Stability of Linear Conservative Gyroscopic Systems

1991 ◽  
Vol 58 (1) ◽  
pp. 229-232 ◽  
Author(s):  
J. A. Walker
Keyword(s):  
Sufficient Conditions ◽  
Design Constraints ◽  
Stability And Instability ◽  
The Stability ◽  
Gyroscopic Systems

Sufficient conditions are obtained for the stability and instability of linear conservative gyroscopic systems. The conditions are nonspectral, involve only the definiteness of certain combinations of the coefficient matrices, and may yield useful design constraints. An example is employed to compare these results with earlier results of the same type.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Eigenvalue problems for the wave equation with strong damping

1997 ◽  
Vol 127 (4) ◽  
pp. 755-771 ◽  
Author(s):  
Pedro Freitas
Keyword(s):  
Stationary Solution ◽  
Wave Equations ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Stationary Solutions ◽  
Linear Operators ◽  
Stability And Instability ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Strongly Damped

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

scholarly journals Stability and Instability Criteria for Idealized Precipitating Hydrodynamics

2015 ◽  
Vol 72 (6) ◽  
pp. 2379-2393 ◽  
Author(s):  
Gerardo Hernandez-Duenas ◽  
Leslie M. Smith ◽  
Samuel N. Stechmann
Keyword(s):  
Full Range ◽  
Potential Temperature ◽  
Sufficient Conditions ◽  
Singular Limit ◽  
Energy Principle ◽  
Equivalent Potential ◽  
Stability And Instability ◽  
Conditional Instability ◽  
The Stability ◽  
Single Boundary

Abstract A linear stability analysis is presented for fluid dynamics with water vapor and precipitation, where the precipitation falls relative to the fluid at speed VT. The aim is to bridge two extreme cases by considering the full range of VT values: (i) VT = 0, (ii) finite VT, and (iii) infinitely fast VT. In each case, a saturated precipitating atmosphere is considered, and the sufficient conditions for stability and instability are identified. Furthermore, each condition is linked to a thermodynamic variable: either a variable θs, which denotes the saturated potential temperature, or the equivalent potential temperature θe. When VT is finite, separate sufficient conditions are identified for stability versus instability: dθe/dz > 0 versus dθs/dz < 0, respectively. When VT = 0, the criterion dθs/dz = 0 is the single boundary that separates the stable and unstable conditions; and when VT is infinitely fast, the criterion dθe/dz = 0 is the single boundary. Asymptotics are used to analytically characterize the infinitely fast VT case, in addition to numerical results. Also, the small-VT limit is identified as a singular limit; that is, the cases of VT = 0 and small VT are fundamentally different. An energy principle is also presented for each case of VT, and the form of the energy identifies the stability parameter: either dθs/dz or dθe/dz. Results for finite VT have some resemblance to the notion of conditional instability: separate sufficient conditions exist for stability versus instability, with an intermediate range of environmental states where stability or instability is not definitive.

Dynamics of Asymmetric Nonconservative Systems

1983 ◽  
Vol 50 (1) ◽  
pp. 199-203 ◽  
Author(s):  
D. J. Inman
Keyword(s):  
Sufficient Conditions ◽  
Symmetric System ◽  
Qualitative Behavior ◽  
Parameter System ◽  
Nonconservative Systems ◽  
Asymmetric System ◽  
Lumped Parameter ◽  
Stability And Instability ◽  
The Stability ◽  
Necessary And Sufficient

This work examines a linear, asymmetric lumped parameter system. Results on the qualitative behavior of a certain subclass of such systems are presented. In particular, necessary and sufficient conditions for the existence of a linear transformation that transforms an asymmetric system into an equivalent symmetric system are derived. Results on the stability and instability of such systems are presented and stated in terms of the original asymmetric system’s coefficient matrices. This work is compared with that of other authors and numerical examples illustrating the utility and correctness of the results are presented.

A Sufficient Condition for the Stability of Conservative Gyroscopic Systems

1988 ◽  
Vol 55 (4) ◽  
pp. 895-898 ◽  
Author(s):  
D. J. Inman
Keyword(s):  
Equations Of Motion ◽  
Physical Parameters ◽  
Sufficient Condition ◽  
Design Constraints ◽  
The Stability ◽  
Gyroscopic Systems

A sufficient condition for the stability of conservative gyroscopic systems with negative definite stiffness is presented. The conditions for stability are stated in terms of the definiteness of certain combinations of the coefficient matrices of the equations of motion. These conditions yield design constraints in terms of the physical parameters of the system. An example is given to illustrate the correctness of the result, as well as to provide a comparison with the results of other researchers.

Stability and instability of solitary waves of Korteweg-de Vries type

1987 ◽  
Vol 411 (1841) ◽  
pp. 395-412 ◽  
Keyword(s):  
Solitary Waves ◽  
Sufficient Conditions ◽  
Solitary Wave Solutions ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Stability And Instability ◽  
Special Cases ◽  
De Vries ◽  
The Stability ◽  
Necessary And Sufficient

Considered herein are the stability and instability properties of solitary-wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear, dispersive long waves. Special cases for which our analysis is decisive include equations of the Korteweg-de Vries and Benjamin-Ono type. Necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable.

Stability, Instability and Bifurcation Modes of a Delayed Small World Network with Excitatory or Inhibitory Short-Cuts

2016 ◽  
Vol 26 (04) ◽  
pp. 1650070 ◽  
Author(s):  
Jing Zhou ◽  
Xu Xu ◽  
Dongyuan Yu ◽  
Zhuoqun Zheng
Keyword(s):  
Field Theory ◽  
Mean Field Theory ◽  
Mean Field ◽  
Sufficient Conditions ◽  
Small World ◽  
System Stability ◽  
Upper And Lower Bounds ◽  
Stability And Instability ◽  
The Mean ◽  
The Stability

This paper presents a detailed analysis on the stability and instability of a coupled oscillator network with small world connections. This network consists of regular connections, excitatory short-cuts or inhibitory short-cuts. By using the perturbation theory of matrix, we give the upper and lower bounds of maximum and minimum eigenvalues of the coupling strength matrix, and then give the generalized sufficient conditions that guarantee the system complete stability or complete instability. In addition, we analyze the effects of the short-cut possibility, excitatory or inhibitory short-cut strength and time delay on the system stability. We also analyze the instability mechanism and bifurcation modes. In addition, the studies on the robustness stability show that the stability of this network is more robust to change of short-cut connections than the regular network. Compared to the mean-field theory, the stability conditions from the proposed method are more conservational. However, the proposed method can guarantee the complete stability even if the randomness is in the system. They are more useful and adaptive than mean-field theory especially when the excitatory and inhibitory connections exist simultaneously.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

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