Study of the Bifurcation of a Multiple Limit Cycle of the Second Kind by Means of a Dulac–Cherkas Function: A Case Study

2016 ◽  
Vol 26 (14) ◽  
pp. 1650229 ◽  
Author(s):  
Alexander Grin ◽  
Klaus R. Schneider
Keyword(s):  
Limit Cycle ◽  
Scalar Parameter ◽  
Pendulum Equation ◽  
Multiple Limit Cycle ◽  
Bifurcation Behavior

We consider a generalized pendulum equation depending on the scalar parameter [Formula: see text] having for [Formula: see text] a limit cycle [Formula: see text] of the second kind and of multiplicity three. We study the bifurcation behavior of [Formula: see text] for [Formula: see text] by means of a Dulac–Cherkas function.

Large scale motions of multiple limit-cycle high Reynolds number annular and toroidal rotor/stator cavities

2017 ◽  
Vol 29 (6) ◽  
pp. 065115 ◽  
Author(s):  
Thibault Bridel-Bertomeu ◽  
L. Y. M. Gicquel ◽  
G. Staffelbach
Keyword(s):  
Reynolds Number ◽  
Limit Cycle ◽  
High Reynolds Number ◽  
Large Scale ◽  
Multiple Limit Cycle ◽  
High Reynolds

Multiple Limit Cycles Shimmy of the Dual-Front Axle Steering Heavy Truck Based on Bisectional Road

2019 ◽  
Vol 14 (5) ◽  
Author(s):  
Daogao Wei ◽  
Yingjie Zhu ◽  
Tong Jiang ◽  
Andong Yin ◽  
Wenhao Zhai
Keyword(s):  
Dynamic Model ◽  
Limit Cycle ◽  
Dry Friction ◽  
Bifurcation Theory ◽  
Steering System ◽  
Front Axle ◽  
Adhesion Coefficient ◽  
The Road ◽  
Heavy Truck ◽  
Multiple Limit Cycle

The shimmy problem causes considerable harm to vehicles and is difficult to solve, especially multiple limit cycle shimmy. Moreover, the dynamic behavior of the multiple limit cycle shimmy of vehicles based on a bisectional road is more complex. Shimmy is practically observed in trucks of cooperative factories during utilization. Thus, we take a heavy truck of a cooperative factory as the prototype and establish a dynamic model of the vehicle-road coupling shimmy system, considering the road adhesion coefficient and dry friction between the suspension and steering system. Based on the dynamic model, the Hopf bifurcation theory is used to qualitatively analyze the existence of the limit cycle for the vehicle shimmy system, and the multiple limit cycle shimmy phenomenon is successfully reproduced using a numerical method. Moreover, the effect of the road adhesion coefficient on the multiple limit cycle shimmy characteristic is studied. Results show that the speed interval and amplitude of the multiple limit cycle shimmy decrease with the road adhesion coefficient; when the coefficient is reduced to a certain extent, the multiple limit cycle shimmy phenomenon is not observed. In addition, the adhesion coefficient of the second axle has a stronger effect on the shimmy characteristic than that of the first axle.

On Multiple Hopf Bifurcations of Airflow Excited Vibration of a Translating String

2007 ◽  
Author(s):  
Yuefang Wang ◽  
Lefeng Lu¨ ◽  
Yingxi Liu
Keyword(s):  
Limit Cycle ◽  
Harmonic Balance ◽  
Equilibrium Configuration ◽  
Transverse Motion ◽  
Hopf Bifurcations ◽  
Flip Bifurcation ◽  
Excited Vibration ◽  
Incremental Harmonic Balance ◽  
The Stability ◽  
Bifurcation Behavior

This paper presents the stability and bifurcation of transverse motion of translating strings excited by a steady wind flowfield. The stability of the equilibrium configuration is presented for loss of stability and generation of limit cycles via the Hopf bifurcation. It is demonstrated that there are single, double and quadruple Hopf bifurcations in the parametric space that lead to the limit cycle motion. The method of Incremental Harmonic Balance is used to solve the limit cycle response of which the stability is determined by computation of the Floquet multipliers. For the forced vibration, it is pointed out that the periodic and quasi-periodic motions exist as parameters are changed. The quench frequency and the Neimark-Sacker (NS) bifurcation and flip bifurcation are obtained. The continuity software MATCONT is adopted and the Resonance 1:1, 1:3 and 1:4 as well as NS to NS bifurcations are presented. The bifurcation behavior reveals the complexity of the string’s motion response induce by aerodynamic excitations.

Diffusive broadening of limit cycle in presence of noise: A case study of reversible Brusselator

1995 ◽  
Vol 39 (1-2) ◽  
pp. 115-124 ◽  
Author(s):  
S.S. Tambe ◽  
S.R. Inamdar ◽  
B.D. Kulkarni
Keyword(s):  
Limit Cycle

scholarly journals The Calculation Process of the Limit Cycle for Lorenz System

2021 ◽  
Author(s):  
Hao Dong ◽  
Bin Zhong
Keyword(s):  
Numerical Method ◽  
Limit Cycle ◽  
Equilibrium Solution ◽  
Lorenz System ◽  
Continuation Method ◽  
Equilibrium Solutions ◽  
Periodical Solution ◽  
Calculation Process ◽  
Continuation Algorithm ◽  
Bifurcation Behavior

This work focuses on the bifurcation behavior before chaos phenomenon happens. Traditional numerical method is unable to solve the unstable limit cycle of nonlinear system. One algorithm is introduced to solve the unstable one, which is based one of the continuation method is called DEPAR approach. Combined with analytic and numerical method, the two stable and symmetrical equilibrium solutions exist through Fork bifurcation and the unstable and symmetrical limit cycles exist through Hopf bifurcation of Lorenz system. With the continuation algorithm, the bifurcation behavior and its phase diagram is solved. The results demonstrate the unstable periodical solution is around the equilibrium solution, besides the trajectory into the unstable area cannot escape but only converge to the equilibrium solution.

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