The stability theory of comparison systems

The stability of incompressible boundary-layer flows on a semi-infinite flat plate and the growth of disturbances in such flows are investigated by numerical integration of the complete Navier–;Stokes equations for laminar two-dimensional flows. Forced time-dependent disturbances are introduced into the flow field and the reaction of the flow to such disturbances is studied by directly solving the Navier–Stokes equations using a finite-difference method. An implicit finitedifference scheme was developed for the calculation of the extremely unsteady flow fields which arose from the forced time-dependent disturbances. The problem of the numerical stability of the method called for special attention in order to avoid possible distortions of the results caused by the interaction of unstable numerical oscillations with physically meaningful perturbations. A demonstration of the suitability of the numerical method for the investigation of stability and the initial growth of disturbances is presented for small periodic perturbations. For this particular case the numerical results can be compared with linear stability theory and experimental measurements. In this paper a number of numerical calculations for small periodic disturbances are discussed in detail. The results are generally in fairly close agreement with linear stability theory or experimental measurements.

Download Full-text

Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems

ISRN Applied Mathematics ◽

10.1155/2014/472371 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 4

Author(s):

Bin Wang ◽

Yuangui Zhou ◽

Jianyi Xue ◽

Delan Zhu

Keyword(s):

Sliding Mode Control ◽

Fractional Order ◽

Wide Class ◽

Stability Theory ◽

Chaotic Systems ◽

Sliding Mode ◽

Order System ◽

Fractional Order Calculus ◽

Simulation Results ◽

The Stability

We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method.

Download Full-text

SYMPLECTIC STABILITY, ANALYTIC STABILITY IN NON-ALGEBRAIC COMPLEX GEOMETRY

International Journal of Mathematics ◽

10.1142/s0129167x04002223 ◽

2004 ◽

Vol 15 (02) ◽

pp. 183-209 ◽

Cited By ~ 10

Author(s):

ANDREI TELEMAN

Keyword(s):

Weight Function ◽

Complex Manifold ◽

Stability Theory ◽

Complex Geometry ◽

Reductive Group ◽

Hamiltonian Action ◽

Maximal Weight ◽

Analytic Stability ◽

The Stability

We give a systematic presentation of the stability theory in the non-algebraic Kählerian geometry. We introduce the concept of "energy complete Hamiltonian action". To an energy complete Hamiltonian action of a reductive group G on a complex manifold one can associate a G-equivariant maximal weight function and prove a Hilbert criterion for semistability. In other words, for such actions, the symplectic semistability and analytic semistability conditions are equivalent.

Download Full-text

Non-local effects in the stability of flow between eccentric rotating cylinders

Journal of Fluid Mechanics ◽

10.1017/s0022112072000758 ◽

1972 ◽

Vol 54 (3) ◽

pp. 393-415 ◽

Cited By ~ 54

Author(s):

R. C. Diprima ◽

J. T. Stuart

Keyword(s):

Differential Equations ◽

Stability Theory ◽

Taylor Number ◽

Circular Cylinders ◽

Taylor Vortex ◽

Local Theory ◽

Local Instability ◽

Clearance Ratio ◽

Linearized Stability ◽

The Stability

In this paper the linear stability of the flow between two long eccentric rotating circular cylinders is considered. The problem, which is of interest in lubrication technology, is an extension of the classical Taylor problem for concentric cylinders. The basic flow has components in the radial and azimuthal directions and depends on both of these co-ordinates. As a consequence the linearized stability equations arepartial differential equationsrather than ordinary differential equations. Thus standard methods of stability theory are not immediately useful. However, there are two small parameters in the problem, namely δ, the clearance ratio, and ε, the eccentricity. By letting these parameters tend to zero in such a way that δ½ is proportional to ε, a global solution to the stability problem is obtained without recourse to the concept of ‘local instability’, or ‘parallel-flow’ approximation, so commonly used in boundary-layer stability theory. It is found that the predictions of the present theory are at variance with what is given by a ‘local’ theory. First, the Taylor-vortex amplitude is found to be largest at about 90° downstream of the region of ‘maximum local instability’. This result is given support by some experimental observations of Vohr (1968) with δ = 0·1 and ε = 0·475, which yield a corresponding angle of about 50°. Second, the critical Taylor number rises with ε, rather than initially decreasing with ε as predicted by local stability theory using the criteria of maximum local instability. The present prediction of the critical Taylor number agrees well with Vohr's experiments for ε up to about 0·5 when δ = 0·01 and for ε up to about 0·2 when δ = 0.1.

Download Full-text

Plasma stability theory including the resistive wall effects

Journal of Plasma Physics ◽

10.1017/s0022377815001403 ◽

2015 ◽

Vol 81 (6) ◽

Cited By ~ 15

Author(s):

V. D. Pustovitov

Keyword(s):

Stability Theory ◽

Nonlinear Function ◽

Thin Wall ◽

Conducting Wall ◽

Resistive Wall ◽

Energy Sink ◽

The Stability ◽

Resistive Wall Modes ◽

Complex Growth Rate ◽

Plasma Stabilization

Plasma stabilization due to a nearby conducting wall can provide access to better performance in some scenarios in tokamaks. This was proved by experiments with an essential gain in${\it\beta}$and demonstrated as a long-lasting effect at sufficiently fast plasma rotation in the DIII-D tokamak (see, for example, Straitet al.,Nucl. Fusion, vol. 43, 2003, pp. 430–440). The rotational stabilization is the central topic of this review, though eventually the mode rotation gains significance. The analysis is based on the first-principle equations describing the energy balance with dissipation in the resistive wall. The method emphasizes derivation of the dispersion relations for the modes which are faster than the conventional resistive wall modes, but slower than the ideal magnetohydrodynamics modes. Both the standard thin wall and ideal-wall approximations are not valid in this range. Here, these are replaced by an approach incorporating the skin effect in the wall. This new element in the stability theory makes the energy sink a nonlinear function of the complex growth rate. An important consequence is that a mode rotating above a critical level can provide a damping effect sufficient for instability suppression. Estimates are given and applications are discussed.

Download Full-text

The stability of the positive solution for a fractional SIR model

International Journal of Biomathematics ◽

10.1142/s1793524517500140 ◽

2016 ◽

Vol 10 (01) ◽

pp. 1750014 ◽

Cited By ~ 1

Author(s):

Yingjia Guo

Keyword(s):

Differential Equation ◽

Positive Solution ◽

Stability Theory ◽

Lyapunov Stability Theory ◽

Sir Model ◽

Differentiable Functions ◽

Classical Calculus ◽

Global Positive Solution ◽

Fractional Differential ◽

The Stability

In order to deal with non-differentiable functions, a modification of the Riemann–Liouville definition is recently proposed which appears to provide a framework for a fractional calculus which is quite parallel with classical calculus. Based on these new results, we study on the fractional SIR model in this paper. First, we give the general solution of the fractional differential equation. And then a unique global positive solution of the SIR model is obtained. Furthermore, we get the Lyapunov stability theory. By using this stability theory, the asymptotic stability of the positive solution is analyzed.

Download Full-text