The local first-return map of a scalar harmonic oscillator near (1,0) is the reflection x->-x

Let $\omega >0$ and let $F:{\mathbb{R}}^2\to {\mathbb{R}}^2$, $F(x,y)=(y,-{\omega }^{2}x)$, be the phase-plane field of the scalar harmonic oscillator $x^{\mathrm{\prime }\mathrm{\prime }}+{\omega }^{2}x=0$, with $y=x^{\mathrm{\prime }}$, with local flow $\mathrm{\Phi }$ (The local flow of an autonomous first-order system). Its solutions are

$\mathrm{\Phi }(t,(x,y))=(x\cos \omega t+\frac{y}{\omega }\sin \omega t,-\omega x\sin \omega t+y\cos \omega t),$

defined for all real $t$. Take the section $\mathrm{\Sigma }=\{y=0\}$, the reference point $x_0=(1,0)$ with $h(x,y):=y$, and $t_0:=\pi \mathrm{/}\omega$, so that $x_1=\mathrm{\Phi }(t_0,x_0)=(-1,0)$. Then $h(x_0)=h(x_1)=0$ and

$Dh(x_0)F(x_0)=-{\omega }^2\mathrm{\ne }0,Dh(x_1)F(x_1)={\omega }^2\mathrm{\ne }0,$

and $\mathrm{\Phi }(s,x_0)$ has second coordinate $-\omega \sin (\omega s)\mathrm{\ne }0$ for $0<s<\pi \mathrm{/}\omega$, so it avoids the closed section on that interval. Thus the first-return clause of A transversal return of a C¹ flow yields a local C¹ Poincaré map, with inverse given by the reversed crossing applies and produces a local $C^1$ Poincaré map near $x_0$. On the section itself the map and its inverse can be computed exactly: for $(x,0)$ with $x$ near $1$, the return time is $\tau (x,0)=\pi \mathrm{/}\omega$ and the return map is

$P(x,0)=(-x,0),$

the reflection in the origin restricted to the section, with inverse $Q(y,0)=(-y,0)$; this is a $C^1$ diffeomorphism between neighbourhoods of $(1,0)$ and $(-1,0)$ in $\mathrm{\Sigma }$.

• The example is the simplest case in which the local Poincaré map can be written in closed form. The proposition's hypothesis that the section be transversal at both crossings is visible in the computation: at the crossing the velocity is purely vertical, with $Dh\cdot F=\mp {\omega }^2\mathrm{\ne }0$.

• The return map is computed on the section only; the general map $\tau (x,y)$ for points off the section is a $C^1$ function near $(1,0)$ by the proposition, but its explicit formula is not needed here. No stability or periodicity of the map is asserted: the reflection is an involution, and every point of the reference orbit is periodic for the flow with period $2\pi \mathrm{/}\omega$.

• At $(0,0)$ the displayed flow is stationary. Every positive time is a hitting time of $\mathrm{\Sigma }$, so there is no least positive hitting time and hence no first-return map there. The asserted local map is near $(1,0)$, with its image near $(-1,0)$.