  Understanding the intra-cavity field dynamics in passive microresonator
systems has already been intriguing. It becomes fascinating when the system is
complex, such as a concentric dual microring resonator that exhibit avoided
mode crossing (AMC). In this work, we present a systematic study of
intra-cavity oscillatory field dynamics near AMC in a concentric silicon
nitride microring resonator with the help of the coupled Lugiato-Lefever
equation (LLE). We identify two regions viz. weakly coupled region (WCR) and
strongly coupled region (SCR) based on eigenfrequency separation of the
two-hybrid modes, which originate from mode coupling near AMC. In WCR, the mode
coupling effect is dominant, leading to intra-cavity power oscillation between
these two modes in a periodic manner and non-identical variation of their
phases. In SCR, the mode coupling effect reduces gradually with nearly
identical characteristics of both the modes. We further verify our numerical
findings with the semi-analytical variational method, leading to an in-depth
understanding of the mode coupling induced dynamics. We finally analyze the
polarization evolving state, and the polarization locked state with the help of
Stokes parameters and Jones vectors in WCR and SCR, respectively.
