A role of Abel's equation in the stability theory of differential equations

1971 ◽  
Vol 6 (1) ◽  
pp. 66-70 ◽  
Author(s):  
F. Neuman
Keyword(s):  
Differential Equations ◽  
Stability Theory ◽  
The Stability

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

1972 ◽  
Vol 54 (3) ◽  
pp. 393-415 ◽  
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.

The Projective Synchronization of Drive-Reponse Complex Dynamical Networks

2014 ◽  
Vol 687-691 ◽  
pp. 2458-2461
Author(s):  
Feng Ling Jia
Keyword(s):  
Differential Equations ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Stability Theory ◽  
Complex Dynamical Networks ◽  
Projective Synchronization ◽  
Dynamical Networks ◽  
Fractional Order Differential Equations ◽  
The Stability ◽  
Response Complex

This paper investigates the projective synchronization of drive-response complex dynamical networks. Based on the stability theory for fractional-order differential equations, controllers are designed torealize the projective synchronization for complex dynamical networks. Morover, some simple synchronization conditions are proposed. Numerical simulations are presented to show the effectiveness of the proposed method.

scholarly journals Stability of Systems of Fractional-Order Differential Equations with Caputo Derivatives

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 914
Author(s):  
Oana Brandibur ◽  
Roberto Garrappa ◽  
Eva Kaslik
Keyword(s):  
Differential Equations ◽  
Detailed Analysis ◽  
Linear Systems ◽  
Fractional Order ◽  
Numerical Experiments ◽  
Limit Case ◽  
Fractional Order Differential Equations ◽  
Fractional Differential ◽  
The Stability

Systems of fractional-order differential equations present stability properties which differ in a substantial way from those of systems of integer order. In this paper, a detailed analysis of the stability of linear systems of fractional differential equations with Caputo derivative is proposed. Starting from the well-known Matignon’s results on stability of single-order systems, for which a different proof is provided together with a clarification of a limit case, the investigation is moved towards multi-order systems as well. Due to the key role of the Mittag–Leffler function played in representing the solution of linear systems of FDEs, a detailed analysis of the asymptotic behavior of this function and of its derivatives is also proposed. Some numerical experiments are presented to illustrate the main results.

On Stability in the Large for Systems of Ordinary Differential Equations

1961 ◽  
Vol 13 ◽  
pp. 480-492 ◽  
Author(s):  
Philip Hartman
Keyword(s):  
Euclidean Space ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Riemannian Manifolds ◽  
Autonomous Systems ◽  
The Stability ◽  
Image Position

Autonomous systems. This note concerns the stability of systems of (real) differential equations in the large on Euclidean space En and on certain Riemannian manifolds Mn. The results will be refinements of those of Krasovski (3), (4), (5) and of Markus and Yamabe (8) and will make clear the role of the various assumptions in the type of theorems under consideration.In this section, the main theorems are stated for autonomous systems(1)Their proofs are given in § 2, 3, 4. In § 5, 6, 7, generalizations to non-autonomous systems are made.

scholarly journals ON THE INTERLACING PROPERTY OF STABLE COMPLEX SYSTEMS OF DIFFERENTIAL EQUATIONS

1994 ◽  
Vol 25 (4) ◽  
pp. 317-320
Author(s):  
ZIAD ZAHREDDINE
Keyword(s):  
Differential Equations ◽  
Complex Systems ◽  
Stability Theory ◽  
Interesting Property ◽  
Stable Complex ◽  
Systems Of Differential Equations ◽  
The Stability ◽  
Stability Results ◽  
Interlacing Property

By exploiting recent stability results, an interesting property known in stability theory as the interlacing property is revisited and reproduced. The approach is straightforward and highlights the central role that positive para-odd functions is currently playing in the stability of complex systems of differential equations.

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