A periodic vibrational resonance in the fractional-order bistable system
Authors: Yang Jian-Hua  , Ma Qiang  , Wu Cheng-Jin  , Liu Hou-Guang  ,   School of Mechatronic Engineering, China University of Mining and Technology, Xuzhou 221116, China;,   Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109, Ameirica;,   Jiangsu Key Laboratory of Mine Mechanical and Electrical Equipment, China University of Mining and Technology, Xuzhou 221116, China;,   College of Mechanical and Equipment Engineering, Hebei University of Engineering, Handan 056038, China
Published: December 2020

Aperiodic signal is widely used in different engineering fields.It is important to detect or enhance a weak aperiodic signal.In this work,we investigate the aperiodic vibrational resonance (AVR) in a fractional-order bistable system excited by an aperiodic binary signal and a square waveform signal simultaneously.The weak aperiodic binary signal is the characteristic signal which usually carries the useful information.The square waveform signal is the auxiliary signal which is used to induce the AVR.By tuning the amplitude of the auxiliary signal,the AVR may occur and the aperiodic binary signal is enhanced.The occurrence of the AVR is measured by the cross-correlated coefficient between the input aperiodic binary signal and the output time series.When the cross-correlated coefficient achieves a large enough value, the AVR may occur and the weak aperiodic signal is enhanced excellently by the auxiliary signal.If the aperiodic binary signal has large pulse width and the system has small parameters (usually on the order of 1),the AVR can be realized by tuning the amplitude of the square waveform.If the aperiodic binary signal has small pulse width,the AVR cannot be realized in the system with small parameters directly.For this case,we realize the AVR by the re-scaled method and the twice sampling method separately.By the re-scaled method,through a scale transformation,the equivalent system with large system parameters can match the input characteristic signal with arbitrary small pulse width.When the re-scaled method is used,the scale parameter is a key factor.By the twice sampling method,the reconstructed characteristic signal after the twice sampling has a large pulse width.Then,it can match the original system with small system parameters.When the twice sampling method is used,the ratio of the twice sampling frequency to the first sampling frequency is a key factor.Although these two methods have different physical processes,they can achieve the same goal. The AVR also depends on the fractional-order value closely.Specifically,with the increase of the fractional-order,the resonance region in the cross-correlated coefficient curve turns wider.Moreover,the amplitude of the square waveform signal which induces the optimal AVR to turn larger.Simultaneously,the similarity between the optimal output and the input binary aperiodic signal is enhanced.The method and the results of this paper not only can be used to enhance the weak aperiodic binary signal but also have a certain reference value in processing other kinds of aperiodic signals, such as the linear or nonlinear frequency modulated signal,etc.Furthermore,the results in this paper also present rich dynamical behaviors of a fractional-order system and may provide reference value in the study of fractional-order systems.