scholarly journals Stochastic Stability of Damped Mathieu Oscillator Parametrically Excited by a Gaussian Noise

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Claudio Floris
Keyword(s):  
Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Averaging ◽  
Critical Values ◽  
Second Moments ◽  
Ode System ◽  
Complex Eigenvalues ◽  
The Matrix ◽  
The Stability ◽  
Necessary And Sufficient

This paper analyzes the stochastic stability of a damped Mathieu oscillator subjected to a parametric excitation of the form of a stationary Gaussian process, which may be both white and coloured. By applying deterministic and stochastic averaging, two Itô’s differential equations are retrieved. Reference is made to stochastic stability in moments. The differential equations ruling the response statistical moment evolution are written by means of Itô’s differential rule. A necessary and sufficient condition of stability in the moments of orderris that the matrixArof the coefficients of the ODE system ruling them has negative real eigenvalues and complex eigenvalues with negative real parts. Because of the linearity of the system the stability of the first two moments is the strongest condition of stability. In the case of the first moments (averages), critical values of the parameters are expressed analytically, while for the second moments the search for the critical values is made numerically. Some graphs are presented for representative cases.

scholarly journals The Mean Stability Criteria in terms of Two Measures for Stochastic Differential Equations with Coefficient’s Uncertainty

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Rui Zhang ◽  
Yinjing Guo ◽  
Xiangrong Wang ◽  
Xueqing Zhang
Keyword(s):  
Optimal Control ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stochastic Systems ◽  
Stability Criteria ◽  
Two Measures ◽  
The Mean ◽  
The Stability

This paper extends the stochastic stability criteria of two measures to the mean stability and proves the stability criteria for a kind of stochastic Itô’s systems. Moreover, by applying optimal control approaches, the mean stability criteria in terms of two measures are also obtained for the stochastic systems with coefficient’s uncertainty.

Instability and Unbalance Response of Dissymmetric Rotor-Bearing Systems

1988 ◽  
Vol 110 (3) ◽  
pp. 288-294 ◽  
Author(s):  
P. M. Guilhen ◽  
P. Berthier ◽  
G. Ferraris ◽  
M. Lalanne
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Transfer Matrix ◽  
Differential System ◽  
Industrial Application ◽  
Floquet Theory ◽  
Unbalance Response ◽  
Stationary Coordinate System ◽  
The Matrix ◽  
The Stability

The study deals with the instability and unbalance response of dissymmetric rotors, when periodic differential equations are impossible to avoid. The method which yields motion instability is based on an extension of the well-known Floquet theory. A transfer matrix over one period of the motion is obtained, and the stability of the system can be tested with the eigenvalues of the matrix. To find the instability and the unbalance response, the Newmark formulation is used. Here, the dissymmetry comes either from the rotor or from the bearings in such a way that it is possible to solve a regular differential system without periodic coefficients, either in the stationary coordinate system or in the rotating one. Three examples are given, including an industrial application. The results show that the method proposed is satisfactory.

scholarly journals The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huanting Li ◽  
Yunfei Peng ◽  
Kuilin Wu
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Initial State ◽  
Switching Line ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

Convergence of Solutions of Third Order Differential Equations*

1969 ◽  
Vol 12 (6) ◽  
pp. 779-792 ◽  
Author(s):  
K. E. Swick
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Higher Order ◽  
Similar Technique ◽  
System Of Differential Equations ◽  
Third Order ◽  
Convergence Of Solutions ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Direct Use

Consider a system of differential equations . Solutions of this system are said to be convergent if, given any pair of solutions x(t), y(t), x(t) - y(t) → 0 as t → ∞. In this case the system is also said to be extremely stable.In [6] a technique was developed which yielded the convergence of solutions of the forced Lienard equation. Here a similar technique i s applied to forced third order equations. A critical step in [6] was to show that a certain matrix was negative definite. This could be done directly in [6] since the matrix was only 2 × 2. With third and higher order equations, direct use of necessary and sufficient conditions is not feasible since the computations become unwieldy.

Discrete-Element Methods for Stability Analysis of Complex Structures

1968 ◽  
Vol 72 (696) ◽  
pp. 1077-1086 ◽  
Author(s):  
J. S. Przemieniecki
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Load Condition ◽  
Buckling Load ◽  
Complex Structures ◽  
Matrix Methods ◽  
The Matrix ◽  
Deflection Analysis ◽  
The Stability ◽  
Discrete Element Methods

The matrix methods of structural analysis developed specifically for use on modern digital computers have now become universally accepted in structural design. These methods provide a means for rapid and accurate stress and deflection analysis of complex structures under static and dynamic loading conditions and they can also be used very effectively for the stability analysis. In the conventional stability analysis two possible approaches are normally used; either the differential equations describing the structural deflections are formulated and the lowest eigenvalue representing the buckling load condition is found for a given set of boundary conditions, or alternatively, if the differential equations are too difficult to prescribe, approximate deflection shapes are used in the strain energy expression for large deflections which is subsequently minimised, leading to the stability determinant whose lowest root represents the instability condition. When designing complex structures the conventional methods of finding buckling load conditions are extremely difficult to apply, and therefore in such cases we have to rely on the matrix methods of stability analysis.

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy, and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Stochastic Stability of Some Hamiltonian Systems

2002 ◽  
Author(s):  
F. Jedrzejewski
Keyword(s):  
Hamiltonian Systems ◽  
Stochastic Stability ◽  
Stochastic Modelling ◽  
Original System ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Van Der Pol Oscillator ◽  
Stochastic Fluctuations ◽  
The Stability ◽  
Averaged Hamiltonian

Stochastic stability plays an important role in modern theories of nonlinear structural dynamics. Recently, more realistic models based on stochastic modelling and Itoˆ calculus, like flow induced vibrations and seismic excitations have been proposed. In this paper, the almost-sure asymptotic stability of some hamiltonian systems subjected to stochastic fluctuations is investigated. Dynamical systems are reduced to Itoˆ stochastic differential equations for the averaged hamiltonian by using a new stochastic averaging method. The stability of the original system is determined approximately by examining the behavior of the averaged hamiltonian. Analytical expressions for the stochastic stability exponents are obtained. The proposed procedure is illustrated on the Rayleigh Van der Pol Oscillator.

scholarly journals A Note on the Periodic Solutions for a Class of Third Order Differential Equations

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 31
Author(s):  
Zouhair Diab ◽  
Juan L. G. Guirao ◽  
Juan A. Vera
Keyword(s):  
Differential Equations ◽  
Periodic Solutions ◽  
Monodromy Matrix ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Averaging Theory ◽  
Third Order ◽  
Existence Of Periodic Solutions ◽  
The Stability ◽  
Necessary And Sufficient

The aim of the present work is to study the necessary and sufficient conditions for the existence of periodic solutions for a class of third order differential equations by using the averaging theory. Moreover, we use the symmetry of the Monodromy matrix to study the stability of these solutions.

scholarly journals Stability of solutions of Caputo fractional stochastic differential equations

2021 ◽  
Vol 26 (4) ◽  
pp. 581-596
Author(s):  
Guanli Xiao ◽  
JinRong Wang
Keyword(s):  
Differential Equations ◽  
Exponential Stability ◽  
Stochastic Differential Equations ◽  
Stochastic Stability ◽  
Stopping Time ◽  
Asymptotical Stability ◽  
Stability Of Solutions ◽  
Numerical Examples ◽  
Fractional Stochastic Differential Equations ◽  
The Stability

In this paper, we study the stability of Caputo-type fractional stochastic differential equations. Stochastic stability and stochastic asymptotical stability are shown by stopping time technique. Almost surly exponential stability and pth moment exponentially stability are derived by a new established Itô’s formula of Caputo version. Numerical examples are given to illustrate the main results.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

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