An Asymptotic Comparison of Differentiable Dynamics and Tropical Geometry

In this paper we introduce a new comparison method to give rough asymptotic estimates of different evolutional dynamics. It uses a kind of scale transform called tropical geometry, which connects automata with real rational dynamics. By this procedure the defining equations are transformed rather than solutions themselves. Real rational dynamics is regarded as an approximation of evolutional dynamics given by partial differential equations (PDEs). Two different evolutional dynamics can be considered when their defining equations are transformed to the same automata at infinity.