scholarly journals Mixed-mode dynamics of certain bistable oscillators: behavioural mapping, approximations for motion and links with van der Pol oscillators

2015 ◽  
Vol 471 (2184) ◽  
pp. 20150638 ◽  
Author(s):  
Ivana Kovacic ◽  
Matthew Cartmell ◽  
Miodrag Zukovic
Keyword(s):  
Mixed Mode ◽  
Autonomous Systems ◽  
Low Frequency ◽  
Harmonic Excitation ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Restoring Force ◽  
Van Der Pol Oscillators ◽  
Flow Oscillations ◽  
Fast Flow

This study is concerned with a new generalized mathematical model for single degree-of-freedom bistable oscillators with harmonic excitation of low-frequency, linear viscous damping and a restoring force that contains a negative linear term and a positive nonlinear term which is a power-form function of the generalized coordinate. Comprehensive numerical mapping of the range of bifurcatory behaviour shows that such non-autonomous systems can experience mixed-mode oscillations, including bursting oscillations (fast flow oscillations around the outer curves of a slow flow), and relaxation oscillations like a classical (autonomous) van der Pol oscillator. After studying the global system dynamics the focus of the investigations is on cubic oscillators of this type. Approximate techniques are presented to quantify their response, i.e. to determine approximations for both the slow and fast flows. In addition, a clear analogy between the behaviour of two archetypical oscillators—the non-autonomous bistable oscillator operating at low frequency and the strongly damped autonomous van der Pol oscillator—is established for the first time.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Dynamics of Two Van Der Pol Oscillators Coupled via a Bath

2003 ◽  
Author(s):  
Erika Camacho ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Software Package ◽  
Bifurcation Theory ◽  
Floquet Theory ◽  
Van Der Pol Oscillator ◽  
Direct Connection ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
The Stability

In this work we study a system of two van der Pol oscillators, x and y, coupled via a “bath” z: x¨−ε(1−x2)x˙+x=k(z−x)y¨−ε(1−y2)y˙+y=k(z−y)z˙=k(x−z)+k(y−z) We investigate the existence and stability of the in-phase and out-of-phase modes for parameters ε > 0 and k > 0. To this end we use Floquet theory and numerical integration. Surprisingly, our results show that the out-of-phase mode exists and is stable for a wider range of parameters than is the in-phase mode. This behavior is compared to that of two directly coupled van der Pol oscillators, and it is shown that the effect of the bath is to reduce the stability of the in-phase mode. We also investigate the occurrence of other periodic motions by using bifurcation theory and the AUTO bifurcation and continuation software package. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We present a simplified model of a circadian oscillator which shows that it can be modeled as a van der Pol oscillator. Although there is no direct connection between the two eyes, they can influence each other by affecting the concentration of melatonin in the bloodstream, which is represented by the bath in our model.

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.

Synchronization or cluster synchronization in coupled Van der Pol oscillators networks with different topological types

2021 ◽  
Author(s):  
Shuai Wang ◽  
Yong Li
Keyword(s):  
Periodic Solutions ◽  
Phase Synchronization ◽  
Van Der Pol Oscillator ◽  
Cluster Synchronization ◽  
Critical Conditions ◽  
Complete Synchronization ◽  
Van Der Pol ◽  
Special Groups ◽  
Different Types ◽  
Van Der Pol Oscillators

Abstract In this paper, we try to discuss the mechanism of synchronization or cluster synchronization in the coupled Van der Pol oscillator networks with different topology types by using the theory of rotating periodic solutions. The synchronous solutions here are transformed into rotating periodic solutions of some dynamical systems. By analyzing the bifurcation of rotating periodic solutions, the critical conditions of synchronous solutions are given in three different networks. We use the rotating periodic matrix in the rotating periodic theory to judge various types of synchronization phenomena, such as complete synchronization, anti-phase synchronization, periodic synchronization, or cluster synchronization. All rotating periodic matrices which satisfy the exchange invariance of multiple oscillators form special groups in these networks. By using the conjugate classes of these groups, we obtain various possible synchronization solutions in the three networks. In particular, we find symmetry has different effects on synchronization in different networks. The network with better symmetry has more elements in the corresponding group, which may have more types of synchronous solutions. However, different types of symmetry may get the same type of synchronous solutions or different types of synchronous solutions, depending on whether their corresponding rotating periodic matrices are similar.

scholarly journals Construction of Approximate Analytical Solutions to Strongly Nonlinear Coupled van der Pol Oscillators

2014 ◽  
Vol 6 ◽  
pp. 817570
Author(s):  
Y. H. Qian ◽  
W. K. Liu ◽  
S. M. Chen
Keyword(s):  
Analytical Solutions ◽  
Degrees Of Freedom ◽  
Van Der Pol Oscillator ◽  
Practical Significance ◽  
Engineering System ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Homotopy Analysis ◽  
Practical Engineering ◽  
Van Der Pol Oscillators

Using nonlinear theory to research vibration model of engineering system has important theoretical and practical significance. Multi-degree-of-freedom (MDOF) coupled van der Pol oscillator is a typical model in the nonlinear vibration; many complex dynamic problems in practical engineering can be simplified as this model to be solved in the end. This paper discusses a class of two-degrees-of-freedom (2-DOF) coupled van der Pol oscillator, which was divided into three parameters of different situations α1≠α2, β1≠β2, and γ1≠γ2 to discuss. Employing symbolic software such as Mathematica for those problems, the explicit analytical solutions of frequency ω and displacements x1( t) and x2( t) are well formulated. Results showed that the homotopy analysis method (HAM) can effectively deal with this kind of parameter of different coupled vibrators, just request the values of some parameters are not too big. Finally, we got four important theorems to simplify the solution of the nonlinear system.

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