In this paper, the dynamics of a bistable structure is investigated. The analysis focuses on the minimum required energy to move a bistable structure between stable equilibrium positions of the system. The investigation is done under a limited scenario for energy and force. The nonlinear behavior of bistable structures have been previously proposed as a method to hold shape with no energy consumption in a variety of applications including common electronic devices such as switches, relays, and in small aerial, land, and underwater vehicles as control surfaces. This paper focuses on the wellknown Duffing-Holmes oscillator as a one-degree-of-freedom representative of a bistable structure. The paper identifies several unique features of the response of the nonlinear system subjected to force and energy constraints. The paper also shows how the required energy for cross-well oscillation varies as a function of damping ratio, frequency ratio, and for different values of excitation force amplitudes. The response of the bistable nonlinear system is compared to a mono-stable linear system with the same parameters. For a linear system, it was observed that the energy function is quantized, and the energy function becomes more continuous and less quantized by increasing the force amplitude. For a bistable structure subjected to a harmonic force amplitude less than the static force, the energy function is scattered and divided into several levels. By increasing the force amplitude to the so-called static force or larger values, the ranges of excitation frequency ratios and damping ratios, which are able to achieve cross-well oscillation, increase significantly, and also the energy requirement becomes less quantized.