Strange attractors in rattleback dynamics

Abstract

This review is dedicated to the dynamics of the rattleback, a phenomenon with curious physical properties that is studied in nonholonomic mechanics. All known analytical results are collected here, and some results of our numerical simulation are presented. In particular, three-dimensional Poin-care maps associated with dynamical systems are systematically investigated for the first time. It is shown that the loss of stability of periodic and quasiperiodic solutions, which gives rise to strange attractors, is typical of the three-dimensional maps related to rattleback dynamics. This explains some newly discovered properties of the rattleback related to the transition from regular to chaotic solutions at certain values of the physical parameters.

Understanding reversals of a rattleback

“…Last decade, due to work of Borisov, Mamaev and their collaborators, it became apparent that global dynamics of a rattleback is very rich. Apart from the well-known stable and unstable stationary equilibrium rotations about the vertical principal inertia axis there may exist limit cycle solutions and even strange attractors [9,12,4]. It appears that reversal phenomena in dynamics of nonholonomic rigid bodies are quite common and have been discovered in motion of Chaplygin ball [6] and Suslov top [2].…”

“…The linearized and/or simplified equations of motion cannot predict all phenomena in ratteback motion and several papers analyzed dynamics of a rattleback using numerical simulations of full equations [15,5,9]. Last decade, due to work of Borisov, Mamaev and their collaborators, it became apparent that global dynamics of a rattleback is very rich.…”

“…In a more general framework of nonholonomic systems on a Lie algebra change of stability of stationary motions and reversion of direction of motion has been discussed in [14]. Further references about rattleback dynamics may be found in review papers [9,7].…”

“…In [3] this analysis has been extended to determine dependence of stability, for vertical spinning solutions, on the moments of inertia and on curvatures of rattlebacks bottom surface. H. Bondi also asserted that slipping and friction is unimportant in understanding dynamics of a rattleback, see also [9]. A rigorous linear stability analysis of vertical spinning solutions has been addressed in [1,13] where the condition for relation between spinning velocity, main curvatures of a convex body (ellipsoid), the moments of inertia and the chirality parameter (angle) have been formulated.…”

Rolling resistance modelling in the Celtic stone dynamics

“…The first scientific work on rattleback was published at the end of the 19th century [17] and it has been an object of interest till today [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], both from the point of view of dynamics and of modelling. During modelling and analysis of the Celtic stone, rolling without sliding is often assumed, leading to non-holonomic constraints [17, 21, 22, 24-26, 29, 31].…”