Asymptotic behavior in the dynamics of a smooth body in an ideal fluid
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O R I G I NA L PA P E R
Evgeny V. Vetchanin
· Ivan S. Mamaev
Asymptotic behavior in the dynamics of a smooth body in an ideal fluid
Received: 27 September 2019 / Revised: 10 June 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract This paper addresses a conservative system describing the motion of a smooth body in an ideal fluid under the action of an external periodic torque with nonzero mean and of an external periodic force. It is shown that, in the case where the body is circular in shape, the angular velocity of the body increases indefinitely (linearly in time), and the projection of the phase trajectory onto the plane of translational velocities is attracted to a circle. Asymptotic orbital stability (or asymptotic stability with respect to part of variables) exists in the system. It is shown numerically that, in the case of an elliptic body, the projection of the phase trajectory onto the plane of translational velocities is attracted to an annular region.
1 Introduction 1. In recent years, there has been much discussion of the problem of the motion of various devices due to periodic actions produced by internal mechanisms or external forces. The study of this problem was motivated by challenges of mobile robotics. A number of mathematical models were proposed to describe the motion of a Chaplygin sleigh actuated by oscillating the internal mass or the rotor [2,3,8], the motion of a ball with a displaced center of mass (the Chaplygin top) and internal rotors [6,11], and the plane-parallel motion in a fluid of a rigid body with an internal mass or a rotor inside [5,9,14,15]. The motion of the Chaplygin sleigh under the action of periodic pulsed torque impacts was dealt with in [7]. The motion of a circular body in a fluid due to an external periodic force and torque was studied in [16]. It is also of interest to study the dynamical effects achieved in some mathematical models: Fermi acceleration in nonholonomic systems [2,3,6,8,11], strange attractors arising due to viscous dissipation [5,9,14], and KAM curves preventing an unbounded acceleration in an ideal fluid [5,9]. 2. This paper is a continuation of the study presented in Ref. [16]. Reference [16] considered the motion in a fluid (ideal or viscous) of a balanced circular cylinder acted upon by external periodic forces and torque whose mean values are zero. In that case, the solution of the equations of motion can be represented as a multiple series. In this paper, we consider the motion of a balanced circular cylinder in an ideal fluid in the case where the external torque has a nonzero mean. In this case, the system preserves a standard invariant measure, i.e., it is conservative, and hence, no attractors, either simple (regular) or strange (chaotic) ones, can arise in its phase The work of E.V. Vetchanin and I.S. Mamaev was carried out within the framework of the state assignment of the Ministry of Science and Higher Education of Russia (Projects FEWS-2020-0009 and FZZN-2020-0011, respectively). Also this work is supported
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