Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations

Козлов, В. В.; Furta, Stanislav D.

doi:10.1007/978-3-642-33817-5

Cited by 42 publications

(57 citation statements)

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“…In accordance with the major results in [14][15][16][17], if (5) allows the existence of the solution q = q(t) with the property q(t) → 0 as t → −∞, then the equilibrium state q = 0,q = 0 of the considered mechanical system is unstable.…”

Section: Introductionsupporting

confidence: 76%

“…Kozlov [14,17] and Kozlov and Furta [15][16] generalized Lyapunov's first method for the strongly nonlinear systems of differential equations. In accordance with the major results in [14][15][16][17], if (5) allows the existence of the solution q = q(t) with the property q(t) → 0 as t → −∞, then the equilibrium state q = 0,q = 0 of the considered mechanical system is unstable.…”

Section: Introductionmentioning

confidence: 99%

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Lyapunov-Kozlov method for singular cases

Čović

Djuric

Vesković

et al. 2011

Appl. Math. Mech.-Engl. Ed.

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Lyapunov's first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov's approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov's generalization of Lyapunov's first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.

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Section: Introductionsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 99%

Lyapunov-Kozlov method for singular cases

Čović

Djuric

Vesković

et al. 2011

Appl. Math. Mech.-Engl. Ed.

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“…Это утверждение и его обобщения играют существенную роль в первом методе Ляпу-нова для сильно нелинейных систем [6]. Укажем кратко основные моменты его доказатель-ства.…”

Section: динамика систем с сервосвязями I 359unclassified

The dynamics of systems with servoconstraints. I

Козлов

2015

Nelin. Dinam.

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“…Known asymptotic methods in nonlinear dynamics such as the Krylov-Bogoliubov averaging method, the Lyapunov method of slow varying parameters, the Poincare and the Van der Pol methods, and KAM-theory are used to improve perturbation theory results when describing small oscillations, see [1,2,3,4]. Even the term itselfnonlinear mechanics -is frequently considered as a synonym to nonlinear oscillations theory, see [1].…”

Section: Introductionmentioning

confidence: 99%

“…Expansions of elliptic functions in series of hyperbolic functions with error estimations for n-mode approximations are presented in Sect. 4. A rearrangement of the expansion of the elliptic function in a form suitable to relate them with the series obtained via the hyperbolic analogue of the averaging method is presented in Sect.5.…”

Section: Introductionmentioning

confidence: 99%

Rolling in the Higgs model and elliptic functions

2012

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Asymptotic methods in nonlinear dynamics are used to improve perturbation theory results in the oscillations regime. However, for some problems of nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and the Friedmann cosmological equations, not only small oscillations regime is of interest but also the regime of rolling (climbing), more precisely the rolling from a top (climbing to a top). In the Friedman cosmology, where the slow rolling regime is often used, the rolling from a top (not necessary slow) is of interest too. In the present work a method for approximate solution to the Higgs equation in the rolling regime is presented. It is shown that in order to improve perturbation theory in the rolling regime turns out to be effective not to use an expansion in trigonometric functions as it is done in case of small oscillations but use expansions in hyperbolic functions instead. This regime is investigated using the representation of the solution in terms of elliptic functions. An accuracy of the corresponding approximation is estimated.

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Asymptotic Solutions of Strongly Nonlinear Systems of Differential Equations

Cited by 42 publications

References 0 publications

Lyapunov-Kozlov method for singular cases

Lyapunov-Kozlov method for singular cases

The dynamics of systems with servoconstraints. I

Rolling in the Higgs model and elliptic functions

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