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

Abstract

The phenomenon of Vibrational Resonance (VR) is explored in the presence of an amplitude-modulated (AM) signal comprising a low-frequency component (ω) and two highfrequency components (Ω + ω) and (Ω ω), where Ω >> ω. 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 (ω) 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.