Moment Lyapunov exponents and stochastic stability for two coupled oscillators

Kozi, Predrag; Janevski, Goran

doi:10.2140/jomms.2009.4.1689

Cited by 11 publications

(5 citation statements)

References 9 publications

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“…and after introdusing suposed expressions (64) for solutions in the system of partial fractional order differential equations ( 63), we obtain system of ordinary fractional order differential equations along eigen time functions T k nm ð Þ ðtÞ, k ¼ 1; 2; 3, n; m ¼ 1; 2; 3; 4 . .…”

Section: Governing Partial Fractional Order Differential Equations Of a Hybrid Multi Deformable Beam System Transversal Oscillationsmentioning

confidence: 99%

“…12,13,20,24,27,31,33,35,[39][40][41] In Hedrih (Stevanovi c) and Simonovi c [42][43][44][45][46][47][48][49][50][51] published in period in 2008-2013, and late published Simonovi c [52][53][54] as well as in monographs published by Kluwer and Springer and other contributions of author and coauthors to linear and nonlinear dynamics of deformable bodies (rods, beams, plates, moving strips) can be classified as systems of coupled subsystems and deformable bodies. [55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73] After the first published scientific papers by Hedrih (Stevanovi c) K.R. in the newly opened field of coupled deformable bodies and the dynamics of hybrid systems of complex structures, Hedrih (Stevanovi c) K.R.…”

Section: Introductionmentioning

confidence: 99%

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Linear and nonlinear dynamics of hybrid systems

Hedrih

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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Discrete continuum method for investigation of linear and nonlinear dynamics of hybrid systems containing coupled multi deformable bodies is presented. By use coupled rods, beams, strings, plates and membranes by discrete continuum mass less layers as well as layers with translator and rotator inertia properties into series of hybrid system dynamics are investigated and phenomenological mappings in dynamics of these different real systems are identified. Expressions of generalized forces of subsystem interactions in hybrid system are presented by component mechanical energies and functions of energy dissipations. A model of dynamical dislocations with inertia properties in plate is presented. Transfer energy between subsystems is investigated. Constitutive relation of standard elements of discrete continuum coupling layers with translator and rolling inertia properties, nonlinear elastic and fractional order properties are presented. Interaction between two coupled linear and nonlinear system, each with one degree of freedom as well as dynamics of discrete no homogeneous chain are considered in the light of mathematical analogy for obtaining eigen time functions of solutions of component deformable body displacements in hybrid system dynamics. For pointing out the major contributions outlined in the manuscript it is necessary to add: The manuscript contains reviews on results obtained of a few scientific problems of nonlinear dynamics, namely, for stochastic stability of vibration modes of a parametrically excited sandwich beam, transversal vibrations of axially moving double belt system, multi deformable bodies coupled by standard light fractional type discrete continuum layers. New model of dynamical dislocation in continuum is proposed and analyzed Series of the original results of author’s doctorates supervised is listed.

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Section: Governing Partial Fractional Order Differential Equations Of a Hybrid Multi Deformable Beam System Transversal Oscillationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Linear and nonlinear dynamics of hybrid systems

Hedrih

2020

Proceedings of the Institution of Mechanical Engineers, Part C:

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“…It was concluded that the ``periodic windows" or intervals of the largest Lyapunov exponent beyond the onset of chaotic motion are gradually diminished when the intensity of bounded noise is increased. Another approach on the two degree-of-freedom linear systems [16] was presented and better response of stability from Lyapunov and moment Lyapunov were recorded. Therefore, so far the disturbance due to colored noise has not been focused.…”

Section: Cdsr 116-2mentioning

confidence: 99%

Adaptive Control of Satellite System using Kalman Estimates in the presence of Disturbance

Dube¹,

Patel²

2019

International Conference of Control, Dynamic Systems, and Robotics

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There has always been an increasing interest in the space related activities. The research on aerospace vehicles have become popular due to the problems and complexity of the systems involved. The main challenge lies in keeping a balance when in motion and hence tracking plays a key role here. The satellite system discussed in this paper is described for the problem arising due to presence of disturbances (colored and white noise). This model is evinced by the augmentation of discrete Kalman filter, a powerful tool for controlling the noisy estimates. Whereas, the regulation of the state dynamics (Euler angles and velocity) is achieved with the adaptive law which focuses on guaranteeing stability of adaptive control scheme using a good choice of Lyapunov argument. The presence of Gaussian White noise is also mentioned where a comparison resembles stability for almost, mean square, third moment and mean value between the satellite and oscillator system. A clear discussion of the system in the presence of colored and white noise with perfect estimates derived from kalman filter and an adaptation law governing the evolution of system around equilibrium is done.

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“…For example, Sri Namachchivaya and Van Roessel [27] studied the moment Lyapunov exponents of two-degrees-of-freedom coupled elastic oscillators under real noise excitation by combining an asymptotic approximation for the moment Lyapunov exponents. Kozić and his associates [28][29][30] developed the first-order perturbation approach to obtain weak noise expansion of moment Lyapunov exponents and Lyapunov exponents for a stochastically coupled doublebeam system and Timoshenko beam system, respectively. Subsequently, Stojanović and Petković [31] used a perturbation method to study the moment Lyapunov exponents and the Lyapunov exponents of the three elastically connected Euler beams.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

Liu

Xie

2018

Mathematical Problems in Engineering

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Nonviscously damped structural system has been raised in many engineering fields, in which the damping forces depend on the past time history of velocities via convolution integrals over some kernel functions. This paper investigates stochastic stability of coupled viscoelastic system with nonviscously damping driven by white noise through moment Lyapunov exponents and Lyapunov exponents. Using the coordinate transformation, the coupled Itô stochastic differential equations of the norm of the response and angles process are obtained. Then the problem of the moment Lyapunov exponent is transformed to the eigenvalue problem, and then the second-perturbation method is used to derive the moment Lyapunov exponent of coupled stochastic system. Lyapunov exponent also can be obtained according to the relationship between moment Lyapunov exponent and Lyapunov exponent. Finally, the effects of various physical quantities of stochastic coupled system on the stochastic stability are discussed in detail. These results are validated by the direct Monte Carlo simulation technique.

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Moment Lyapunov exponents and stochastic stability for two coupled oscillators

Cited by 11 publications

References 9 publications

Linear and nonlinear dynamics of hybrid systems

Linear and nonlinear dynamics of hybrid systems

Adaptive Control of Satellite System using Kalman Estimates in the presence of Disturbance

Stochastic Stability Analysis of Coupled Viscoelastic Systems with Nonviscously Damping Driven by White Noise

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