Asymptotic expansions of the solutions of nonlinear evolution equations

Lukomsky, V. P.; Bobkov, V. B.

doi:10.1109/mmet.1996.565745

Cited by 3 publications

(5 citation statements)

References 2 publications

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“…A remarkable example here is the embedding of the Newton procedure in the harmonic balance method [2,3,6,8,9,10]. The last motivation to the actual interest is simply to propose novel methods [1,11,13,15]. Some of these are specially designed and work very efficiently for systems with uncommon restoring force, such as those with fractional powers [13].…”

Section: Accepted Manuscriptmentioning

confidence: 99%

“…The development of methods applicable to the determination of approximate analytical solutions to oscillator equations of the form x + f (x,ẋ) = 0, x(0) = ρ 0 ,ẋ(0) = 0 (1) has witnessed a renewed interest in recent years. Here, single and double overdots denote respectively first and second derivatives with respect to time t, x = x(t) ∈ R, and f is a nonlinear function of its arguments.…”

Section: Introductionmentioning

confidence: 99%

“…[1,2,4,3,5,6,7,8,9,10,11,12,13,14,15] for a sample of the related works. We can identify three valuable motivations that sustain this interest.…”

Section: Introductionmentioning

confidence: 99%

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Linearized harmonic balance based derivation of slow flow for some class of autonomous single degree of freedom oscillators

Yamgoué

Kofané

2008

International Journal of Non-Linear Mechanics

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Cite this article as: Serge Bruno Yamgoué and Timoléon Crépin Kofané, Linearized harmonic balance based derivation of slow flow for some class of autonomous single degree of freedom oscillators, International Journal of Non-Linear Mechanics, doi:10.1016/j.ijnonlinmec.2008 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Section: Accepted Manuscriptmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linearized harmonic balance based derivation of slow flow for some class of autonomous single degree of freedom oscillators

Yamgoué

Kofané

2008

International Journal of Non-Linear Mechanics

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“…We remark that these developments are mainly focused to the determination of analytical approximations to periodic or limit-cycle solutions of oscillator equations [5,6,7,8,9,10,11,12,13,14,15,16]. Comparatively, little attention is granted to the determination of approximate solutions to autonomous damped equations.…”

Section: Introductionmentioning

confidence: 99%

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

Yamgoué¹,

Kofané²

2007

International Journal of Non-Linear Mechanics

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This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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“…where , [20], and their explicit form is given in Appendix A. Using these recurrence formulae and complex exponents in Fourier expansions is the key novel step in the formulation of the harmonic balance method developed in this paper.…”

Section: Introductionmentioning

confidence: 99%

Cascades of subharmonic stationary states in strongly non-linear driven planar systems

Lukomsky¹,

Gandzha²

2004

Journal of Sound and Vibration

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The dynamics of a one-degree of freedom oscillator with arbitrary polynomial non-linearity subjected to an external periodic excitation is studied. The sequences (cascades) of harmonic and subharmonic stationary solutions to the equation of motion are obtained by using the harmonic balance approximation adapted for arbitrary truncation numbers, powers of non-linearity, and orders of subharmonics. A scheme for investigating the stability of the harmonic balance stationary solutions of such a general form is developed on the basis of the Floquet theorem. Besides establishing the stable/unstable nature of a stationary solution, its stability analysis allows obtaining the regions of parameters, where symmetry-breaking and period-doubling bifurcations occur. Thus, for perioddoubling cascades, each unstable stationary solution is used as a base solution for finding a subsequent stationary state in a cascade. The procedure is repeated until this stationary state becomes stable provided that a stable solution can finally be achieved. The proposed technique is applied to calculate the sequences of subharmonic stationary states in driven hardening Duffing's oscillator. The existence of stable subharmonic motions found is confirmed by solving the differential equation of motion numerically by means of a time-difference method, with initial conditions being supplied by the harmonic balance approximation.

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Asymptotic expansions of the solutions of nonlinear evolution equations

Cited by 3 publications

References 2 publications

Linearized harmonic balance based derivation of slow flow for some class of autonomous single degree of freedom oscillators

Linearized harmonic balance based derivation of slow flow for some class of autonomous single degree of freedom oscillators

Application of the Krylov–Bogoliubov–Mitropolsky method to weakly damped strongly non-linear planar Hamiltonian systems

Cascades of subharmonic stationary states in strongly non-linear driven planar systems

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