The dipteran flight mechanism of the insects is commonly used to design the nonlinear flight robot system. However, the dynamic response of the click mechanism of the nonlinear robot system with multiple stability still unclear. In this paper, a novel dipteran robot model with click mechanism proposed based on the multiple stability of snap-through buckling. The motion of equation of the nonlinear flight robot system is obtained by using the Euler-Lagrange equation. The nonlinear potential energy, the elastic force, equilibrium bifurcation, as well as equilibrium stability are investigated to show the multiple stability characteristics. The transient sets of bifurcation and persistent set of regions in the system parameter plane and the corresponding phase portraits are obtained with multiple stability of single and double well behaviors. Then, the periodic free vibration response are defined by the analytical solution of three kinds of elliptical functions, as well as the amplitude frequency responses are investigated by numerical integration. Based on the topological equivalent method, the chaotic thresholds of the homo-clinic orbits for the chaotic vibration of harmonic forced robot system are derived to show the chaotic parametric condition. Finally, the prototype of nonlinear flapping robot is manufactured and the experimental system is setup. The nonlinear static moment of force curves, periodic response and dynamic flight vibration of dipteran robot system are carried out. It is shown that the test results are agree well with the theoretical analysis and numerical simulation. Those result have the potential application for the structure design of the efficient flight robot.