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.

Period-m Motions in a Periodically Forced van der Pol Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Fourier Series ◽  
Periodic Motion ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Stable Period ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
The Stability

Period-m motions in a periodically forced, van der Pol oscillator are investigated through the Fourier series expression, and the stability and bifurcation analysis of such periodic motions are carried out. To verify the approximate solutions of period-m motions, numerical illustrations are given. Period-m motions are separated by quasi-periodic motion or chaos, and the stable period-m motions are in independent periodic motion windows.

A Numerical Investigation of Phase-Locked and Chaotic Behavior of Coupled Van Der Pol Oscillators

1995 ◽  
Author(s):  
Per G. Reinhall ◽  
Duane W. Storti
Keyword(s):  
Parameter Space ◽  
Chaotic Behavior ◽  
Van Der Pol Oscillator ◽  
Stable Phase ◽  
Phase Lag ◽  
Van Der Pol ◽  
Dimensional Parameter ◽  
Existence And Stability ◽  
Van Der Pol Oscillators ◽  
Parameter Values

Abstract This paper presents the results of numerical simulations of the dynamics of a pair of linearly coupled van der Pol oscillators. A four-dimensional parameter space (including the displacement and velocity coupling strengths and the detuning in addition to the usual non-linearity parameter of the uncoupled van der Pol oscillator) is explored. In addition to corroboration of analytical results for the existence and stability of the in-phase and out-of-phase modes, regions in the parameter space are obtained where stable phase-locked motions exist with phase differences other than 0° or 180°. The dependence of stable phase lag on parameter values is presented for representative portions of the parameter space. A region is also located where trajectories are obtained which provide the first evidence of chaotic behavior and strange at tractors in this system of unforced non-conservative oscillators.

Vibration of Harmonically Excited Oscillators With Asymmetric Constraints

1992 ◽  
Vol 59 (2S) ◽  
pp. S284-S290 ◽  
Author(s):  
S. Natsiavas ◽  
H. Gonzalez
Keyword(s):  
Piecewise Linear ◽  
Periodic Motions ◽  
Van Der Pol ◽  
System Parameters ◽  
State Response ◽  
Damping Coefficients ◽  
Restoring Forces ◽  
Van Der Pol Oscillators ◽  
The Stability ◽  
Stiffness And Damping

investigation is carried out for a class of piecewise linear oscillators with asymmetric characteristics. The damping and restoring forces are general trilinear functions of the system velocity and displacement, respectively, while the excitation is harmonic in time. First, an analysis is presented which determines harmonic and subharmonic steady-state response. Then, a special formulation is employed in examining the stability of located periodic motions. Finally, numerical results are presented for several representative sets of the system parameters. Effects of asymmetries in the response due to unequal gaps as well as unequal stiffness and damping coefficients are analyzed in detail. Asymmetric response of a system with symmetric technical characteristics is also investigated. The behavior of the systems examined resembles response of similar nonlinear systems with continuous characteristics, like the response of the Duffing and van der Pol oscillators. Complicated nonperiodic response is also encountered and analyzed.

scholarly journals Dynamics of Three Coupled Van der Pol Oscillators With Application to Circadian Rhythms

2005 ◽  
Author(s):  
Kevin Rompala ◽  
Richard Rand ◽  
Howard Howland
Keyword(s):  
Circadian Rhythms ◽  
Pineal Gland ◽  
Multiple Scales ◽  
Van Der Pol Oscillator ◽  
Slow Flow ◽  
Direct Connection ◽  
Van Der Pol ◽  
Software Products ◽  
Van Der Pol Oscillators ◽  
The Brain

In this work we study a system of three van der Pol oscillators, x, y and w, coupled as follows: x¨−ε(1−x2)x˙+x=εμ(w−x)y¨−ε(1−y2)y˙+y=εμ(w−y)w¨−ε(1−w2)w˙+p2w=εμ(x−w)+εμ(y−w) Here the x and y oscillators are identical, and are not directly coupled to each other, but rather are coupled via the w oscillator. We investigate the existence of the in-phase mode x = y for ε ≪ 1. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the software products MACSYMA and AUTO. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator (x and y). Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator (w).

Dynamics of a Simplified van der Pol Oscillator Revisited

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Arun Rajendran
Keyword(s):  
Bifurcation Analysis ◽  
Local Stability ◽  
Van Der Pol Oscillator ◽  
Nonsmooth Dynamics ◽  
Mapping Technique ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Chaotic Motions ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.

On the Delayed van der Pol Oscillator with Time-Varying Feedback Gain

2014 ◽  
Vol 706 ◽  
pp. 149-158 ◽  
Author(s):  
Mustapha Hamdi ◽  
Mohamed Belhaq
Keyword(s):  
Numerical Simulation ◽  
Type System ◽  
Stationary Solutions ◽  
Van Der Pol Oscillator ◽  
Feedback Gain ◽  
Time Varying ◽  
Van Der Pol ◽  
Existence Region ◽  
The Stability ◽  
Time Delayed Feedback

This work studies the effect of time delayed feedback on stationary solutions in a van derPol type system. We consider the case where the feedback gain is harmonically modulated with a resonantfrequency. Perturbation analysis is conducted to obtain the modulation equations near primaryresonance, the stability analysis for stationary solutions is performed and bifurcation diagram is determined.It is shown that the modulated feedback gain position can influence significantly the steadystates behavior of the delayed van der Pol oscillator. In particular, for appropriate values of the modulateddelay parameters, the existence region of the limit cycle (LC) can be increased or quenched.Moreover, new regions of quasiperiodic vibration may emerge for certain values of the modulatedgain. Numerical simulation was conducted to validate the analytical predictions.

An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Approximate Solution ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Amplitude Spectrum ◽  
Van Der Pol Oscillator ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Series Expression

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (N) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., AN≤ɛ and ɛ=10-8). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

scholarly journals Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol

2019 ◽  
Vol 3 (2) ◽  
pp. 156
Author(s):  
Yuni Yulida ◽  
Muhammad Ahsar Karim
Keyword(s):  
Periodic Solution ◽  
Limit Cycle ◽  
Multiple Scale ◽  
Multiple Scale Method ◽  
Van Der Pol Equation ◽  
Damping Force ◽  
Van Der Pol ◽  
Scale Method ◽  
Existence And Stability ◽  
The Stability

Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two.

scholarly journals Calculating periodic vibrations of nonlinear autonomous multi-degree-of-freedom systems by the incremental harmonic balance method

2004 ◽  
Vol 26 (3) ◽  
pp. 157-166
Author(s):  
Nguyen Van Khang ◽  
Thai Manh Cau
Keyword(s):  
Harmonic Balance ◽  
Floquet Theory ◽  
Monodromy Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Incremental Harmonic Balance ◽  
The Stability

In this paper the incremental harmonic balance method is used to calculate periodic vibrations of nonlinear autonomous multip-degree-of-freedom systems. According to Floquet theory, the stability of a periodic solution is checked by evaluating the eigenvalues of the monodromy matrix. Using the programme MAPLE, the authors have studied the periodic vibrations of the system multi-degree van der Pol form.

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