Nature of solutions of differential equations associated with a class of one-dimensional maps

Discrete time evolution of one-dimensional maps is embedded in continuous time by truncating the Taylor series expansion of the time evolution operator to a finite order N. The fixed points of the ordinary differential equations thus obtained are unstable whenever N > 4 regardless of the details of the underlying map, so long as it is continuous and differentiable. Generalization of the truncated equations with N = 3 and 4 shows dynamical behaviour characteristic of systems with a riddled parameter space.