Experimental studies [Hauser, M. J. B.; Olsen, L. F. J. Chem. Sec., Faraday Trans. 1996, 92, 2857-2863] of the peroxidase-oxidase (PO) reaction at pH values in excess of 5.4 suggest the existence of narrow regions of complex dynamics between adjacent mixed-mode oscillations (MMOs) that occur in period-adding sequences. Previously [Hauser, M. J. B.; Olsen, L. F.; Bronnikova, T. V.; Schaffer, W. M. J. Phys. Chern. B 1997, 101, 5075-5083], it was argued that both the period-adding sequences and the transitional regions between neighboring MMOs are predictable by a detailed model of the reaction called BFSO [Bronnikova, T. V.; Fed'kina, V. R.; Schaffer, W. M.; Olsen, L. F. J. Phys. Chem. 1995, 99, 9309-9312]. In the present paper, we study the transitional regions via computer simulation. Our investigations indicate that the motion therein may be periodic, quasiperiodic, or chaotic. In greater detail, we observe a quasiperiodic route to chaos whereby period-doubled cycles give rise to doubled tori that, in turn, undergo homoclinic bifurcations to chaos. Because the latter transitions are a consequence of progressive fattening of the tori, we propose calling this scenario the "fat torus" route to chaos, and the homoclinic bifurcations "fat torus" bifurcations (FTBs). The numerical results are qualitatively consistent with the experimental findings reported to date. FTBs and the resultant period-doubled, fractal tori may provide a criterion for discriminating among alternative models of the PO reaction.