Mode-Locking and Chaos in a Jeffcott Rotor With Bearing Clearances

1994 ◽  
Vol 61 (1) ◽  
pp. 131-138 ◽  
Author(s):  
Sang-Kyu Choi ◽  
Sherif T. Noah
Keyword(s):  
Fixed Point ◽  
Periodic Solutions ◽  
Period Doubling ◽  
Mode Locking ◽  
Rotor System ◽  
Winding Number ◽  
Fixed Point Algorithm ◽  
Jeffcott Rotor ◽  
Multiple Periodic Solutions ◽  
The Stability

A complex mode-locking (or entrainment) structure underlying the nonlinear whirling phenomenon of a horizontal Jeffcott rotor with a discontinuous nonlinearity (bearing clearance) was identified. A winding number is introduced as a measure of the ratio between two frequencies involved in the aperiodic whirling motions of the rotor system considered. Utilizing the winding number map, it was revealed that the alternating periodic and quasi-periodic responses take place according to the Farey number tree. The winding number varies in the form of the so-called “Devil’s staircase” as a certain system parameter varies. From the mode-locking pattern in the parameter space of the forcing amplitude and frequency, it was observed that as the forcing amplitude increases, the size of each locking interval increases so that its growth takes place in the form of “Arnol’d tongues,” where the winding number remains a rational number. Moreover, inside each locking zone, i.e., each “Arnol’d tongue,” there exist many smaller tongues similar to the main tongue, in which a sequence of period-doubling bifurcations leading to chaos occurred. The boundaries of each locking zone was obtained using a fixed-point algorithm along with the Floquet theory for checking the stability of the periodic solutions. The winding numbers were estimated utilizing a fixed-point algorithm modified to obtain quasi-periodic responses. A jump phenomenon was also observed by tracking multiple periodic solutions for several parameters of the rotor system.

Nonlinear Dynamics of a Magnetically Supported Rotor on Safety Auxiliary Bearings

1998 ◽  
Vol 120 (2) ◽  
pp. 596-606 ◽  
Author(s):  
X. Wang ◽  
S. Noah
Keyword(s):  
Steady State ◽  
Periodic Solutions ◽  
Nonlinear Dynamic System ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Fixed Point Algorithm ◽  
Friction Forces ◽  
Flexible Supports ◽  
Multiple Periodic Solutions ◽  
The Stability

This study concerns the dynamic response of a rotor landed on auxiliary (catcher) bearings in an Active Magnetic Bearing (AMB) supported rotor, following postulated loss of power or overload of the AMB. An analytical model involving a disk, a shaft and auxiliary bearings on damped flexible supports is constructed and appropriate equations of the nonlinear dynamic system are developed. The equations include a switch function to indicate contact/non-contact events and determine the existence of contact normal forces and tangential friction forces between the shaft and the bearings. Steady state solutions are obtained. An analytical method was formulated and used to yield solutions for cases with well balanced rotors, in absence of any side forces. The Fixed Point Algorithm (FPA) is used to obtain steady state periodic solutions of the unbalanced rotor for various parameters. The FPA is used to determine the stability of the periodic solutions and the type of bifurcation involved. Multiple periodic solutions, quasi-periodic and chaotic responses are detected and discussed. A set of preliminary guidelines for selection of the parameters of the catcher bearings is given.

scholarly journals Multiple Periodic Solutions for Discrete Nicholson’s Blowflies Type System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Hui-Sheng Ding ◽  
Julio G. Dix
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Type System ◽  
Functional Difference ◽  
Difference System ◽  
Multiple Periodic Solutions

This paper is concerned with the existence of multiple periodic solutions for discrete Nicholson’s blowflies type system. By using the Leggett-Williams fixed point theorem, we obtain the existence of three nonnegative periodic solutions for discrete Nicholson’s blowflies type system. In order to show that, we first establish the existence of three nonnegative periodic solutions for then-dimensional functional difference systemyk+1=Akyk+fk, yk-τ, k∈ℤ, whereAkis not assumed to be diagonal as in some earlier results. In addition, a concrete example is also given to illustrate our results.

scholarly journals Further Investigations on the Dynamics and Multistability Coexisted in a Memory-Based Cobweb Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
S. S. Askar
Keyword(s):  
Fixed Point ◽  
Equilibrium Point ◽  
Profit Maximization ◽  
Period Doubling ◽  
Speed Of Adjustment ◽  
Discrete Dynamic ◽  
Cobweb Model ◽  
Numerical Studies ◽  
Market Clearing Price ◽  
The Stability

Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.

Period-1 to Period-8 Motions in a Nonlinear Jeffcott Rotor System

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C.J. Luo
Keyword(s):  
Period Doubling ◽  
Pitchfork Bifurcation ◽  
Mapping Method ◽  
Rotor System ◽  
Period Doubling Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Period 2 ◽  
Jeffcott Rotor

Abstract In this paper, a bifurcation tree of period-1 to period-8 motions in a nonlinear Jeffcott rotor system is obtained through the discrete mapping method. The bifurcations and stability of periodic motions on the bifurcation tree are discussed. The quasi-periodic motions on the bifurcation tree are caused by two (2) Neimark bifurcations of period-1 motions, one (1) Neimark bifurcation of period-2 motions and four (4) Neimark bifurcations of period-4 motions. The specific quasi-periodic motions are mainly based on the skeleton of the corresponding periodic motions. One stable and one unstable period-doubling bifurcations exist for the period-1, period-2 and period-4 motions. The unstable period-doubling bifurcation is from an unstable period-m motion to an unstable period-2m motion, and the unstable period-m motion becomes stable. Such an unstable period-doubling bifurcation is the 3rd source pitchfork bifurcation. Periodic motions on the bifurcation tree are simulated numerically, and the corresponding harmonic amplitudes and phases are presented for harmonic effects on periodic motions in the nonlinear Jeffcott rotor system. Such a study gives a complete picture of periodic and quasi-periodic motions in the nonlinear Jeffcott rotor system in the specific parameter range. One can follow the similar procedure to work out the other bifurcation trees in the nonlinear Jeffcott rotor systems.

Period-1 Motion to Chaos in a Nonlinear Flexible Rotor System

2020 ◽  
Vol 30 (05) ◽  
pp. 2050077 ◽  
Author(s):  
Yeyin Xu ◽  
Zhaobo Chen ◽  
Albert C. J. Luo
Keyword(s):  
Period Doubling ◽  
Rotor System ◽  
Harmonic Amplitude ◽  
Flexible Rotor ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Saddle Node ◽  
Jeffcott Rotor ◽  
And Control

In this paper, a bifurcation tree of period-1 motion to chaos in a flexible nonlinear rotor system is presented through period-1 to period-8 motions. Stable and unstable periodic motions on the bifurcation tree in the flexible rotor system are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are analyzed by eigenvalue analysis. On the bifurcation tree, the appearance and vanishing of jumping phenomena of periodic motions are generated by saddle-node bifurcations, and quasi-periodic motions are induced by Neimark bifurcations. Period-doubling bifurcations of periodic motions are for developing cascaded bifurcation trees, however, the birth of new periodic motions are based on the saddle-node bifurcation. For a better understanding of periodic motions on the bifurcation tree, nonlinear harmonic amplitude characteristics of periodic motions are presented. Numerical simulations of periodic motions are performed for the verification of semi-analytical predictions. From such a study, nonlinear Jeffcott rotor possesses complex periodic motions. Such results can help one detect and control complex motions in rotor systems for industry.

ON THE NUMERICAL DETECTION OF CODIMENSION-3 BIFURCATIONS OF PERIODIC SOLUTIONS (WITH APPLICATION TO OPTICAL BISTABILITY)

1994 ◽  
Vol 04 (06) ◽  
pp. 1425-1446
Author(s):  
KLAUS-GEORG NOLTE ◽  
IVAN L’HEUREUX
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Optical Bistability ◽  
Continuation Method ◽  
Period Doubling ◽  
Bistable System ◽  
Dimensional System ◽  
Systems Of Equations ◽  
The Stability ◽  
Stationary Behavior

Based upon the combination of the pseudo-arclength continuation method and the Poincaré map defined on a variable return plane, systems of equations are constructed that trace a Takens-Bogdanov bifurcation, a cusp, an isola formation/perturbed bifurcation point and a degenerate period-doubling/secondary Hopf bifurcation of periodic solutions of autonomous ordinary differential equations. The implementation of these ideas into a collection of FORTRAN codes and its application to a five-dimensional system describing an optical bistable system lead to the detection of interesting codimension-3 bifurcations away from the stationary behavior. A winged cusp, a swallow tail, a degenerate hysteresis point, an isola formation point for a codimension-1 loop and two kinds of degenerate Takens-Bogdanov bifurcations of periodic solutions are presented. Finally, based upon the computation of the stability coefficient “a”, attractive tori are found in a systematic way and briefly discussed.

Stability and existence of multiple periodic solutions for a quasilinear differential equation with maxima

2000 ◽  
Vol 130 (5) ◽  
pp. 1103-1118 ◽  
Author(s):  
Manuel Pinto ◽  
Sergei Trofimchuk
Keyword(s):  
Differential Equation ◽  
Periodic Solutions ◽  
Global Attractivity ◽  
Variational Equation ◽  
Periodic Forcing ◽  
Multiple Periodic Solutions ◽  
Halanay’S Inequality ◽  
The Stability ◽  
Image Position ◽  
Delay Differential

We study the stability of periodic solutions of the scalar delay differential equation where f(t) is a periodic forcing term and δ,p∈R. We study stability in the first approximation showing that the non-smooth equation (*) can be linearized along some ‘non-singular’ periodic solutions. Then the corresponding variational equation together with the Krasnosel'skij index are used to prove the existence of multiple periodic solutions to (*). Finally, we apply a generalization of Halanay's inequality to establish conditions for global attractivity in equations with maxima.

scholarly journals Bifurcation Analysis of Piecewise Smooth Bimodal Maps Using Normal Form

2020 ◽  
Vol 24 (3) ◽  
pp. 137-151
Author(s):  
Z. T. Zhusubaliyev ◽  
D. S. Kuzmina ◽  
O. O. Yanochkina
Keyword(s):  
Fixed Point ◽  
Normal Form ◽  
Bifurcation Analysis ◽  
Stability Region ◽  
Piecewise Linear ◽  
Period Doubling ◽  
Piecewise Smooth ◽  
Continuous Map ◽  
The Stability ◽  
Smooth Map

Purpose of reseach. Studyof bifurcations in piecewise-smooth bimodal maps using a piecewise-linear continuous map as a normal form. Methods. We propose a technique for determining the parameters of a normal form based on the linearization of a piecewise-smooth map in a neighborhood of a critical fixed point. Results. The stability region of a fixed point is constructed numerically and analytically on the parameter plane. It is shown that this region is limited by two bifurcation curves: the lines of the classical period-doubling bifurcation and the “border collision” bifurcation. It is proposed a method for determining the parameters of a normal form as a function of the parameters of a piecewise smooth map. The analysis of "border-collision" bifurcations using piecewise-linear normal form is carried out. Conclusion. A bifurcation analysis of a piecewise-smooth irreversible bimodal map of the class Z1–Z3–Z1 modeling the dynamics of a pulse–modulated control system is carried out. It is proposed a technique for calculating the parameters of a piecewise linear continuous map used as a normal form. The main bifurcation transitions are calculated when leaving the stability region, both using the initial map and a piecewise linear normal form. The topological equivalence of these maps is numerically proved, indicating the reliability of the results of calculating the parameters of the normal form.

Dynamic Response of a Rotor Landed on Auxiliary Bearings

1995 ◽  
Author(s):  
Xinchao Wang ◽  
Sherif Noah
Keyword(s):  
Steady State ◽  
Dynamic Response ◽  
Periodic Solutions ◽  
Nonlinear Dynamic System ◽  
Magnetic Bearing ◽  
Active Magnetic Bearing ◽  
Fixed Point Algorithm ◽  
Friction Forces ◽  
Flexible Supports ◽  
Multiple Periodic Solutions

Abstract This study concerns the dynamic response of a rotor landed on auxiliary (catcher) bearings in an Active Magnetic Bearing (AMB) supported rotor, following postulated loss of power or overload of the AMB. An analytical model involving disk, shaft and auxiliary bearings on damped flexible supports is constructed and appropriate equations of the nonlinear dynamic system are developed. The equations include a switch function to indicate contact/noncontact events and determine the existence of contact normal forces and tangential friction forces between the shaft and the bearings. Steady state solutions are obtained. An analytical method was formulated and used to yield solutions for cases with well balanced rotors, in absence of any side forces. The Fixed Point Algorithm (FPA) is used to obtain steady state periodic solutions for various parameters. The FPA is used to determine the stability of the periodic solution and the type of bifurcations involved. Multiple periodic solutions, quasiperiodic and chaotic responses are detected and discussed. A set of preliminary guidelines for selection of the parameters of the catcher bearings are given.

Study on Nonlinear Dynamic Characters of Rotor-Bearing System

2001 ◽  
Author(s):  
Songbo Xia ◽  
Xinjiang Zhang ◽  
Xinhua Wu ◽  
Genfa Xu
Keyword(s):  
Integral Method ◽  
Period Doubling ◽  
Poincaré Maps ◽  
Poincare Maps ◽  
Stable Operating ◽  
Jeffcott Rotor ◽  
Typical Parameter ◽  
The Stability ◽  
Chaos Motion ◽  
Numerical Integral

Abstract The stability of a rigid Jeffcott rotor system based on short-bearing model is study in a relatively wide parameter range using the Poincaré maps and numerical integral method. The results of calculation show that the period doubling bifurcation, quasi-periodic and chaos motions may be occurred. In some typical parameter regions the bifurcation diagrams, phase portrait, Poincaré maps and the frequency spectrums of the system are acquired with numerical integral method. They demonstrate some motion state of the system. The fractal dimension concept is used to determine whether the system is in a state of chaos motion. The analysis result of this paper provides the theoretical bases for qualitatively controlling the stable operating states of rotors.

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