Quantifying information dynamics in a nonlinear system is crucial in complex dynamics. The Mutual Information Matrix (MIM) method was developed to study nonlinear interactions in high-dimensional time series. In this paper, MIM analysis is extended from a Shannon entropy approach to a Rényi entropy one. Specifically, this paper presents the MIM for Rényi entropy with emphasis on the quadratic case. A global measure to quantify the total information shared between time series is defined and compared with the Shannon case. The main properties are illustrated by numerical simulations related to vector autoregressive fractionally integrated moving-average and sinusoidal models, and the methodology was applied to ozone multivariate time series of a meteorological monitoring network. Comparing both global measures, the proposed one gave different results in the nonlinear interaction quantification between stations than those based on Shannon entropy in the ozone monitoring network.