Integer Ratio Self-Synchronization in Pairs of Limit Cycle Oscillators

2021 ◽  
Author(s):  
Aditya Bhaskar ◽  
B. Shayak ◽  
Alan T. Zehnder ◽  
Richard H. Rand
Keyword(s):  
Numerical Analysis ◽  
Limit Cycle ◽  
Higher Order ◽  
Strong Nonlinearity ◽  
Stable States ◽  
Van Der Pol ◽  
Periodic Force ◽  
Mems Oscillators ◽  
Van Der Pol Oscillators ◽  
Multiple Stable States

Abstract The existence of multiple stable states of higher order m:n locking in coupled limit cycle oscillators has been studied by prior authors in the context of injection-locking in systems driven by an external periodic force. The current work builds on this concept to study the higher order locking characteristics of pairs of limit cycle oscillators self-synchronizing under coupling forces. To this end we analyze three oscillator systems: Van der Pol oscillators using numerical analysis, a simplified model for MEMS oscillators using numerical analysis as well as perturbation theory, and a full model of thermo-optically driven MEMS oscillators using numerical analysis. For the Van der Pol system, higher order locking is observed for the strongly nonlinear case corresponding to relaxation oscillations and the transition from weak to strong nonlinearity is studied using a parameter sweep. Additionally, coupling of a different nature such as quadratic coupling is also capable of inducing higher order coupling in Van der Pol oscillators. For the MEMS systems with linear coupling, higher order locking is observed when a strong cubic stiffness nonlinearity exists. Devil’s staircase-like structures are obtained for the coupling strength-frequency ratio parameter space which suggest overlapping Arnold locking regions for m:n locks corresponding to different integers.

scholarly journals Comment on the Chaotic Behaviour of van der Pol Equations with an External Periodic Excitation

1989 ◽  
Vol 44 (2) ◽  
pp. 160-162
Author(s):  
W.-H. Steeb ◽  
Jeun Chyuan Huang ◽  
Yih Shun Gou
Keyword(s):  
Limit Cycle ◽  
Chaotic Motion ◽  
Chaotic Behaviour ◽  
Surprising Result ◽  
Periodic Excitation ◽  
Van Der Pol ◽  
Periodic Force ◽  
Cycle System ◽  
Van Der Pol Equations

Abstract The limit cycle system with an external periodic force d2u/dt2 - a( 1 - u2)du/dt + un = kcos(Ωf) (n = 1, 3, 5,...) can show chaotic behaviour for certain values of a, k and Ω. We study the influence of n on the chaotic behaviour. For n = 1 we select values which result in chaotic motion of the system. Then we investigate the behaviour of the system for n = 3, 5 and 7. Introducing the nonlinearity un(n - 3, 5, 7) gives the surprising result that the chaotic motion ceases to exist.

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.

Analog Simulation of Phase Locked Modes of Diffusively Coupled van der Pol Oscillators

1999 ◽  
Author(s):  
Lesley Ann Low ◽  
Per G. Reinhall ◽  
Duane W. Storti
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Physical System ◽  
Coupled Oscillators ◽  
Electric Circuit ◽  
Van Der Pol ◽  
Analog Simulation ◽  
Coupling Parameters ◽  
Van Der Pol Oscillators ◽  
The Stability

Abstract Limit cycle oscillators arise in a wide variety of mechanical, electrical and biological systems. Recently, emphasis has been placed on the study of systems of coupled limit cycles, such as cardiac oscillations. Synchronization criteria have remained a focus of most investigations. One area of investigation in the field of coupled limit cycles is studying the behavior of a pair of linearly coupled van der Pol oscillators (Low, 1998; Rand, 1980; Sliger, 1997). Previous investigations (Storti, 1993; Storti, 1996) found the stability regions of the coupled oscillators for their in-phase and out-of-phase modes numerically. The coupled oscillators can be viewed as a mechanical system, where the coupling parameters are equivalent to a spring and damper attached between two masses. With positive coupling (positive damping) a region was found (Storti, 1993; Storti, 1996) where the in-phase mode is unstable. This counter intuitive result is yet to be discovered in a physical system of two coupled limit cycle oscillators. This research focuses on finding the region with positive coupling parameters where the in-phase mode is unstable using a physical model of two linearly coupled van der Pol oscillators. The coupled van der Pol oscillators were modeled using an analog electric circuit.

Asymptotic Wave Solutions for the Model of a Medium with Van Der Pol Oscillators

2014 ◽  
Vol 59 (9) ◽  
pp. 932-938
Author(s):  
V.A. Danylenko ◽  
◽  
S.I. Skurativskyi ◽  
I.A. Skurativska ◽  
◽  
...  
Keyword(s):  
Van Der Pol ◽  
Wave Solutions ◽  
Van Der Pol Oscillators

scholarly journals Modal synchronization of coupled bistable van der Pol oscillators

2021 ◽  
Vol 143 ◽  
pp. 110555
Author(s):  
I.B. Shiroky ◽  
O.V. Gendelman
Keyword(s):  
Van Der Pol ◽  
Van Der Pol Oscillators

scholarly journals Uncovering Droop Control Laws Embedded Within the Nonlinear Dynamics of Van der Pol Oscillators

2017 ◽  
Vol 4 (2) ◽  
pp. 347-358 ◽  
Author(s):  
Mohit Sinha ◽  
Florian Dorfler ◽  
Brian B. Johnson ◽  
Sairaj V. Dhople
Keyword(s):  
Nonlinear Dynamics ◽  
Droop Control ◽  
Van Der Pol ◽  
Control Laws ◽  
Van Der Pol Oscillators

scholarly journals Dynamics of hierarchical weighted networks of van der Pol oscillators

2020 ◽  
Vol 30 (12) ◽  
pp. 123146
Author(s):  
Daniel Monsivais-Velazquez ◽  
Kunal Bhattacharya ◽  
Rafael A. Barrio ◽  
Philip K. Maini ◽  
Kimmo K. Kaski
Keyword(s):  
Van Der Pol ◽  
Weighted Networks ◽  
Van Der Pol Oscillators

Analytical and experimental study on Van der Pol-type and Rayleigh-type equations for modeling nonlinear aeroelastic instabilities

2021 ◽  
pp. 136943322110220
Author(s):  
Guangzhong Gao ◽  
Ledong Zhu ◽  
Hua Bai ◽  
Wanshui Han ◽  
Feng Wang
Keyword(s):  
Aerodynamic Force ◽  
Early Stage ◽  
Higher Order ◽  
Empirical Modeling ◽  
Identification Algorithm ◽  
Van Der Pol ◽  
Single Degree Of Freedom ◽  
Section Shape ◽  
Vibration Responses ◽  
Aerodynamic Parameters

An empirical modeling of nonlinear aerodynamic force during aeroelastic instabilities, that is, vortex-induced vibration (VIV), galloping and flutter, is necessary in the estimation of vibration responses. Previous works on single-degree-of-freedom (SDOF) models suggest that nonlinear forms (Van der Pol or Rayleigh types) differ from section to section, which causes difficulty in practical application. Analytical evidences in this study have clarified that Van der Pol-type and Rayleigh-type models are equivalent in the amplitude-dependent aerodynamic damping; their difference lies in the higher-order harmonic responses. An identification algorithm of aerodynamic parameters is proposed to improve the robustness of aerodynamic parameters and guarantee the equivalence of both model types. Wind-tunnel tests of typical aeroelastic instabilities indicate that higher-order harmonic responses are small for VIV, galloping, and early-stage flutter instability when compared with the fundamental components due to weak nonlinearity. Van der Pol-type and Rayleigh-type models are both applicable until the flutter amplitude grows excessively large. It is clear that both model types are suitable for any section shape when use the proposed method of aerodynamic identification, and thus can be treated as a universal model for estimating the vibration amplitudes of nonlinear aeroelastic instabilities.

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