On ultradifferentiable functions

We give complete and exact descriptions of spaces of ultradifferentiable functions that are closed under composition with either holomorphic or ultradifferentiable functions -- which are two distinct cases. The proof works by considering formal power series, and stability under differentiation is not required. As an application of the power series approach we reprove regularity results for solutions of ode's and pde's without employing tedious estimates imploying the Fa\`a di Bruno formula for higher derivatives of composite maps.