Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Mikhlin, Yuri V.; Shmatko, Tetyana; Manucharyan, G.V.

doi:10.1016/j.compstruc.2004.03.082

Cited by 9 publications

(7 citation statements)

References 20 publications

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“…3. The Runge-Kutta test for the equation in variations (10) shows limited/ unlimited solutions when parameters are chosen from the stability/ instability regions obtained earlier. The same fixed parameters as were used above are used in the calculations.…”

Section: Eejp 2 2019mentioning

confidence: 83%

“…Namely, the so-called Ince algebraization [8] is used. Note that this approach was successfully used earlier in study of the stability of nonlinear normal vibration modes in nonlinear systems with a finite number of degrees of freedom [9][10][11]. Besides, this procedure is similar to one proposed in the paper [5] for the stability of traveling waves problem, but results on such stability problem for concrete systems are not presented in this publication.…”

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confidence: 99%

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Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation

2019

EEJP

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Nonlinear traveling waves of the Klein-Gordon equation with cubic nonlinearity are considered. These waves are described by the nonlinear ordinary differential equation of the second order having the energy integral. Linearized equation for variation obtained for such waves is transformed to the ordinary one using separation of variables. Then so-called algebraization by Ince is used. Namely, a new independent variable associated with the solution under consideration is introduced to the equation in variations. Integral of energy for the stationary waves is used in this transformation. An advantage of this approach is that an analysis of the stability problem does no need to use the specific form of the solution under consideration. As a result of the algebraization, the equation in variations with variable in time coefficients is transformed to equation with singular points. Indices of the singularities are found. Necessary conditions of the waves stability are obtained. Solutions of the variational equation, corresponding to boundaries of the stability/instability regions in the system parameter space, are constructed in power series by the new independent variable. Infinite recurrent systems of linear homogeneous algebraic equations to determine coefficients of the series can be written. Non-trivial solutions of these systems can be obtained if their determinants are equal to zero. These determinants are calculated up to the fifth order inclusively, then relations connecting the system parameters and corresponding to boundaries of the stability/ instability regions in the system parameter place are obtained. Namely, the relation between parameters of anharmonicity and energy of the waves are constructed. Analytical results are illustrated by numerical simulation by using the Runge-Kutta procedure for some chosen parameters of the system. A correspondence of the numerical and analytical results is observed.

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Section: Eejp 2 2019mentioning

confidence: 83%

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confidence: 99%

Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation

2019

EEJP

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“…Semi-analytical approach of the NNMs stability analysis is proposed in Ref. [150]. This approach is based on the definition of Lyapunov stability.…”

Section: Numerical Methods For Nnms Calculationsmentioning

confidence: 99%

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

Mikhlin

Аврамов

2010

Applied Mechanics Reviews

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Two principal concepts of nonlinear normal vibrations modes (NNMs), namely the Kauderer-Rosenberg and Shaw-Pierre concepts, are analyzed. Properties of the NNMs and methods of their analysis are presented. NNMs stability and bifurcations are discussed. Combined application of the NNMs and the Rauscher method to analyze forced and parametric vibrations is discussed. Generalization of the NNMs to continuous systems dynamics is also described.

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“…Often a problem of stability of modes or stationary waves with complex behavior in time has no analytical solution. It particular, it concerns to stability problem in postbuckling behavior of elastic systems, such as roads, plates, shells, when chaotic in time behavior can be observed [18,19]. The stability of NNMs with complex behavior in time is considered here by use of the numerical-analytical approach based on the known Lyapunov definition of stability [7].…”

Section: Stability Of Nonlinear Normal Mode With Complex Behavior In Timementioning

confidence: 99%

“…Thus one can formulate a problem of the stability of periodic/ chaotic vibration mode in the higher-dimensional spaces. Taking into account that analytical approaches in a case of chaotic motions are absent, moreover, an analysis of stability of stationary states with complex behavior in time, is difficult, the numerical-analytical test which is based on the known Lyapunov definition of stability [7] is used in such stability problem [13]. This test can be used also in a problem of the nonlinear standing waves stability.…”

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confidence: 99%

Stability of Steady States with Complex Behavior in Time

Mikhlin

Goloskubova

2021

Springer Proceedings in Mathematics &Amp; Statistics

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A stability of the NNMs and standing or traveling waves is analyzed by two approaches. One of them is the method of Ince algebraization (IA), when a new independent variable associated with the unperturbed solution is used as independent one. In this case equations in variations transform to equations with singular points. A problem of determination of solutions, corresponding to boundaries of the stability/ instability regions, is reduced here to problem of determination of ones that have singularity at the mentioned points. An advantage of the IA is that in the stability problem we do not need in use of the unperturbed solution time-presentation. Other approach of the steady states stability investigation is associated with the classical Lyapunov definition of stability. An implementation of this definition permits to obtain boundaries between the stability/instability regions in the system parameter space. Such analytical-numerical test can be used in stability problem for periodic vibrations or waves with complex behavior in time when the stability problem has no analytical solution.

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Lyapunov definition and stability of regular or chaotic vibration modes in systems with several equilibrium positions

Cited by 9 publications

References 20 publications

Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation

Algebraization in Stability Problem for Stationary Waves of the Klein-Gordon Equation

Nonlinears Normal Modes for Vibrating Mechanical Systems. Review of Theoretical Developments

Stability of Steady States with Complex Behavior in Time

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