Second method of Lyapunov and comparison principle for impulsive differential–difference equations

In the present paper questions related to stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations are considered. The investigations are carried out by means of piecewise-continuous functions which are analogues of the classical Lyapunov's functions. By means of a vectorial comparison equation and differential inequalities for piecewise-continuous functions, theorems are proved on stability and boundedness with respect to manifolds of solutions of impulsive differential-difference equations with impulse effect at fixed moments.

On the stability of viscous flow between rotating cylinders Part 1. Asymptotic analysis

The stability of Couette flow is discussed in the case in which the cylindes rotate in opposite directions by an asymptotic method in which the Taylor number is treated as a large parameter. On assuming the principle of exchange of stabilities to hold, the problem is then governed by a sixth-order differential equation with a simple turning point. It is shown how the solutions of this equation can be represented asymptotically in terms of the solutions of the comparison equation yvi = xy. The solutions of this comparison equation have recently been tabulated and we thus have an explicit representation of the solution of the stability problem in terms of tabulate functions. Detailed results for the critical Taylor number and wave-number at the onset of instability and the associated eigenfunctions are given for the limiting case μ → − ∞, where μ = Ω2/Ω1, and Ω1 and Ω2 are the angular velocities of the inner and outer cylinders respectively. In this limiting case it is found that there exists and infinite number of cells between the cylinders, but that the amplitude of the secondary motion in all but the innermost cell is small.