Monotonicity of the period map for the equation $-{\phi }^{\mathrm{\prime }\mathrm{\prime }}+\phi -{\phi }^k=0$

In this paper, we establish the monotonicity of the period map in terms of the energy levels for certain periodic solutions of the equation $-{\phi }^{\mathrm{\prime }\mathrm{\prime }}+\phi -{\phi }^k=0$, where $k>1$ is a real number. We present a new approach to demonstrate this property, utilizing spectral information of the corresponding linearized operator around the periodic solution and tools related to Floquet theory.