Stability region

A set in the space of values of the parameter on which the Cauchy problem depends. This set (which is not a domain, in general) is the union of the connected components of the interior of the set $S$ of values of the parameter for which the solution of the Cauchy problem is Lyapunov stable (cf. Lyapunov stability) and the set of all points on the boundary of these components which belong to $S$. The definition quoted is an attempt to give a precise meaning to a concept usually described in a more or less diffuse manner (cf. [1], p. 194, 195, 197).

Example. The null solution of the Mathieu equation

$$\ddot y+(\delta+\epsilon\cos t)y=0,$$

depending on the parameters $(\delta,\epsilon)\in\mathbf R^2$, has a countable set of stability regions (cf. [1], Figure 78). Among these are regions meeting the half-plane $\delta<0$, which explains the possibility of stabilization of the upper position of equilibrium of a pendulum by means of periodic (sinusoidal) oscillations of the point of the hanger in the vertical direction (cf. [1], Chapt. VI, Sect. 1.4).

References

[1]	J.J. Stoker, "Nonlinear vibrations in mechanical and electrical systems" , Interscience (1950)
[2]	N.N. Bautin, "The behaviour of dynamical systems near the border of their region of stability" , Moscow (1984) (In Russian)

Comments

For a difference method used to solve an ordinary differential equation numerically, discrete analogues of this region exist. In particular, such a set $S$ then refers to a part of the complex plane; often a model problem like $\dot x=\lambda x$ ($\lambda$ complex) is used for this.

References

[a1]	E. Harter, S. Nörgett, G. Wanner, "Solving ordinary differential equations" , I , Springer (1987)