Research Article Analysis of Rattleback Chaotic Oscillations Michael Hanias, 1 Stavros G. Stavrinides, 2 and Santo Banerjee 3 1 Department of Electronics, Computers, Telecommunications and Control, Faculty of Physics, National and Kapodistrian University of Athens, Athens, Greece 2 Electrical and Computer Engineering Department, University of Cyprus, Nicosia, Cyprus 3 Laboratory of Cryptography, Analysis & Structure, Institute for Mathematical Research, University Putra Malaysia, Malaysia Correspondence should be addressed to Santo Banerjee; santoban@gmail.com Received 30 August 2013; Accepted 7 October 2013; Published 8 January 2014 Academic Editors: S. Anita, G. Dai, and Z. Mukandavire Copyright © 2014 Michael Hanias et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Rattleback is a canoe-shaped object, already known from ancient times, exhibiting a nontrivial rotational behaviour. Although its shape looks symmetric, its kinematic behaviour seems to be asymmetric. When spun in one direction it normally rotates, but when it is spun in the other direction it stops rotating and oscillates until it fnally starts rotating in the other direction. It has already been reported that those oscillations demonstrate chaotic characteristics. In this paper, rattleback’s chaotic dynamics are studied by applying Kane’s model for diferent sets of (experimentally decided) parameters, which correspond to three diferent experimental prototypes made of wax, gypsum, and lead-solder. Te emerging chaotic behaviour in all three cases has been studied and evaluated by the related time-series analysis and the calculation of the strange attractors’ invariant parameters. 1. Introduction Behaviour of dynamical systems is always of great interest, especially when these dynamics reveal a nonlinear-chaotic behavior. Rattleback is such a case and there have been numerous analyses of its peculiar behaviour, since the 1890’s [1, 2]. Many other papers have been published on the issue, especially during the 1980’s [3–5]. Rattleback is a canoe-shaped body, that is, a semiellipsoid object, known from the ancient years named as “celt” or “anagyre.” It demonstrates the very interesting property of spin asymmetry leading to a peculiar kinematic behaviour. Tis property seems to be unexpected in frst sight, due to its symmetrical shape. Te demonstrated behaviour consists of a reasonably smooth spin in one direction, while in the oppo- site direction it develops a pitching instability that leads to spin reversal, in an apparent defance of the principle of con- servation of angular momentum. It is apparent that rattleback provides a prototype of chiral dynamics, where in lack of mirror-symmetry it leads to unconventional dynamics. Te frst mathematical model was introduced by Walker [6], who studied the linearized rat- tleback equations of motion and concluded that the com- pletely stable motion is possible in only one (clockwise) spin direction. It has already been reported that rattleback is demonstrating a chaotic behaviour during the procedure of reversing its spin [7–9]. In this paper, this chaotic behaviour for diferent variables is studied and evaluated by utilizing a mathematical model, whose parameters have been experimentally defned by three diferent rattlebacks made of wax, gypsum, and solder. Time-series analysis and the corresponding chaotic evalua- tion reveal the global dynamical features of this interesting object dynamical behaviour. Te paper is organized in three sections. In the frst section, rattleback’s dynamics are intro- duced, utilizing Kane’s model [4]. It is shown, by means of numerical solutions of full, non-linear motion equations that one can construct a realistic mathematical model by assum- ing rolling without slipping and employing a torque propor- tional to the angular velocity in order to provide for energy dissipation. Rattlebacks made of diferent materials such as wax, gypsum, and lead-solder have been constructed and Kane’s model parameter values were experimentally deter- mined, in order to study their dynamical behaviour. In the second section, time-series presentation for three (of the six) variables appears. Tese time-series have been numerically calculated and they demonstrate irregular behaviour, hinting chaos. Finally in the third section, time-series analysis is Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 569386, 15 pages http://dx.doi.org/10.1155/2014/569386