Can two chaotic systems give rise to order?

The recently discovered Parrondoʹs paradox claims that two losing games can result, under random or periodic alternation of their dynamics, in a winning game: “losing + losing = winning”. In this paper we follow Parrondoʹs philosophy of combining different dynamics and we apply it to the case of one-dimensional quadratic maps. We prove that the periodic mixing of two chaotic dynamics originates an ordered dynamics in certain cases. This provides an explicit example (theoretically and numerically tested) of a different Parrondian paradoxical phenomenon: “chaos + chaos = order”.