scholarly journals Asymptotic Stability of Periodic Solutions for Nonsmooth Differential Equations with Application to the Nonsmooth van der Pol Oscillator

2009 ◽  
Vol 40 (6) ◽  
pp. 2478-2495 ◽  
Author(s):  
Adriana Buică ◽  
Jaume Llibre ◽  
Oleg Makarenkov
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

scholarly journals Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Olivia Bundău
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Periodic Solutions ◽  
General Class ◽  
Least Squares Method ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Problems ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

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 On a certain family of periodic solutions of differential equations, with an application to the triode oscillator

1923 ◽  
Vol 103 (722) ◽  
pp. 516-524 ◽  
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Power Series ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Constant Parameter ◽  
Anode Potential ◽  
Van Der Pol ◽  
Period 2

In this paper we shall do concerned with the simultaneous differential equations dx / dt = μξ dy / dt = λ ( x ) + μη , (1) where λ ( x ) is a function of x only and ξ and η are functions of x and y , periodic in y with period 2 π and expressible as Fourier series in sines and cosines of multiples of y , the coefficients being functions of x not involving t explicitly, μ is simply a constant parameter. These equations are a generalisation of the equation for the triode oscillator. Appleton and van der Pol have shown that, in the free triode, the anode potential v is determined by a differential equation of the type d 2 v / dt 2 + f ( v ) dv / dt + ω 2 v = 0, (i) where f ( v ) is expressible as a power series in v of the type f ( v ) = - α + βv + γv 2 + ... , (ii) the coefficients α, β, γ , etc., being small compared with ω .

scholarly journals Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Hector Vazquez-Leal ◽  
Arturo Sarmiento-Reyes ◽  
Yasir Khan ◽  
Uriel Filobello-Nino ◽  
Alejandro Diaz-Sanchez
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Homotopy Perturbation ◽  
Physical Phenomena

The fact that most of the physical phenomena are modelled by nonlinear differential equations underlines the importance of having reliable methods for solving them. This work presents the rational biparameter homotopy perturbation method (RBHPM) as a novel tool with the potential to find approximate solutions for nonlinear differential equations. The method generates the solutions in the form of a quotient of two power series of different homotopy parameters. Besides, in order to improve accuracy, we propose the Laplace-Padé rational biparameter homotopy perturbation method (LPRBHPM), when the solution is expressed as the quotient of two truncated power series. The usage of the method is illustrated with two case studies. On one side, a Ricatti nonlinear differential equation is solved and a comparison with the homotopy perturbation method (HPM) is presented. On the other side, a nonforced Van der Pol Oscillator is analysed and we compare results obtained with RBHPM, LPRBHPM, and HPM in order to conclude that the LPRBHPM and RBHPM methods generate the most accurate approximated solutions.

scholarly journals Normal form of double-Hopf singularity with 1:1 resonance for delayed differential equations

2019 ◽  
Vol 24 (2) ◽  
pp. 241-260
Author(s):  
Xiaoqin P. Wu ◽  
Liancheng Wang
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Delayed Feedback ◽  
Van Der Pol Oscillator ◽  
Third Order ◽  
Van Der Pol ◽  
The Third ◽  
Delayed Differential Equations ◽  
Theoretical Results

In this manuscript, we provide a framework for the double-Hopf singularity with 1:1 resonance for general delayed differential equations (DDEs). The corresponding normal form up to the third-order terms is derived. As an application of our framework, a double-Hopf singularity with 1:1 resonance for a van der Pol oscillator with delayed feedback is investigated to illustrate the theoretical results.

scholarly journals STUDY OF THE SINGULAR POINTS OF THE FRACTIONAL OSCILLATOR VAN DER POL-DUFFING

2019 ◽  
pp. 47-54
Author(s):  
Е.Р. Новикова
Keyword(s):  
Asymptotic Stability ◽  
Differential Equations ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Oscillatory System ◽  
Memory Effects ◽  
Oscillatory Process ◽  
Van Der Pol ◽  
Phase Trajectories ◽  
Fractional Oscillator

В работе проводится исследование на асимптотическую устойчивость точек покоя дробного осциллятора Ван дер ПоляДуффинга. Дробный осциллятор Ван дер Поля Дуффинга представляет собой колебательную систему двух дифференциальных уравнений с производными дробных порядков в смысле ГерасимоваКапуто. Порядки дробных производных характеризуют свойства среды (эффекты памяти), в которой происходит колебательный процесс и могут быть одинаковыми (соизмеримыми) или разными (несоизмеримыми). С помощью теорем для соизмеримой и несоизмеримой систем на конкретных примерах исследуется асимптотическая устойчивость точек покоя дробного осциллятора Ван дер ПоляДуффинга. Результаты исследований были подтверждены с помощью построения соответствующих осциллограмм и фазовых траекторий A study is conducted on the asymptotic stability of the rest points of the fractional oscillator Van der PolDuffing. The fractional van der PolDuffing oscillator is an oscillatory system of two differential equations with fractional order derivatives in the sense of GerasimovCaputo. The orders of fractional derivatives characterize the properties of the medium (memory effects) in which the oscillatory process takes place and can be the same (commensurate) or different (incommensurable). Using theorems for commensurable and incommensurable systems, the asymptotic stability of the rest points of the fractional van der PolDuffing oscillator is investigated with concrete examples. The results of the studies were confirmed by constructing the appropriate waveforms and phase trajectories.

ON THE STUDY OF BIFURCATIONS IN DELAY-DIFFERENTIAL EQUATIONS: A FREQUENCY-DOMAIN APPROACH

2012 ◽  
Vol 22 (06) ◽  
pp. 1250137 ◽  
Author(s):  
FRANCO S. GENTILE ◽  
JORGE L. MOIOLA ◽  
EDUARDO E. PAOLINI
Keyword(s):  
Differential Equations ◽  
Frequency Domain ◽  
Delay Differential Equations ◽  
Van Der Pol Oscillator ◽  
Van Der Pol ◽  
Delay Dependence ◽  
Collateral Findings ◽  
Delay Differential ◽  
Controllability And Observability ◽  
Frequency Domain Approach

An improved version of a frequency-domain approach to study bifurcations in delay-differential equations is presented. The proposed methodology provides information about the frequency, amplitude, and stability of the orbit emerging from Hopf bifurcation. We apply this method to different schemes of the delayed van der Pol oscillator. The time-delay dependence can appear intrinsically because of the system dynamics or can be intentionally introduced in a feedback loop. Also, a discussion about system controllability and observability is given for a proper and rigorous application of the frequency domain technique. Collateral findings involving some types of static bifurcations are included for completeness.

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