Let $f(x,\lambda)$ be of $C^1$ class in $\mathbb{R}^n\times\mathbb{R}^m$, sucht that $x'=f(x,0)$ has a single periodic solution $p(t)$ (not constant). If $\tau$ is the period of this solution, suppose that the only solutions of $$y'=D_1f(p(t),0)y \qquad y(0)=y(\tau)$$
are functions of the form $ap'(t)$ with $a\in\mathbb{R}$.
Prove that there is a $\delta>0$ and a $C^1$ function $\omega(\lambda)$ in $|\lambda|<\delta$ such that $\omega(0)=\tau$ and $$x'=f(x,\lambda)$$
has a periodic $C^1$ solution $p(t,\lambda)$ with period $\omega(\lambda)$ and $p(t,0)=p(t)$.
I'm pretty sure that I have to apply the Implicit Function Theorem to get a poincaré transformation, but I'm having trouble formalizing it.
I think I have to consider hyperplane $H$, normal to the curve $p(t$) at point $p(0)$...