STRONG SENSITIVITY OF THE VIBRATIONAL RESONANCE INDUCED BY FRACTAL STRUCTURES

2013 ◽  
Vol 23 (07) ◽  
pp. 1350129 ◽  
Author(s):  
ALVAR DAZA ◽  
ALEXANDRE WAGEMAKERS ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Phase Space ◽  
Nonlinear System ◽  
Oscillatory Regime ◽  
Fractal Structures ◽  
Periodic Forcing ◽  
Vibrational Resonance ◽  
Extreme Sensitivity

We consider a nonlinear system perturbed by two harmonic forcings of different frequencies. The slow forcing drives the system into an oscillatory regime while the fast perturbation enhances the effect of the slow periodic drive. The vibrational resonance occurs when this enhancement is optimal, usually when the fast perturbation has an amplitude much higher than the slow periodic forcing. We show that this resonance can also happen when the amplitude of the fast perturbation is far below the amplitude of the slow periodic forcing due to a peculiar condition of the phase space. Moreover, this resonance presents an extreme sensitivity to small variations of the fast perturbation. We explore here this phenomenon that we call ultrasensitive 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.

TO ESCAPE OR NOT TO ESCAPE, THAT IS THE QUESTION — PERTURBING THE HÉNON–HEILES HAMILTONIAN

2012 ◽  
Vol 22 (06) ◽  
pp. 1230010 ◽  
Author(s):  
FERNANDO BLESA ◽  
JESÚS M. SEOANE ◽  
ROBERTO BARRIO ◽  
MIGUEL A. F. SANJUÁN
Keyword(s):  
Phase Space ◽  
Hamiltonian Systems ◽  
Survival Probability ◽  
Crucial Role ◽  
Physical Space ◽  
Periodic Forcing

In this work, we study the Hénon–Heiles Hamiltonian, as a paradigm of open Hamiltonian systems, in the presence of different kinds of perturbations as dissipation, noise and periodic forcing, which are very typical in different physical situations. We focus our work on both the effects of these perturbations on the escaping dynamics and on the basins associated to the phase space and to the physical space. We have also found, in presence of a periodic forcing, an exponential-like decay law for the survival probability of the particles in the scattering region where the frequency of the forcing plays a crucial role. In the bounded regions, the use of the OFLI2 chaos indicator has allowed us to characterize the orbits. We have compared these results with the previous ones obtained for the dissipative and noisy case. Finally, we expect this work to be useful for a better understanding of the escapes in open Hamiltonian systems in the presence of different kinds of perturbations.

Localization Method of Compact Invariant Sets with Application to the Chua System

2016 ◽  
Vol 26 (05) ◽  
pp. 1650073 ◽  
Author(s):  
A. F. Gribov ◽  
A. N. Kanatnikov ◽  
A. P. Krishchenko
Keyword(s):  
Phase Space ◽  
Nonlinear System ◽  
Bifurcation Analysis ◽  
Invariant Set ◽  
Nonempty Intersection ◽  
Invariant Sets ◽  
Localization Method ◽  
Poincaré Sections ◽  
Systems Of Inequalities ◽  
Parametric Families

In this paper, we consider the problem of compact invariant sets localization for the Chua system. To obtain our results we develop and apply a localization method. This method allows us to find two types of subsets in the phase space of a nonlinear system. The first type consists of Poincaré sections having a nonempty intersection with any compact invariant set of the system. The second type consists of localizing sets containing all compact invariant sets of the system. The considered localization method produces systems of inequalities describing the localizing sets and specifies the equations of the appropriate global sections. These inequalities and equations depend on parameters of the system and, therefore, the obtained localization results can be used in the bifurcation analysis. We find one-parametric families of both compact global sections and nontrivial localizing sets for the Chua system. These localizing sets are compact or unbounded. The intersection of unbounded localizing sets in some cases is a compact localizing set. We indicate the domains where trajectories of the Chua system go to infinity.

Interpolated Cell Mapping of Dynamical Systems

1988 ◽  
Vol 55 (2) ◽  
pp. 461-466 ◽  
Author(s):  
B. H. Tongue ◽  
K. Gu
Keyword(s):  
Dynamical System ◽  
Steady State ◽  
Phase Space ◽  
Nonlinear System ◽  
Basins Of Attraction ◽  
Mapping Technique ◽  
Cell Mapping ◽  
Simple Cell Mapping ◽  
Steady State Solutions ◽  
Continuous Problem

A method is proposed to efficiently determine the basins of attraction of a nonlinear system’s different steady-state solutions. The phase space of the dynamical system is spacially discretized and the continuous problem in time is converted to an iterative mapping. By means of interpolation procedures, an improvement in the system accuracy over the Simple Cell Mapping technique is achieved. Both basins of attraction for a representative nonlinear system and characteristic system trajectories are generated and compared to exact solutions.

scholarly journals Enhanced Bearing Fault Detection Using Step-Varying Vibrational Resonance Based on Duffing Oscillator Nonlinear System

2017 ◽  
Vol 2017 ◽  
pp. 1-14 ◽  
Author(s):  
Yongbin Liu ◽  
Zhijia Dai ◽  
Siliang Lu ◽  
Fang Liu ◽  
Jiwen Zhao ◽  
...  
Keyword(s):  
Nonlinear System ◽  
Fault Detection ◽  
Background Noise ◽  
Duffing Oscillator ◽  
Normal Operation ◽  
Vibrational Resonance ◽  
Gaussian Potential ◽  
Rotary Machines ◽  
The Status ◽  
Status Information

Bearing is a key part of rotary machines, and its working condition is critical in normal operation of rotary machines. Vibrational signals are usually analyzed to monitor the status of bearing. However, information on the status of bearing is always buried in heavy background noise; that is, status information of bearing is weaker than the background noise. Extracting the status features of bearing from signals buried in noise is difficult. Given this, a step-varying vibrational resonance (SVVR) method based on Duffing oscillator nonlinear system is proposed to enhance the weak status feature of bearing by tuning different parameters. Extraction ability of SVVR was verified by analyzing simulation signal and practical bearing signal. Experimental results show that SVVR is more effective in extracting weak characteristic information than other methods, including multiscale noise tuning stochastic resonance (SR), Woods–Saxon potential-based SR, and joint Woods–Saxon and Gaussian potential-based SR. Two evaluation indices are investigated to qualitatively and quantitatively assess the fault detection capability of the SVVR method. The results show that the SVVR can effectively identify the weak status information of bearing.

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