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The optomechanical systems produce chaotic behaviour due to nonlinear interaction between photons and phonons, and the same systems are used to understand the synthetic fields as well. Here, we report on the study of chaotic behaviour in the presence of a phononic synthetic magnetic field in a closed loop configuration consisting of a single optical mode and two mechanical modes. The modulation phase of the mechanical coupling between the two mechanical modes plays a critical role in determining the mechanical and optical intensity dynamics in the nonlinear regime. Our study shows the dark mode breaking effect in the presence of a synthetic magnetic field, which brings about a complex way of mechanical energy exchange that causes the cavity field to alternate between chaotic and regular behaviour periodically in temporal domain. However in the stronger nonlinear regime the temporal dynamics demonstrate predominantly chaotic behaviour. Besides, with the advent of advanced fabrication technologies, this study holds promises in developing phase tunable integrated low-power chaotic light sources to support efficient optical secure communication systems.