Growing through chaos in the Matsuyama map via subcritical flip bifurcation and bistability

Recent publications revisit the growth model proposed by Matsuyama (”Growing through cycles”, Econometrica 1999), presenting new economic interpretations of the system as well as new results on its dynamics described by a one-dimensional piecewise smooth map (also called M-map). The goal of the present paper is to give the rigorous proof of some results which were remaining open, related to the dynamics of M-map. We prove that an attracting 2-cycle appears via border collision bifurcation, give the explicit flip bifurcation value at which this cycle looses stability, as well as the explicit coordinates of its points at the bifurcation value, proving that the flip bifurcation is always of subcritical type. We show that this leads to the existence of a region of bistability associated with an attracting 2-cycle coexisting with attracting 4-cyclic chaotic intervals. This means that the effects of the destabilization of the 2-cycle, related to a corridor stability, are catastrophic and irreversible. We give also the conditions related to the sharp transition to chaos, proving that the cascade of stable cycles of even periods cannot occur. The parameter region in which repelling cycles of odd period exist is further investigated, namely, we give an explicit boundary of this region and show its relation to the non existence of cycles of period three.