Investigations of Stochastic Layers in Nonlinear Dynamics

1999 ◽  
Vol 122 (1) ◽  
pp. 36-41 ◽  
Author(s):  
Albert C. J. Luo ◽  
Ray P. S. Han
Keyword(s):  
Nonlinear Dynamics ◽  
Numerical Investigation ◽  
Duffing Oscillator ◽  
Minimum Energy ◽  
Energy Approach ◽  
Energy Spectra ◽  
Poincaré Mapping ◽  
Periodically Driven ◽  
Poincare Mapping ◽  
Stochastic Layer

The onset of a new resonance in the stochastic layer is predicted numerically through the maximum and minimum energy spectra when the energy jump in the spectra occurs. The incremental energy approach among all the established, analytic approaches gives the best prediction of the onset of resonance in the stochastic layer compared to numerical investigation. The stochastic layers in the periodically-driven pendulum are discussed as another example. Illustrations of stochastic layers in the twin-well Duffing oscillator and the periodically-driven pendulum are given through the Poincare´ mapping sections. [S0739-3717(00)00701-7]

Resonant Layers in Nonlinear Dynamics

1998 ◽  
Vol 65 (3) ◽  
pp. 727-736 ◽  
Author(s):  
R. P. S. Han ◽  
A. C. J. Luo
Keyword(s):  
Nonlinear Dynamics ◽  
Renormalization Group ◽  
Numerical Simulations ◽  
Duffing Oscillator ◽  
Energy Approach ◽  
New Method ◽  
Mapping Technique ◽  
Standard Mapping ◽  
Left And Right ◽  
Group Technique

A new method based on an incremental energy approach and the standard mapping technique is proposed for the study of resonant layers in nonlinear dynamics. To demonstrate the procedure, the method is applied to four types of Duffing oscillators. The appearance, disappearance and accumulated disappearance strengths of the resonant layers for each type of oscillator are derived. A quantitative check of the appearance strength is performed by computing its value using three independent methods: Chirikov overlap criterion, renormalization group technique, and numerical simulations. It is also observed that for the case of the twin-well Duffing oscillator, its perturbed left and right wells are asymmetric.

GLOBAL TANGENCY AND TRANSVERSALITY OF PERIODIC FLOWS AND CHAOS IN A PERIODICALLY FORCED, DAMPED DUFFING OSCILLATOR

2008 ◽  
Vol 18 (01) ◽  
pp. 1-49 ◽  
Author(s):  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
First Integral ◽  
Poincaré Mapping ◽  
Nonlinear Dynamical ◽  
Periodic Flows ◽  
Poincare Mapping ◽  
Integral Quantity ◽  
Energy Increment

This paper presents how to apply a newly developed general theory for the global transversality and tangency of flows in n-dimensional nonlinear dynamical systems to a 2-D nonlinear dynamical system (i.e. a periodically forced, damped Duffing oscillator). The global tangency and transversality of the periodic and chaotic motions to the separatrix for such a nonlinear system are discussed to help us understand the complexity of chaos in nonlinear dynamical systems. This paper presents the concept that the global transversality and tangency to the separatrix are independent of the Melnikov function (or the energy increment). Chaos in nonlinear dynamical systems makes the exact energy increment quantity to be chaotic no matter if the nonlinear dynamical systems have separatrices or not. The simple zero of the Melnikov function cannot be used to simply determine the existence of chaos in nonlinear dynamical systems. Through this paper, the expectation is that, from now on, one can use the alternative aspect to look into the complexity of chaos in nonlinear dynamical systems. Therefore, in this paper, the analytical conditions for global transversality and tangency of 2-D nonlinear dynamical systems are presented. The first integral quantity increment (i.e. the energy increment) for a certain time interval is achieved for periodic flows and chaos in the 2-D nonlinear dynamical systems. Under the perturbation assumptions and convergent conditions, the Melnikov function is recovered from the first integral quantity increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are obtained and verified by numerical simulations. The switching planes and the corresponding local and global mappings are defined on the separatrix. The mapping structures are developed for local and global periodic flows passing through the separatrix. The mapping structures of global chaos in the damped Duffing oscillator are also discussed. Bifurcation scenarios of the damped Duffing oscillator are presented through the traditional Poincaré mapping section and the switching planes. The first integral quantity increment (i.e. L-function) is presented to observe the periodicity of flows. In addition, the global tangency of periodic flows in such an oscillator is measured by the G-function and G(1)-function, and is verified by numerical simulations. The first integral quantity increment of periodic flows is zero for their complete periodic cycles. Numerical simulations of chaos in such a Duffing oscillator are carried out through the Poincaré mapping sections. The conservative energy distribution, G-function and L-function along the displacement of Poincaré mapping points are presented to observe the complexity of chaos. The first integral quantity increment (i.e. L-function) of chaotic flows at the Poincaré mapping points is nonzero and chaotic. The switching planes of chaos are presented on the separatrix for a better understanding of the global transversality to the separatrix. The switching point distribution on the separatrix is presented and the switching G-function on the separatrix is given to show the global transversality of chaos on the separatrix. The analytical conditions are obtained from the new theory rather than the Melnikov method. The new conditions for the global transversality and tangency are more accurate and independent of the small parameters.

scholarly journals An Improved Method for Estimating the Domain of Attraction of Passive Biped Walker

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Yu Wang ◽  
Heng Cao ◽  
JinLin Jiang
Keyword(s):  
Domain Of Attraction ◽  
Mapping Method ◽  
Simple Cell ◽  
Iteration Algorithm ◽  
Poincaré Mapping ◽  
Cell Mapping ◽  
Improved Method ◽  
Stable Domain ◽  
Simple Cell Mapping ◽  
Poincare Mapping

An indicator of a passive biped walker’s global stability is its domain of attraction, which is usually estimated by the simple cell mapping method. It needs to calculate a large number of cells’ Poincare mapping result in the estimating process. However, the Poincare mapping is usually computationally expensive and time-consuming due to the complex dynamical equation of the passive biped walker. How to estimate the domain of attraction efficiently and reliably is a problem to be solved. Based on the simple cell mapping method, an improved method is proposed to solve it. The proposed method uses the multiple iteration algorithm to calculate a stable domain of attraction and effectively decreases the total number of Poincare mappings. Through the simulation of the simplest passive biped walker, the improved method can obtain the same domain of attraction as that calculated using the simple cell mapping method and reduce calculation time significantly. Furthermore, this improved method not only proposes a way of rapid estimating the domain of attraction, but also provides a feasible tool for selecting the domain of interest and its discretization level.

scholarly journals Study on Clearance-Rubbing Dynamic Behavior of 2-DOF Supporting System of Magnetic-Liquid Double Suspension Bearing

Processes ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 973
Author(s):  
Jianhua Zhao ◽  
Weidong Yan ◽  
Ziqi Wang ◽  
Dianrong Gao ◽  
Guojun Du
Keyword(s):  
Stable Operation ◽  
Poincaré Mapping ◽  
Eccentricity Ratio ◽  
Magnetic Liquid ◽  
Rotating Speed ◽  
Single Period ◽  
Operating Stability ◽  
Poincare Mapping ◽  
Assembly Error ◽  
Mapping Point

As a new type of suspension bearing, Magnetic-Liquid Double Suspension Bearing (MLDSB) is mainly supported by electromagnetic suspension and supplemented by hydrostatic supporting. Its bearing capacity and stiffness can be greatly improved. Because of the small liquid film thickness (it is smaller 10 times than air gap), the eccentricity, crack, bending of the rotor, and the assembly error, it is easy to cause a clearance-rubbing fault between the rotor and stator. The coating can be worn and peeled, the operating stability can be reduced, and then it is one of the key problems of restricting the development and application of MLDSB. Therefore, the clearance-rubbing dynamic equation of 2-DOF system of MLDSB is established and converted into Taylor Series form and the nonlinear components are retained. Dimensionless treatment is carried out by dimensional normalization method. Finally, the rotor displacement response under different rotor eccentricity ratio and rotating speeds is numerically simulated. The studies show that the trajectory of the rotor is periodic elliptic without clearance-rubbing phenomenon when the eccentricity ratio is less than 0.2, while the rotor is greatly affected by the rotation speed and a variety of motions, such as single-period, quasi-period, double-period and chaos, are presented when greater than 0.3. Within the largest range of rotating speed and eccentricity ratio, the rotor presents the single-period trajectory, and then the number of Poincare mapping point is 1, without a clearance-rubbing fault. When the rotational speed is in the scope of (9, 13) krpm and the eccentricity ratio is in the scope of (0.27, 0.4), the number of Poincare mapping point is more than one, the maximum dimensionless rubbing force is −5.7, and then clearance-rubbing fault occurs. The research can provide a theoretical basis for the safe and stable operation of MLDSB.

Sign in / Sign up

Export Citation Format

Share Document