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Three periods are distinguished: the first period \u201cthe dawn of chaotic dynamical systems\u201d from 1885 to 1970 which includes the Kolmogorov\u2013Arnold\u2013Moser theorem on the persistence of quasi-periodic motions under small perturbations extended by Moser in 1962, with its proof achieved in 1963 by Arnold, the discovery of the Lorenz attractor in 1963, the Sharkovsky order (1964), the Shilnikov bifurcation (1965), the horseshoe and the fractals in 1967. The second period \u201cthe intermediate period 1971\u20131989\u201d, begins with the definition of the strange attractors by Ruelle and Takens, and ends just before the introduction of the synchronization of chaotic systems in 1990 by Pecora and Carroll. The third period extends from 1990 to the present day, which is a flourishing time of applications in several sciences of this mathematical theory. This period corresponds to the publication of 35 volumes (415 issues) of the International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, since the 35 years of existence of this journal. This history of chaotic dynamical systems is presented in a reasoned manner, which provides some insight into the driving forces behind their evolution. The people who shaped them are also featured. An epistemological analysis structures the analysis of these three periods. 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