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We study the existence of global canard surfaces for a wide class of real singular perturbation problems. These surfaces define families of solutions which remain near the slow curve as the singular parameter goes to zero.

Let $ℰ$ be a disjoint decomposition of ${ℝ}^n$ and let $X$ be a vector field on ${ℝ}^n$, defined to be linear on each cell of the decomposition $ℰ$. Under some natural assumptions, we show how to associate a semiflow to $X$ and prove that such semiflow belongs to the o-minimal structure ${ℝ}_{\mathrm{an},exp}$. In particular, when $X$ is a vector field and $\Gamma$ is an invariant subset of $X$, our result implies that if $\Gamma$ is then the Poincaré first return map associated $\Gamma$ is also in ${ℝ}_{\mathrm{an},exp}$.