DYNAMICS AND ACTIVE CONTROL OF MOTION OF A DRIVEN MULTI-LIMIT-CYCLE VAN DER POL OSCILLATOR

2007 ◽  
Vol 17 (04) ◽  
pp. 1343-1354 ◽  
Author(s):  
R. YAMAPI ◽  
B. R. NANA NBENDJO ◽  
H. G. ENJIEU KADJI
Keyword(s):  
Limit Cycle ◽  
Active Control ◽  
Coupling Parameter ◽  
Van Der Pol Oscillator ◽  
External Excitation ◽  
Van Der Pol ◽  
Chaotic Vibrations ◽  
Van Der Pol Model ◽  
Nonautonomous Case ◽  
Selection Of

This paper deals with the dynamics and active control of a driven multi-limit-cycle Van der Pol oscillator. The amplitude of the oscillatory states both in the autonomous and nonautonomous case are derived. The interaction between the amplitudes of the external excitation and the limit-cycles are also analyzed. The domain of the admissible values on the amplitude for the external excitation is found. The effects of the control parameter on the behavior of a driven multi-limit-cycle Van der Pol model are analyzed and it appears that with the appropriate selection of the coupling parameter, the quenching of chaotic vibrations takes place.

Suppression of limit cycle oscillations in the van der Pol oscillator by means of passive non-linear energy sinks

2006 ◽  
Vol 13 (1) ◽  
pp. 41-75 ◽  
Author(s):  
Young S. Lee ◽  
Alexander F. Vakakis ◽  
Lawrence A. Bergman ◽  
D. Michael McFarland
Keyword(s):  
Limit Cycle ◽  
Van Der Pol Oscillator ◽  
Limit Cycle Oscillations ◽  
Van Der Pol ◽  
Non Linear

scholarly journals Asymptotic Limit-cycle Analysis of Oscillating Chemical Reactions

2021 ◽  
Author(s):  
Alain Brizard ◽  
Samuel Berry
Keyword(s):  
Limit Cycle ◽  
Chemical Reactions ◽  
Relaxation Oscillation ◽  
Three Dimensional ◽  
Van Der Pol Oscillator ◽  
Asymptotic Limit ◽  
Two Dimensional ◽  
Van Der Pol ◽  
Parameter Values ◽  
Analytical Expressions

Abstract The asymptotic limit-cycle analysis of mathematical models for oscillating chemical reactions is presented. In this work, after a brief presentation of mathematical preliminaries applied to the biased Van der Pol oscillator, we consider a two-dimensional model of the Chlorine dioxide Iodine Malonic-Acid (CIMA) reactions and the three-dimensional and two-dimensional Oregonator models of the Belousov-Zhabotinsky (BZ) reactions. Explicit analytical expressions are given for the relaxation-oscillation periods of these chemical reactions that are accurate within 5% of their numerical values. In the two-dimensional CIMA and Oregonator models, we also derive critical parameter values leading to canard explosions and implosions in their associated limit cycles.

Perturbation Solution of an Oscillator With Periodic van der Pol Damping

1993 ◽  
Author(s):  
Duane W. Storti ◽  
Cornelius Nevrinceanu ◽  
Per G. Reinhall
Keyword(s):  
Periodic Solution ◽  
Limit Cycle ◽  
Variational Equation ◽  
Phase Locking ◽  
Perturbation Solution ◽  
Van Der Pol Oscillator ◽  
Transition Curve ◽  
Van Der Pol ◽  
Additional Stability ◽  
Van Der Pol Oscillators

Abstract We present a perturbation solution for a linear oscillator with a variable damping coefficient involving the limit cycle of the van der Pol equation (van der Pol 1926). This equation arises as the variational equation governing the stability of in-phase vibration in a pair of identical van der Pol oscillators with linear coupling. The van der Pol oscillator has served as the classic example of a limit cycle oscillator, and coupled limit cycle oscillators appear in mathematical models of self-excited systems ranging from tube rows in cross flow heat exchangers to arrays of stomates in plant leaves. As in many systems modeled by coupled oscillators, criteria for phase-locking or synchronization are of fundamental importance in understanding the dynamics. In this paper we study a simple but interesting problem consisting of a pair of identical van der Pol oscillators with linear diffusive coupling which corresponds, in the mechanical analogy, to a spring connecting the masses of the two oscillators. Intuition and earlier first-order analyses suggest that the spring will pull the two masses together causing stable in-phase locking. However, previous results of a relaxation limit study (Storti and Rand 1986) indicate that the in-phase mode is not always stable and suggest the existence of an additional stability boundary. To resolve the apparent discrepancy, we obtain a new periodic solution of the variational equation as a power series in ε, the small parameter in the sinusoidal van de Pol oscillator. This approach follows Andersen and Geer’s (1982) solution for the limit cycle of an isolated van der Pol oscillator. The coupling strength corresponding to the periodic solution of the variational equation defines an additional stability transition curve which has only been observed previously in the relaxation limit. We show that this transition curve, which provides a consistent connection between the sinusoidal and relaxation limits, is O(ε2) and could not have been delected in O(ε) analyses. We determine the analytical expression for this stability transition curve to O(ε31) and show very favorable agreement with numerical results we obtained using an Adams-Gear method.

Developing a Reduced-Order Model to Understand Non-Synchronous Vibration (NSV) in Turbomachinery

2012 ◽  
Author(s):  
Stephen T. Clark ◽  
Robert E. Kielb ◽  
Kenneth C. Hall
Keyword(s):  
Stable Limit Cycle ◽  
Forced Response ◽  
Van Der Pol Oscillator ◽  
Future Research ◽  
Limit Cycle Oscillation ◽  
Preliminary Design ◽  
Stable Limit ◽  
Van Der Pol ◽  
Reduced Order ◽  
Van Der Pol Model

This paper demonstrates the potential of using a multi-degree-of-freedom, traditional van der Pol oscillator to model Non-Synchronous Vibration (NSV) in turbomachinery. It is shown that the two main characteristics of NSV are captured by the reduced-order, van der Pol model. First, a stable limit cycle oscillation (LCO) is maintained for various conditions. Second, the lock-in phenomenon typical of NSV is captured for various fluid-structure frequency ratios. The results also show the maximum amplitude of the LCO occurs at an off-resonant condition, i.e., when the natural shedding frequency of the aerodynamic instability is not coincident with the natural modal frequency of the structure. This conclusion is especially relevant in preliminary design in industry because it suggests that design engineers cannot treat NSV as a normal Campbell-diagram crossing as they would for preliminary design for forced response; it is possible that by redesigning the blade, the response amplitude of the blade may actually be higher. The goal of future research will be to identify values and significance of the coupling parameters used in the van der Pol model, to match these coefficients with confirmed instances of experimental NSV, and to develop a preliminary design tool that engineers can use to better design turbomachinery for NSV. Proper Orthogonal Decomposition (POD) CFD techniques and coefficient tuning from experimental instances of NSV have been considered to identify the unknown coupling coefficients in the van der Pol model. Both the modeling of experimental NSV and preliminary design development will occur in future research.

Analysis on limit cycle of fractional-order van der Pol oscillator

2014 ◽  
Vol 67 ◽  
pp. 94-102 ◽  
Author(s):  
Yongjun Shen ◽  
Shaopu Yang ◽  
Chuanyi Sui
Keyword(s):  
Limit Cycle ◽  
Fractional Order ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

AMPLITUDE CONTROL OF LIMIT CYCLE IN VAN DER POL SYSTEM

2006 ◽  
Vol 16 (02) ◽  
pp. 487-495 ◽  
Author(s):  
JIASHI TANG ◽  
ZILI CHEN
Keyword(s):  
Limit Cycle ◽  
Multiple Scales ◽  
Numerical Study ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Amplitude Control ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
Controlled Systems ◽  
Strongly Nonlinear System

The feedback controllers are designed to modify the amplitude of limit cycles in van der Pol oscillator and generalized van der Pol oscillator. Bifurcation control equations of weakly nonlinear systems are obtained by using the method of multiple scales. Gain-amplitude curves of controlled systems are drawn. Based on numerical study, the brief results of controlling amplitude of limit cycle are given for strongly nonlinear system.

Suppression of Limit Cycle Oscillations in a Van der Pol Oscillator by Means of a Passive Nonlinear Energy Sink

2005 ◽  
Author(s):  
Young S. Lee ◽  
Alexander F. Vakakis ◽  
Lawrence A. Bergman ◽  
D. Michael McFarland
Keyword(s):  
Vibration Control ◽  
Limit Cycle ◽  
Nonlinear Energy Sink ◽  
Van Der Pol Oscillator ◽  
Numerical Continuation ◽  
Parameter Study ◽  
Limit Cycle Oscillations ◽  
Van Der Pol ◽  
Energy Sink ◽  
Nonlinear Energy

We present a study of passive but efficient vibration control, wherein a so-called nonlinear energy sink (NES) completely eliminates the limit cycle oscillations (LCOs) of a van der Pol oscillator. We first perform a parameter study in order to get overall understanding of responses with respect to parameters. Then, we establish a slow flow dynamics model to perform analytical study of the suppression mechanism which corresponds to classical nonlinear energy pumping, i.e., passive, broadband, and targeted energy transfer through 1:1 resonance capture. Utilizing the method of numerical continuation of equilibrium, we also study the bifurcation of the steady state solutions. It turns out that the system may have either subcritical or supercritical LCOs, and that for some parameter domain the LCOs are completely eliminated. This suggests applicability of the NES to vibration control in self-excited systems.

The Symmetrical Van Der Pol Oscillator

2002 ◽  
Vol 39 (2) ◽  
pp. 128-137
Author(s):  
B. Z. Kaplan ◽  
Y. Horen
Keyword(s):  
Limit Cycle ◽  
Nonlinear Damping ◽  
Van Der Pol Oscillator ◽  
Great Power ◽  
State Equations ◽  
Van Der Pol ◽  
Present Treatment

The limit cycle is associated in the Van der Pol oscillator with damping introduced to merely one of its state equations. In the present oscillator nonlinear damping is added to both equations. A unique behaviour is observed in spite of the smallness of the departure from the Van der Pol case. The present treatment is due to classical Poincaré-Bendixson and Liapunov methods. It illuminates in a surprisingly simple and elegant manner the great power of the methods.

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