Chaotic dynamics of a continuous and discrete generalized Ziegler pendulum

TL;DR

This paper investigates the transition to chaos in a generalized Ziegler pendulum through analytical and numerical methods, considering various external forces and both continuous and discrete dynamics.

Contribution

It provides new insights into how external forces and dissipation affect integrability and chaos in the generalized Ziegler pendulum, including discrete map analysis.

Findings

Presence of external forces can break integrability

Discrete map lacks dense periodic points, indicating limited chaos

Dissipative forces influence the transition to chaotic behavior

Abstract

We present analytical and numerical results on integrability and transition to chaotic motion for a generalized Ziegler pendulum, a double pendulum subject to an angular elastic potential and a follower force. Several variants of the original dynamical system, including the presence of gravity and friction, are considered, in order to analyze whether the integrable cases are preserved or not in presence of further external forces, both potential and non-potential. Particular attention is devoted to the presence of dissipative forces, that are analyzed in two different formulations. Furthermore, a study of the discrete version is performed. The analysis of periodic points, that is presented up to period 3, suggests that the discrete map associated to the dynamical system has not dense sets of periodic points, so that the map would not be chaotic in the sense of Devaney for a choice of the parameters that corresponds to a general case of chaotic motion for the original system.