses to the variation of two parameters. In order to describe and compare the frequency-amplitude response curves conveniently, an encoding rule is proposed to describe their jump phenomena as the frequency sweeps. The above results can guide directly the design of frequency response mode of the system.