Extremal Points of Integral Curves of Second-Order Ordinary Differential Equations and Their Local Stability

Abstract

In [1--3] an extension of the solution of the equation $a\left( {x,\dot x} \right)\ddot x = 1,x \in \mathbb{R},a\left( {x,\dot x} \right) \in C^1$, to the singular set $S = \left\{ {\left( {x,y} \right) \in \mathbb{R}^2 :a\left( {x,y} \right) = 0} \right\},y = \dot x$, is defined in terms of the first integral. In this case all stationary points and all local extrema of the integral curve $x\left( y \right)$ such that the function $x\left( y \right)$ has a derivative at the extreme point belong to a set $S \cup Y$, where Y is the line $y = 0$. We study the local stability of local extrema of different types in the families of equations $\left[ {a\left( {x,y} \right) + \varepsilon b\left( {x,y} \right)} \right]\dot y = 1,{\text{ }}b\left( {x,y} \right) \in C^1 ,{\text{ for }}\left| \varepsilon \right|$ small enough. Introduce the notation $S^* = \left\{ {\left( {x,y} \right) \in \mathbb{R}^2 :a\left( {x,y} \right) + \varepsilon b\left( {x,y} \right) = 0} \right\}$. By abuse of language, we talk about the stability of local extrema when S is replaced with $S^*$. Some sufficient conditions for stability and instability are found.