scholarly journals Quasi-Periodic Oscillations of Roll System in Corrugated Rolling Mill in Resonance

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3201
Author(s):  
Dongping He ◽  
Huidong Xu ◽  
Tao Wang ◽  
Zhihua Wang
Keyword(s):  
Rolling Mill ◽  
Poincaré Map ◽  
Nonlinear Damping ◽  
Power Series Solution ◽  
Poincare Map ◽  
Periodic Oscillations ◽  
Existence And Stability ◽  
Roll System ◽  
Roller System ◽  
Two Degree Of Freedom

This paper investigates quasi-periodic oscillations of roll system in corrugated rolling mill in resonance. The two-degree of freedom vertical nonlinear mathematical model of roller system is established by considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill. In order to investigate the quasi-periodic oscillations at the resonance points, the Poincaré map is established by solving the power series solution of dynamic equations. Based on the Poincaré map, the existence and stability of quasi-periodic oscillations from the Neimark-Sacker bifurcation in the case of resonance are analyzed. The numerical simulation further verifies the correctness of the theoretical analysis.

Nonlinear time-delay feedback controllability for vertical parametrically excited vibration of roll system in corrugated rolling mill

2020 ◽  
Vol 117 (2) ◽  
pp. 210
Author(s):  
Dongping He ◽  
Huidong Xu ◽  
Tao Wang ◽  
Zhongkai Ren
Keyword(s):  
Time Delay ◽  
Rolling Mill ◽  
Nonlinear Damping ◽  
Feedback Controller ◽  
Nonlinear Stiffness ◽  
Parametrically Excited ◽  
Excited Vibration ◽  
Roll System ◽  
Roller System ◽  
Parametrically Excited Vibration

This paper investigates vibration characteristics of the corrugated roll system and designs a time-delay feedback controller to control the parametrically excited vibration of system. The model of parametrically excited nonlinear vertical vibration of roller system is established by considering the nonlinear damping and nonlinear stiffness within corrugated interface of corrugated rolling mill. The approximate analytical solution and amplitude-frequency characteristic equations of principal resonance and sub-resonance of roller system are obtained by using the multiple-scale method. The influences of nonlinear stiffness coefficient, nonlinear damping coefficient, system damping coefficient and rolling force amplitude on vibration are further analyzed. The time-delay feedback controller is designed to eliminate the jump and hysteresis phenomenon of the roll system and numerical simulation results demonstrate the effectiveness of the controller. The analysis results provide some theoretical guidance for vibration suppression of roller system of corrugated rolling mill.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

scholarly journals Poincaré Map Approach to Global Dynamics of the Integrated Pest Management Prey-Predator Model

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Zhenzhen Shi ◽  
Qingjian Li ◽  
Weiming Li ◽  
Huidong Cheng
Keyword(s):  
Periodic Solution ◽  
Integrated Pest Management ◽  
Pest Management ◽  
Poincaré Map ◽  
Global Dynamics ◽  
Poincare Map ◽  
Existence And Stability ◽  
Sufficient And Necessary Conditions ◽  
Predator Model ◽  
Order One Periodic Solution

An integrated pest management prey-predator model with ratio-dependent and impulsive feedback control is investigated in this paper. Firstly, we determine the Poincaré map which is defined on the phase set and discuss its main properties including monotonicity, continuity, and discontinuity. Secondly, the existence and stability of the boundary order-one periodic solution are proved by the method of Poincaré map. According to the Poincaré map and related differential equation theory, the conditions of the existence and global stability of the order-one periodic solution are obtained when ΦyA<yA, and we prove the sufficient and necessary conditions for the global asymptotic stability of the order-one periodic solution when ΦyA>yA. Furthermore, we prove the existence of the order-kk≥2 periodic solution under certain conditions. Finally, we verify the main results by numerical simulation.

BIFURCATION OF QUASI-PERIODIC OSCILLATIONS IN MUTUALLY COUPLED HARD-TYPE OSCILLATORS: DEMONSTRATION OF UNSTABLE QUASI-PERIODIC ORBITS

2012 ◽  
Vol 22 (06) ◽  
pp. 1230022 ◽  
Author(s):  
KYOHEI KAMIYAMA ◽  
MOTOMASA KOMURO ◽  
TETSURO ENDO
Keyword(s):  
Periodic Orbits ◽  
Poincaré Map ◽  
Averaging Method ◽  
Computer Algorithm ◽  
Bisection Method ◽  
Poincare Map ◽  
Periodic Oscillations ◽  
Closed Curves ◽  
Hard Type

In this paper, we obtain bifurcations of quasi-periodic orbits occuring in mutually coupled hard-type oscillators by using our recently developed computer algorithm to directly determine the unstable quasi-periodic orbits. So far, there is no computer algorithm to draw unstable invariant closed curves on a Poincare map representing quasi-periodic orbits. Recently, we developed a new algorithm to draw unstable invariant closed curves by using the bisection method. The results of this new algorithm are compared with the previously obtained averaging method results. Several new results are found, which could not be clarified by the averaging method.

scholarly journals Poincaré Map and Periodic Solutions of First-Order Impulsive Differential Equations on Moebius Stripe

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Yefeng He ◽  
Yepeng Xing
Keyword(s):  
Periodic Orbit ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Poincaré Map ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Poincare Map ◽  
First Order ◽  
Existence And Stability ◽  
Stability And Bifurcations

This paper is mainly concerned with the existence, stability, and bifurcations of periodic solutions of a certain scalar impulsive differential equations on Moebius stripe. Some sufficient conditions are obtained to ensure the existence and stability of one-side periodic orbit and two-side periodic orbit of impulsive differential equations on Moebius stripe by employing displacement functions. Furthermore, double-periodic bifurcation is also studied by using Poincaré map.

scholarly journals Holling-Tanner Predator-Prey Model with State-Dependent Feedback Control

2018 ◽  
Vol 2018 ◽  
pp. 1-18 ◽  
Author(s):  
Jin Yang ◽  
Guangyao Tang ◽  
Sanyi Tang
Keyword(s):  
Limit Cycle ◽  
Poincaré Map ◽  
Impulsive Control ◽  
Phase Portraits ◽  
Poincare Map ◽  
Complicated Shape ◽  
Predator Prey ◽  
Predator Prey Model ◽  
State Dependent ◽  
Existence And Stability

In this paper, we propose a novel Holling-Tanner model with impulsive control and then provide a detailed qualitative analysis by using theories of impulsive dynamical systems. The Poincaré map is first constructed based on the phase portraits of the model. Then the main properties of the Poincaré map are investigated in detail which play important roles in the proofs of the existence of limit cycles, and it is concluded that the definition domain of the Poincaré map has a complicated shape with discontinuity points under certain conditions. Subsequently, the existence of the boundary order-1 limit cycle is discussed and it is shown that this limit cycle is unstable. Furthermore, the conditions for the existence and stability of an order-1 limit cycle are provided, and the existence of order-k(k≥2) limit cycle is also studied. Moreover, numerical simulations are carried out to substantiate our results. Finally, biological implications related to the mathematical results which are beneficial for successful pest control are addressed in the Conclusions section.

Periodic Solution Bifurcation and Spiking Dynamics of Impacting Predator–Prey Dynamical Model

2018 ◽  
Vol 28 (12) ◽  
pp. 1850147 ◽  
Author(s):  
Sanyi Tang ◽  
Xuewen Tan ◽  
Jin Yang ◽  
Juhua Liang
Keyword(s):  
Periodic Solution ◽  
Poincaré Map ◽  
Complex Dynamics ◽  
Poincare Map ◽  
Predator Prey ◽  
Threshold Conditions ◽  
Existence And Stability ◽  
The Stability ◽  
Nonmonotonic Functional Response ◽  
Solution Bifurcation

A planar predator–prey impacting system model with a nonmonotonic functional response function is proposed and analyzed. The existence and stability of a boundary order-1 periodic solution were investigated and the threshold conditions for a transcritical bifurcation and stable switching were obtained, and also the definition and properties of the Poincaré map are discussed. The main results indicate that multiple discontinuous points of the Poincaré map could induce the coexistence of multiple order-1 periodic solutions. Numerical analyses reveal the complex dynamics of the model including periodic adding and halving bifurcations, which could result in multiple active phases, among them rapid spiking and quiescence phases which can switch from one to another and consequently create complex bursting patterns. The main results reveal that it is beneficial to restore the stability and balance of a ecosystem for species with group defence by moderately reducing population densities and the group defence capacity.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

BIFURCATION ANALYSIS OF CURRENT COUPLED BVP OSCILLATORS

2007 ◽  
Vol 17 (03) ◽  
pp. 837-850 ◽  
Author(s):  
SHIGEKI TSUJI ◽  
TETSUSHI UETA ◽  
HIROSHI KAWAKAMI
Keyword(s):  
Periodic Solutions ◽  
Poincaré Map ◽  
Dynamical Behavior ◽  
Coupled Systems ◽  
Poincare Map ◽  
Circuit Implementation ◽  
Coupling Structure ◽  
Bifurcation Phenomena ◽  
Junctional Coupling ◽  
Current Coupling

The Bonhöffer–van der Pol (BVP) oscillator is a simple circuit implementation describing neuronal dynamics. Lately the diffusive coupling structure of neurons attracts much attention since the existence of the gap-junctional coupling has been confirmed in the brain. Such coupling is easily realized by linear resistors for the circuit implementation, however, there are not enough investigations about diffusively coupled BVP oscillators, even a couple of BVP oscillators. We have considered several types of coupling structure between two BVP oscillators, and discussed their dynamical behavior in preceding works. In this paper, we treat a simple structure called current coupling and study their dynamical properties by the bifurcation theory. We investigate various bifurcation phenomena by computing some bifurcation diagrams in two cases, symmetrically and asymmetrically coupled systems. In symmetrically coupled systems, although all internal elements of two oscillators are the same, we obtain in-phase, anti-phase solution and some chaotic attractors. Moreover, we show that two quasi-periodic solutions disappear simultaneously by the homoclinic bifurcation on the Poincaré map, and that a large quasi-periodic solution is generated by the coalescence of these quasi-periodic solutions, but it disappears by the heteroclinic bifurcation on the Poincaré map. In the other case, we confirm the existence a conspicuous chaotic attractor in the laboratory experiments.

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