Animals select gait which is suitable for their moving speeds, environments and their purposes. Such behavior transition is carried out to minimize energy consumption or to adapt to the environment. The transition of animals from a gait to another gait is discontinuous generally. However, the changes of moving speed, energy consumption, and etc. are continuous. In this research, we focus our discussion on the discontinuous dynamic behavior transition of a walking robot. The robot transits from biped walking to quadruped walking to adapt to the environment. For the transition, bifurcation of potential function is utilized. We demonstrate the behavior transition of the walking robot between biped and quadruped walking. The behavior transition which is an adaptive behavior to the environment depending on gradient of a slope is carried out by using bifurcation of parameters used for control. We also show the transition using the hysteresis on a slope where the robot with simple transition between biped and quadruped walking cannot walk. The hysteresis is generated by using a potential function which changes dynamically depending on some parameters.