Vibrational resonance in Duffing systems with fractional-order damping

2012 ◽  
Vol 22 (1) ◽  
pp. 013112 ◽  
Author(s):  
J. H. Yang ◽  
H. Zhu
Keyword(s):  
Fractional Order ◽  
Vibrational Resonance

Vibrational Resonance in an Overdamped System with a Fractional Order Potential Nonlinearity

2018 ◽  
Vol 28 (07) ◽  
pp. 1850082 ◽  
Author(s):  
Jianhua Yang ◽  
Dawen Huang ◽  
Miguel A. F. Sanjuán ◽  
Houguang Liu
Keyword(s):  
Numerical Simulation ◽  
Nonlinear System ◽  
Theoretical Analysis ◽  
Fractional Order ◽  
Low Frequency ◽  
Response Amplitude ◽  
Resonance Phenomenon ◽  
Vibrational Resonance ◽  
Frequency Excitation ◽  
Overdamped System

We investigate the vibrational resonance by the numerical simulation and theoretical analysis in an overdamped system with fractional order potential nonlinearities. The nonlinearity is a fractional power function with deflection, in which the response amplitude presents vibrational resonance phenomenon for any value of the fractional exponent. The response amplitude of vibrational resonance at low-frequency is deduced by the method of direct separation of slow and fast motions. The results derived from the theoretical analysis are in good agreement with those of numerical simulation. The response amplitude decreases with the increase of the fractional exponent for weak excitations. The amplitude of the high-frequency excitation can induce the vibrational resonance to achieve the optimal response amplitude. For the overdamped systems, the nonlinearity is the crucial and necessary condition to induce vibrational resonance. The response amplitude in the nonlinear system is usually not larger than that in the corresponding linear system. Hence, the nonlinearity is not a sufficient factor to amplify the response to the low-frequency excitation. Furthermore, the resonance may be also induced by only a single excitation acting on the nonlinear system. The theoretical analysis further proves the correctness of the numerical simulation. The results might be valuable in weak signal processing.

Saddle-Node Bifurcation and Vibrational Resonance in a Fractional System with an Asymmetric Bistable Potential

2015 ◽  
Vol 25 (02) ◽  
pp. 1550023 ◽  
Author(s):  
J. H. Yang ◽  
Miguel A. F. Sanjuán ◽  
F. Tian ◽  
H. F. Yang
Keyword(s):  
Potential Function ◽  
Fractional Order ◽  
Nonlinear Dynamical Systems ◽  
Technical Problem ◽  
Vibrational Resonance ◽  
High Frequency Signal ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Bistable Potential ◽  
Fractional System

We investigate the saddle-node bifurcation and vibrational resonance in a fractional system that has an asymmetric bistable potential. Due to the asymmetric nature of the potential function, the response and its amplitude closely depend on the potential well where the motion takes place. And consequently for numerical simulations, the initial condition is a key and important factor. To overcome this technical problem, a method is proposed to calculate the bifurcation and response amplitude numerically. The numerical results are in good agreement with the analytical predictions, indicating the validity of the numerical and theoretical analysis. The results show that the fractional-order of the fractional system induces one saddle-node bifurcation, while the asymmetric parameter associated to the asymmetric nature of the potential function induces two saddle-node bifurcations. When the asymmetric parameter vanishes, the saddle-node bifurcation turns into a pitchfork bifurcation. There are three kinds of vibrational resonance existing in the system. The first one is induced by the high-frequency signal. The second one is induced by the fractional-order. The third one is induced by the asymmetric parameter. We believe that the method and the results shown in this paper might be helpful for the analysis of the response problem of nonlinear dynamical systems.

Pitchfork bifurcation and vibrational resonance in a fractional-order Duffing oscillator

Pramana ◽  
2013 ◽  
Vol 81 (6) ◽  
pp. 943-957 ◽  
Author(s):  
J H YANG ◽  
M A F SANJUÁN ◽  
W XIANG ◽  
H ZHU
Keyword(s):  
Fractional Order ◽  
Duffing Oscillator ◽  
Pitchfork Bifurcation ◽  
Vibrational Resonance

Amplification of the LFM signal by using piecewise vibrational methods

2018 ◽  
Vol 25 (1) ◽  
pp. 141-150 ◽  
Author(s):  
Pengxiang Jia ◽  
Jianhua Yang ◽  
Chengjin Wu ◽  
Miguel A.F. Sanjuán
Keyword(s):  
Fractional Order ◽  
Bistable System ◽  
Optimal Model ◽  
Vibrational Resonance ◽  
Linear Frequency

We propose the piecewise re-scaled vibrational resonance (VR) method and the piecewise twice sampling VR method to amplify the weak linear frequency-modulated (LFM) signal. The system used to amplify the weak LFM signal is a typical bistable system with fractional-order deflection nonlinearity. The concrete procedures of both the piecewise re-scaled VR method and the piecewise twice sampling VR method are explained in detail. Through studying the effect of the factional-order exponent on VR, we find that the traditional bistable system is not the optimal model to improve the weak LFM signal. By investigating different parameters on the VR phenomenon, we verify the effectiveness of the two proposed methods.

Enhancing the Weak Signal With Arbitrary High-Frequency by Vibrational Resonance in Fractional-Order Duffing Oscillators

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
J. H. Yang ◽  
Miguel A. F. Sanjuán ◽  
H. G. Liu
Keyword(s):  
Fractional Order ◽  
High Frequency ◽  
Low Frequency ◽  
Original System ◽  
Scale Transformation ◽  
Weak Signal ◽  
Vibrational Resonance ◽  
High Frequency Signal ◽  
System A ◽  
Auxiliary Signal

When the traditional vibrational resonance (VR) occurs in a nonlinear system, a weak character signal is enhanced by an appropriate high-frequency auxiliary signal. Here, for the harmonic character signal case, the frequency of the character signal is usually smaller than 1 rad/s. The frequency of the auxiliary signal is dozens of times of the frequency of the character signal. Moreover, in the real world, the characteristic information is usually indicated by a weak signal with a frequency in the range from several to thousands rad/s. For this case, the weak high-frequency signal cannot be enhanced by the traditional mechanism of VR, and as such, the application of VR in the engineering field could be restricted. In this work, by introducing a scale transformation, we transform high-frequency excitations in the original system to low-frequency excitations in a rescaled system. Then, we make VR to occur at the low frequency in the rescaled system, as usual. Meanwhile, the VR also occurs at the frequency of the character signal in the original system. As a result, the weak character signal with arbitrary high-frequency can be enhanced. To make the rescaled system in a general form, the VR is investigated in fractional-order Duffing oscillators. The form of the potential function, the fractional order, and the reduction scale are important factors for the strength of VR.

scholarly journals A periodic vibrational resonance in the fractional-order bistable system

2018 ◽  
Vol 67 (5) ◽  
pp. 054501
Author(s):  
Yang Jian-Hua ◽  
Ma Qiang ◽  
Wu Cheng-Jin ◽  
Liu Hou-Guang
Keyword(s):  
Fractional Order ◽  
Bistable System ◽  
Vibrational Resonance

scholarly journals Vibrational Resonance and Electrical Activity Behavior of a Fractional-Order FitzHugh–Nagumo Neuron System

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 87
Author(s):  
Jia-Wei Mao ◽  
Dong-Liang Hu
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Neuron Model ◽  
Low Frequency ◽  
Order Model ◽  
Vibrational Resonance ◽  
Frequency Signal ◽  
Neuron System ◽  
Order Neuron ◽  
Activity Behavior

Making use of the numerical simulation method, the phenomenon of vibrational resonance and electrical activity behavior of a fractional-order FitzHugh–Nagumo neuron system excited by two-frequency periodic signals are investigated. Based on the definition and properties of the Caputo fractional derivative, the fractional L1 algorithm is applied to numerically simulate the phenomenon of vibrational resonance in the neuron system. Compared with the integer-order neuron model, the fractional-order neuron model can relax the requirement for the amplitude of the high-frequency signal and induce the phenomenon of vibrational resonance by selecting the appropriate fractional exponent. By introducing the time-delay feedback, it can be found that the vibrational resonance will occur with periods in the fractional-order neuron system, i.e., the amplitude of the low-frequency response periodically changes with the time-delay feedback. The weak low-frequency signal in the system can be significantly enhanced by selecting the appropriate time-delay parameter and the fractional exponent. In addition, the original integer-order model is extended to the fractional-order model, and the neuron system will exhibit rich dynamical behaviors, which provide a broader understanding of the neuron system.

On the Aperiodic Signal Amplification by Rescaled Vibrational Resonance in Fractional-Order Duffing Oscillators

2020 ◽  
Vol 15 (7) ◽  
Author(s):  
Pengxiang Jia ◽  
Yonggang Leng ◽  
Jianhua Yang
Keyword(s):  
Fractional Order ◽  
High Frequency ◽  
Signal Amplification ◽  
Vibrational Resonance ◽  
System Parameters ◽  
Amplification Method ◽  
Signal Parameters ◽  
Practical Engineering ◽  
Common Signal ◽  
Aperiodic Signal

Abstract In the former works of Yang et al., they put forward two vibrational resonance (VR) methods in fractional Duffing oscillators to amplify the weak harmonic character signal with arbitrary high-frequency. Moreover, the aperiodic character signal is also a common signal form in practical engineering. For the “fast-changing” aperiodic character signal in time domain, the classical VR method is difficult to amplify it effectively. To solve this problem, an aperiodic signal amplification method based on rescaled VR in fractional-order Duffing oscillators is proposed. Take the bipolar binary random signals with arbitrary minimal pulse width as an example, the rescaled VR under the overdamped and underdamped conditions can be realized by matching the signal parameters and system parameters effectively, and the amplification of aperiodic signal can be achieved.

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