Weakly nonlinear oscillations of gas column driven by self-sustained sources

Self-sustained sources coupled to some sort of resonator have drawn attention recently as a subject of nonlinear dynamics with many practical applications as well as interesting mathematical problems from the chaos theory and the theory of synchronizations. In order to mimic the self-sustainability arising from physical background the van der Pol equation is commonly used as a model (e.g. vortex induced noise, flowstructure interactions, vocal folds motion etc.). In many cases the sound field inside the resonator is strong enough for weakly nonlinear formulation based on the Kuznetsov model equation to be employed. An array of sources governed by the inhomogeneous van der Pol equation coupled to the nonlinear acoustic wave equation is studied. The one dimensional constant cross-section open resonator with zero radiation impedance is assumed. The focus is on the main features such as mode-locking, harmonics generation and build-up from infinitesimal fluctuations.

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Application of the Hori Method in the Theory of Nonlinear Oscillations

Mathematical Problems in Engineering ◽

10.1155/2012/239357 ◽

2012 ◽

Vol 2012 ◽

pp. 1-32

Author(s):

Sandro da Silva Fernandes

Keyword(s):

Differential Equations ◽

Generating Function ◽

Nonlinear Oscillations ◽

System Of Differential Equations ◽

Van Der Pol Equation ◽

Van Der Pol ◽

Identity Transformations ◽

Hori Method ◽

New System ◽

Made In

Some remarks on the application of the Hori method in the theory of nonlinear oscillations are presented. Two simplified algorithms for determining the generating function and the new system of differential equations are derived from a general algorithm proposed by Sessin. The vector functions which define the generating function and the new system of differential equations are not uniquely determined, since the algorithms involve arbitrary functions of the constants of integration of the general solution of the new undisturbed system. Different choices of these arbitrary functions can be made in order to simplify the new system of differential equations and define appropriate near-identity transformations. These simplified algorithms are applied in determining second-order asymptotic solutions of two well-known equations in the theory of nonlinear oscillations: van der Pol equation and Duffing equation.

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Simplification of weakly nonlinear systems and analysis of cardiac activity using them

Discrete & Continuous Dynamical Systems - B ◽

10.3934/dcdsb.2021191 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Irada Dzhalladova ◽

Miroslava Růžičková

Keyword(s):

Nonlinear System ◽

Normal Form ◽

Cardiac Activity ◽

Original Method ◽

System Of Differential Equations ◽

Van Der Pol Equation ◽

Van Der Pol ◽

Established Method ◽

Weakly Nonlinear ◽

Bogolyubov Method

<p style='text-indent:20px;'>The paper deals with the transformation of a weakly nonlinear system of differential equations in a special form into a simplified form and its relation to the normal form and averaging. An original method of simplification is proposed, that is, a way to determine the coefficients of a given nonlinear system in order to simplify it. We call this established method the degree equalization method, it does not require integration and is simpler and more efficient than the classical Krylov-Bogolyubov method of normalization. The method is illustrated with several examples and provides an application to the analysis of cardiac activity modelled using van der Pol equation.</p>

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Secure communications via chaotic synchronization in Chua's circuit and Bonhoeffer-Van der Pol equation: numerical analysis of the errors of the recovered signal

Proceedings of ISCAS'95 - International Symposium on Circuits and Systems ◽

10.1109/iscas.1995.521606 ◽

2002 ◽

Cited By ~ 6

Author(s):

R. Lozi

Keyword(s):

Numerical Analysis ◽

Chaotic Synchronization ◽

Secure Communications ◽

Chua’S Circuit ◽

Van Der Pol Equation ◽

Van Der Pol ◽

Chua's Circuit

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On Periodic Perturbations of Asymmetric Duffing–Van-der-Pol Equation

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127414500618 ◽

2014 ◽

Vol 24 (05) ◽

pp. 1450061 ◽

Cited By ~ 4

Author(s):

Albert D. Morozov ◽

Olga S. Kostromina

Keyword(s):

Saddle Point ◽

Limit Cycles ◽

Small Neighborhood ◽

Integrable Equation ◽

Van Der Pol Equation ◽

Van Der Pol ◽

Periodic Perturbations ◽

Autonomous Equation ◽

Nonautonomous Equation ◽

Time Periodic

Time-periodic perturbations of an asymmetric Duffing–Van-der-Pol equation close to an integrable equation with a homoclinic "figure-eight" of a saddle are considered. The behavior of solutions outside the neighborhood of "figure-eight" is studied analytically. The problem of limit cycles for an autonomous equation is solved and resonance zones for a nonautonomous equation are analyzed. The behavior of the separatrices of a fixed saddle point of the Poincaré map in the small neighborhood of the unperturbed "figure-eight" is ascertained. The results obtained are illustrated by numerical computations.

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On Integrating Factors and a Perturbation Method

Volume 3A: 15th Biennial Conference on Mechanical Vibration and Noise — Vibration of Nonlinear, Random, and Time-Varying Systems ◽

10.1115/detc1995-0327 ◽

1995 ◽

Author(s):

W. T. van Horssen

Keyword(s):

Differential Equations ◽

Perturbation Method ◽

Ordinary Differential Equations ◽

First Integrals ◽

Van Der Pol Equation ◽

Van Der Pol ◽

Order Ordinary Differential Equation ◽

First Order ◽

Integrating Factors ◽

Complete Solutions

Abstract In this paper the fundamental concept (due to Euler, 1734) of how to make a first order ordinary differential equation exact by means of integrating factors, is extended to n-th order (n ≥ 2) ordinary differential equations and to systems of first order ordinary differential equations. For new classes of differential equations first integrals or complete solutions can be constructed. Also a perturbation method based on integrating factors can be developed. To show how this perturbation method works the method is applied to the well-known Van der Pol equation.

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