Stability

Abstract

In the previous chapter, we explained the behavior of solutions of linear systems as $t \to + \infty$.In this chapter, we look into similar problems for nonlinear systems. To start with, in §VIII-1, we introduce the concepts of stability and asymptotic stability of a given particular solution as $t \to + \infty$.We illustrate those concepts with simple examples. Reducing the given solution to the trivial solution by a simple transformation, we concentrate our explanation on the stability property of the trivial solution. It is well known that the trivial solution is asymptotically stable as $t \to + \infty$ if real parts of eigenvalues of the leading matrix of the given system are all negative. This basic result is given as Theorem VIII-2-1 in §VIII-2. The case when some of those real parts are not negative is treated in §VIII-3. In particular, we discuss the stable and unstable manifolds. In §VIII-4, we look into the structure of stable manifolds more closely for analytic differential equations. First we change a given system by an analytic transformation to a simple standard form. By virtue of such a simplification, we can construct the stable manifold in a simple analytic form. This idea is applied to analytic systems in ℝ² in §VIII-6. In §§VIII-7-VIII-10, using the polar coordinates, we explain continuous perturbations of linear systems in ℝ². In §VIII-5, we summarize some known facts concerning linear systems with constant coefficients in ${\mathbb{R}^2}$ .The topics discussed in this chapter are also found in [CL, pp. 371–388], [Har2, pp. 160–161, 220–227], and [SC, pp. 49–96]. The materials in §§VIII-4 and VIII-6 are also found in [Du], [Huk5], and [Si2].