Analysis of Vibrational Resonance in Certain One Dimensional Nonlinear Maps Driven by an Amplitude Modulated Signal

The phenomenon of Vibrational Resonance (VR) is
explored in the presence of an amplitude-modulated (AM) signal
comprising a low-frequency component ($\omega$) and two
high-frequency components ($\Omega+\omega$) and ($\Omega-\omega$),
where $\Omega >> \omega$. This investigation is conducted within
the framework of certain one-dimensional nonlinear maps,
specifically the Bellows map and the Bountis map. These maps are
of particular interest due to their appearance in diverse physical
contexts and their complex dynamical behaviors. In both the
Bellows and Bountis maps, VR manifests at the low-frequency
($\omega$) of the amplitude-modulated signal as we vary the
amplitude ($g$) of the high-frequency signal. By carefully
selecting suitable values for the high-frequency signal's
amplitude ($g$), it becomes possible to significantly enhance the
response to a weak signal. Furthermore, the maximum resonance peak
is contingent upon the amplitude ($g$) of the amplitude-modulated
signal. We characterize the VR phenomenon through the analysis of
response amplitude and trajectory plots. Additionally, we explore
the impact of the AM signal on various attractors within these
maps, accompanied by appropriate bifurcation diagrams and Lyapunov
exponent diagrams.