scholarly journals Lyapunov Stability and Attractors of Some Systems of Nonlinear Oscillators

1986 ◽  
Vol 41 (8) ◽  
pp. 987-988 ◽  
Author(s):  
H. Tasso
Keyword(s):  
Phase Space ◽  
Limit Cycle ◽  
Lyapunov Stability ◽  
Lyapunov Functions ◽  
Nonlinear Oscillators ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Single Oscillator

Lyapunov functions valid in the greater part of phase space were found for a system of nonlinear oscillators of an extended Van der Pol type. They yield a good estimate of the location of attractors. For a particular single oscillator the appropriately modified Van der Pol equation delivers an ellipse as limit cycle.

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.

Two timescale harmonic balance. I. Application to autonomous one-dimensional nonlinear oscillators

1992 ◽  
Vol 340 (1659) ◽  
pp. 473-501 ◽  
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Evolution Equations ◽  
Multiple Scales ◽  
Phase Locking ◽  
Period Doubling ◽  
Nonlinear Oscillators ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Wide Range

Two timescale harmonic balance is a semi-analytical/numerical method for deriving periodic solutions and their stability to a class of nonlinear autonomous and forced oscillator equations of the form ẍ + x = f(x,ẋ,λ) and ẍ + x = f(x,ẋ,λ,t) , where λ is a control parameter. The method incorporates salient features from both the method of harmonic balance and multiple scales, and yet does not require an explicit small parameter. Essentially periodic solutions are formally derived on the basis of a single assumption: ‘that an N harmonic, truncated, Fourier series and its first two derivatives can represent x(t) , ẋ(t) and ẍ(t) respectively’. By seeking x(t) as a series of superharmonics, subharmonics, and ultrasubharmonics it is found that the method works over a wide range of parameter space provided the above assumption holds which, in practice, imposes some ‘problem dependent’ restriction on the magnitude of the nonlinearities. Two timescales, associated with the amplitude and phase variations respectively, are introduced by means of an implicit parameter Є . These timescales permit the construction of a set of amplitude evolution equations together with a corresponding stability criterion. In Part I the method is formulated and applied to three autonomous equations, the van der Pol equation, the modified van der Pol equation, and the van der Pol equation with escape. In this case an expansion in superharmonics is sufficient to reveal Hopf, saddle node and homoclinic bifurcations which are compared with results obtained by numerical integration of the equations. In Part II the method is applied to forced nonlinear oscillators in which the solution for x(t) includes superharmonics, subharmonics, and ultrasubharmonics. The features of period doubling, symmetry breaking, phase locking and the Feigenbaum transition to chaos are examined.

An Optimal Analytic Approximate Solution for the Limit Cycle of Duffing–van der Pol Equation

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Mustafa Turkyilmazoglu
Keyword(s):  
Limit Cycle ◽  
Asymptotic Methods ◽  
Computational Cost ◽  
Perturbation Parameter ◽  
Van Der Pol Equation ◽  
Van Der Pol ◽  
Analysis Technique ◽  
Homotopy Analysis ◽  
Asymptotic Parameter ◽  
Analytical Expressions

The present paper is concerned with the accurate analytic solution of the limit cycle of the Duffing–van der Pol equation. Instead of the traditional Taylor series or asymptotic methods, the homotopy analysis technique is employed, which does not require a small perturbation parameter or a large asymptotic parameter. It is known that such a method is extremely powerful in gaining the exact solution of the physical problem in terms of purely trigonometric functions, yet the computational cost of the method is considerably high. We propose here an approach that not only greatly reduces the computational efforts but also presents an easy to implement task of application of the homotopy analysis method to the Duffing–van der Pol equation. The explicit analytical expressions obtained using the proposed approach generates the displacement, amplitude, and frequency of the limit cycle that compare excellently with the numerically computed ones.

scholarly journals Perturbation Analysis of the Limit Cycle of the Free van der Pol Equation

1984 ◽  
Vol 44 (5) ◽  
pp. 881-895 ◽  
Author(s):  
Mohammad B. Dadfar ◽  
James Geer ◽  
Carl M. Andersen
Keyword(s):  
Limit Cycle ◽  
Perturbation Analysis ◽  
Van Der Pol Equation ◽  
Van Der Pol
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