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We study the formation of chaos and strange attractors in the order parameter space of a system of two coupled, non-resonantly driven exciton-polariton condensates. The typical scenario of bifurcations experienced by the system with increasing external pumping consists of (i) formation of $π$-synchronized condensates at low pumping, (ii) symmetry breaking pitchfork bifurcation leading to unequal occupations of the condensates with non-trivial phase difference between them, (iii) loss of the stability of all fixed points in the system resulting in chaotic dynamics, (iv) limit cycle dynamics of the order parameter, which ends up in (v) in-phase synchronized condensates via the Hopf bifurcation from a limit cycle. The chaotic dynamics of the order parameter is evidenced by calculating the maximal Lyapunov exponent. The presence of a chaotic domain is studied as a function of polariton-polariton interaction and the Josephson coupling between the condensates. At some values of the parameters the bifurcation route is more complex and the strange attractor can coexist with the stable fixed-point lasing. We also investigate how the chaotic dynamics is reflected in the light emission spectrum from the microcavity.