Phantom Attractors in a Single-Degree-of-Freedom Smooth System Under Additive Stochastic Excitation

Phantom attractors in nonlinear systems under additive stochastic excitation have been recently discovered. This paper uncovers the existence of phantom attractors in a single-degree-of-freedom smooth nonlinear equation, which characterizes the vibration of an inextensible beam subjected to lateral stochastic excitation. It also elucidates that the stochastic averaging method, in this context, may lead to qualitatively erroneous probability density functions, identified as one of the reasons why these attractors were previously overlooked. The study then proceeds to analyze the formation process of the phantom attractor and the critical noise intensity associated with it. Subsequently, the key nonlinear term related to the emergence of phantom attractors is identified by observing whether the system still exhibits phantom attractors after the corresponding nonlinear terms are removed. It is revealed that in this system, the presence of phantom attractors is closely linked to the inertia nonlinearity of the hardening type. The system investigated in this paper is simpler compared to previously identified systems capable of generating phantom attractors. This simplicity aids in facilitating research focused on unraveling the general principles behind the formation of phantom attractors.