We study the global solvability of a class of differential complexes on the product manifold \(\mathbb {T}^m \times \mathbb {R}^n\) associated with systems of evolution operators of the form \(L_r = \partial _{t_r} + ia_r(t)P(x,D_x), r=1,\ldots ,m,\) where the coefficients \(a_r\) are real-valued Gevrey functions on the torus and \(P(x,D_x)\) is a globally elliptic normal differential operator on \(\mathbb {R}^n\) . Within the framework of time-periodic Gelfand-Shilov spaces, we introduce a natural differential complex generated by these operators and investigate its solvability in both functional and ultradistributional settings. We provide a complete characterization of global solvability in terms of a Diophantine condition involving the constant part of the associated 1-form and the spectrum of P. We also analyze global hypoellipticity of the complex. These results extend previous works on scalar operators and constant coefficient systems to the setting of differential complexes with time-dependent real coefficients.
