Abstract This paper is focused on studying Parrondo’s paradox in non-linear dynamics, specifically how the periodic combination of the individual maps f and g can give rise to chaos or order. We construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics. The effect of altering the order of the combination, i.e. considering f ∘ g or g ∘ f , is analyzed, as well as the robustness of the Parrondo effect under small perturbations. Conditions for avoiding Parrondian dynamics are also obtained, placing special emphasis on the notion of chaos given by turbulence.