Asymptotic Method for Analysis of Nonlinear Systems With Two Parameters

Xu, Zhao; Zhang, Lingqi

doi:10.1016/s0252-9602(18)30505-8

Cited by 9 publications

(5 citation statements)

References 3 publications

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“…Without loss of generality, we assume the two saddle points of (27) which linked by the heteroclinic orbits as H 1 (h 1 , 0) and H 2 (h 2 , 0). From (22) and (23) we get…”

Section: The Generalized Harmonic Functions Perturbation Methodsmentioning

confidence: 99%

“…where a, b ∈ R. This solution can be considered as a class of quadratic sinusoidal generalized harmonic function which is different with the functions constructed in references [22][23][24][25][26]. Multiplying two sides of (7) by x (ϕ) = 2a sin ϕ cos ϕ and integrating with respect to ϕ, we can obtain…”

Section: Homo-heteroclinic Orbits Of Generalized Duffing-harmonic Oscmentioning

confidence: 99%

“…(4). Then a and b can be calculated from (20), (22) and (23). From (9) we obtain (14), we know that there also does not exist a finite time period for a heteroclinic solution.…”

Section: Homo-heteroclinic Orbits Of Generalized Duffing-harmonic Oscmentioning

confidence: 99%

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Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator

Tang

Cai

2015

Qual. Theory Dyn. Syst.

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In this paper, a novel construction of solutions of nonlinear oscillators are proposed which can be called as the quadratic generalized harmonic function. Based on this novel solution, a modified generalized harmonic function LindstedtPoincaré method is presented which call the quadratic generalized harmonic function perturbation method. Via this method, the homoclinic and heteroclinic bifurcations of Duffing-harmonic-van de Pol oscillator are investigated. The critical value of the homoclinic and heteroclinic bifurcation parameters are predicted. Meanwhile, the analytical solutions of homoclinic and heteroclinic orbits of this oscillator are also attained. To illustrate the accuracy of the present method, all the above-mentioned results are compared with those of Runge-Kutta method, which shows that the proposed method is effective and feasible. In addition, the present method can be utilized in study many other oscillators.

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“…Without loss of generality, we assume the two saddle points of (27) which linked by the heteroclinic orbits as H 1 (h 1 , 0) and H 2 (h 2 , 0). From (22) and (23) we get…”

Section: The Generalized Harmonic Functions Perturbation Methodsmentioning

confidence: 99%

Section: Homo-heteroclinic Orbits Of Generalized Duffing-harmonic Oscmentioning

confidence: 99%

See 1 more Smart Citation

Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator

Tang

Cai

2015

Qual. Theory Dyn. Syst.

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“…For small ), and 9(x) is linear, the classical perturbation method [1] can be applied to the problem of determining limit cycles approximately. Generalizations have been obtained for the cases where g(x) is linear plus cubic polynomial terms [2][3][4][5] and where 9 (x) is an arbitrary function [6][7][8][9][10]. For moderately large A and 9(x) is an arbitrary nonlinear function, there are the two timescale harmonic balance method [11] and the perturbation-iterative method [12].…”

Section: Introductionmentioning

confidence: 99%

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

Chan²,

Chung³

1996

Nonlinear Dyn

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The perturbation-incremental method is applied to determine the separatrices and limit cycles of strongly nonlinear oscillators. Conditions are derived under which a limit cycle is created or destroyed. The latter case may give rise to a homoclinic orbit or a pair of heteroclinic orbits. The limit cycles and the separatrices can be calculated to any desired degree of accuracy. Stability and bifurcations of limit cycles will also be discussed.

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“…The third category can be termed the generalized harmonic function perturbation procedure. It includes the generalized harmonic average method [19], the generalized harmonic L-P method [20], the generalized harmonic KBM method [21], the generalized harmonic multiple scales method [22], etc. These methods are applicable when the generating equation contains an arbitrary nonlinear function g (x).…”

mentioning

confidence: 99%

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method

Chen

2009

Nonlinear Dyn

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The hyperbolic perturbation method is applied to determining the homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators of the formẍ + c 1 x + c 3 x 3 = εf (μ, x,ẋ), in which the hyperbolic functions are employed instead of the periodic functions in the usual perturbation method. The generalized Liénard oscillator with f (μ, x,ẋ) = (μ − μ 1 x 2 − μ 2ẋ 2 )ẋ is studied in detail. Comparisons with the numerical simulations obtained by using R-K method are made to show the efficacy and accuracy of the present method.

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Asymptotic Method for Analysis of Nonlinear Systems With Two Parameters

Cited by 9 publications

References 3 publications

Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator

Predicting Homoclinic and Heteroclinic Bifurcation of Generalized Duffing-Harmonic-van de Pol Oscillator

Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method

Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method

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