The catastrophe mechanisms of thermal performance characteristics of the firebox gas combustion system were analyzed from the viewpoint of catastrophe theory. The mathematical models of cusp catastrophe were established. The relationship between the thermal performance characteristics and the changing of system control variables was studied. The cusp catastrophe mechanisms of typical performance characteristics, such as kicking and lagging, and those of transition from quenching to igniting were explained. It was illustrated that discontinuity behavior of thermal systems with an "S" motion feature curve and lagging feature may be equivalently classified according to the topology of cusp catastrophe, influenced by two groups of independent control variables.