On dynamical connection between continuous and tropical discretized dynamical systems in one-dimensional (Mathematical structures of integrable systems, their developments and applications)

Abstract

We discuss the relationship of stability and the local bifurcations between one-dimensional differential equations and their tropically discretized ones. The discretized time interval is introduced as a bifurcation parameter in the tropical discretized equation, and emergence condition of an additional bifurcation, flip bifurcation, is revealed. By reviewing the continuous dynamical systems with saddle node and pitchfork bifurcations treated in our previous study (S. Ohmori and Y. Yamazaki, J. Math. Phys. 61 122702 (2020)), correspondence of their dynamics to their tropical discrete equations is discussed.