The Second-Order Differential Equation $$\frac{{{{d}^{2}}x}}{{d{{t}^{2}}}} + h\left( x \right)\frac{{dx}}{{dt}} + g\left( x \right) = 0$$

Abstract

In this chapter, we explain the basic results concerning the behavior of solutions of a system $$\frac{{{{d}^{2}}x}}{{d{{t}^{2}}}} + \epsilon \left( {{{x}^{2}} - 1} \right)\frac{{dx}}{{dt}} + x = 0$$ (E) \(t \to + \infty\) In §X-2, using results given in §IX-2, we show the boundedness of solutions and apply these results to the van der Pol equation $$\frac{{{{d}^{2}}x}}{{d{{t}^{2}}}} + \epsilon \left( {{{x}^{2}} - 1} \right)\frac{{dx}}{{dt}} + x = 0$$ (cf. Example X-2–5). The boundedness of solutions and the instability of the unique stationary point imply that the van der Pol equation has a nontrivial periodic solution. This is a consequence of the Poincare-Bendixson Theorem (cf. Theorem IX-4-1). In §X-3, we prove the uniqueness of periodic orbits in such a way that it can be applied to equation (E). In §X-4, we show that the absolute value of one of the two multipliers of the unique periodic solution of (E) is less than 1. The argument in §X-4 gives another proof of the uniqueness of periodic orbit of (E). In §X-5, we explain how to approximate the unique periodic solution of (E) in the case when e is positive and small. This is a typical problem of regular perturbations. In §X-6, we explain how to locate the unique periodic solution of (E) geometrically as \(\epsilon \to + \infty\). In §X-8, we explain how to find an approximation of the periodic solution of (E) analytically as \(\epsilon \to + \infty\). This is a typical problem of singular perturbations. Concerning singular perturbations, we also explain a basic result due to M. Nagumo [Na6] in §X-7. In §X-1, we look at a boundary-value problem $$ \frac{{{d^2}y}}{{d{t^2}}}{\text{ }}F(t,y,\frac{{dy}}{{dt}}){\text{ }}y(a) = a,{\text{ }}y(b) = \beta . $$