Complex dynamics in generalizations of the Chaplygin sleigh

TL;DR

This paper investigates the complex behaviors of a generalized Chaplygin sleigh in potential wells, revealing chaotic dynamics and strange attractors, thus highlighting the rich and intermediate nature of nonholonomic systems.

Contribution

It introduces a new analysis of Chaplygin sleigh dynamics with external potential forces, demonstrating diverse behaviors including chaos and attractors in specific potential configurations.

Findings

Behavior varies with energy, from regular to chaotic dynamics.

Chaotic phenomena and strange attractors are observed in the models.

Nonholonomic systems can exhibit both conservative and dissipative features.

Abstract

The article considers Chaplygin sleigh on a plane in potential well, assuming that an external potential force is supplied at the mass center. Two particular cases are studied in some detail, namely, a one-dimensional potential valley and a potential with rotational symmetry; in both cases the models reduce to four-dimensional differential equations conserving mechanical energy. Assuming the potential functions quadratic, various behaviors are observed numerically depending of the energy, from those characteristic to conservative dynamics (regularity islands and chaotic sea) to strange attractors. This is another example of nonholonomic system manifesting these phenomena (similar to those for Celtic stone or Chaplygin top), which reflects a fundamental nature of these systems occupying intermediate position between conservative and dissipative dynamics.