A classification of slow convergence near parametric periodic points of discrete dynamical systems

Abstract

We study the phenomenon of slow convergence in families of discrete dynamical systems where the iteration function has a Puiseux series representation. Such occurrence consists in the slow convergence of orbits near non-hyperbolic parametric periodic points. We provide a precise new definition of the slowness of convergence which is based on literature results for the critical exponents associated with parametric periodic points. Such exponents establish a general classification for slow systems and provide a measure of rates of convergence. For dynamical systems whose iteration functions have Taylor series expansions, the new definition is natural with wider applicability. However, it can be also used for iteration functions where a more sophisticated approach, such as a Lagrange expansion, is needed. In addition, we show that even for such iteration functions, the critical exponent can be easily computed. The presented theoretical results are illustrated by numerical examples having different rates of convergence.

Keywords:

• slow convergence
• discrete dynamical systems
• difference equations
• iteration function
• critical exponent

2010 AMS Subject Classifications:

• 37M05
• 39A30
• 65P40
• 65Q10

Disclosure statement

No potential conflict of interest was reported by the authors.