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

In [1–3] an extension of the solution of the equation $a(x,\dot x)\ddot x=1$, $x\in \mathbb R$, $a(x,\dot x)\in C^1$, to the singular set $S=\{(x,y)\in \mathbb R^2:a(x,y)=0\}$, $y=\dot x$, is defined in terms of the first integral. In this case all stationary points and all local extrema of the integral curve $x(y)$ such that the function $x(y)$ 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 $[a(x,y)+\varepsilon b(x,y)]\dot y=1$, $b(x,y)\in C^1$ for $|\varepsilon |$ small enough. Introduce the notation $S^*=\{(x,y)\in \mathbb R^2:a(x,y)+\varepsilon b(x,y)=0\}$. 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.