Stability of differential equations associated with a class of one dimensional maps

TL;DR

This paper investigates the stability of differential equations derived from one-dimensional maps by truncating their Taylor series, revealing instability for higher truncation orders and complex dynamics for lower orders.

Contribution

It introduces a generalized approach to truncated models with N=3 and 4, showing their connection to systems with riddled parameter spaces.

Findings

Truncations with N > 4 are unconditionally unstable.

Models with N=3 and 4 exhibit dynamics characteristic of riddled parameter spaces.

Higher-order truncations lead to instability in the continuous-time embedding.

Abstract

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. Truncations with N > 4 leads to unconditional instability. Generalization of the truncated models with N = 3 and 4 shows dynamical behaviour characteristic of systems with a riddled parameter space.