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We comprehensively analyze the dynamics for the gravitational field equationsfor the Chiral-Quintom theory in a Friedman-Lema\^itre-Robertson-Walkercosmology with an additional matter source. We consider a new set ofdimensionless variables and write the field equations in the equivalent form ofan algebraic-differential system. Specifically, we consider two families ofquintom models where the two scalar fields interact in the kinetic sector. Wemathematically focus on the dynamical effect of spatial curvature. Physically,we find two periods of inflation related to the Universe's early and late-timeacceleration phases.

Global dynamics of two models for Quintom Friedman-Lema\^itre-Robertson-Walker Universes