Periodic solutions induced by an upright position of small oscillations of a sleeping symmetrical gyrostat

Abstract

Abstract. The aim of this paper is to provide sufficient conditions for the existence of periodic solutions emerging from an upright position of small oscillations of a sleeping symmetrical gyrostat with equations of motionbeing α and β parameters satisfying ∆ = α 2 − 4β > 0 and β − α 2 2 ± α √ ∆ 2 < 0, ε a small parameter and, F 1 and F 2 smooth periodic maps in the variable t in resonance p:q with some of the periodic solutions of the system for ε = 0, where p and q are positive integers relatively prime. The main tool used is the averaging theory.

On the periodic solutions of a rigid dumbbell satellite in a circular orbit

“…Our objective is to provide sufficient conditions for the existence of periodic motions about the satellite's center of mass that are asymptotic to translational motion in an absolute coordinate system. This type of motion, denoted, cylindrical equilibrium, is well known in the astrophysics literature on satellite's dynamics (see for instance [4,7,8]).…”

“…The objective of this work is to provide, using as a main tool the averaging theory, a system of nonlinear equations whose simple zeros provide periodic solutions of the perturbed dumbbell satellite in circular orbit with equations of motion given by (2). Some other works using similar techniques (see Llibre et al [1] for more details), are [2,3] or [4] where the method is used for action angle variables. In order to present our results we need some preliminary definitions and notations.…”

“…we obtain the functions given by (4). Then, by Theorem 6 we have that for every simple zero (X * 0 , Y * 0 ) ∈ V of the system of nonlinear functions (10)…”

On the periodic orbits of the perturbed Wilberforce pendulum

“…For doing this we shall use the averaging theory of dynamical systems (see the Appendix for more details). We have been inspired by other works where these techniques have been used for studying other perturbed dynamical problems; see for instance Basu and Gewei (2011), Bustos et al (2013), Bustos et al (2014), Gao et al (2013), Guirao et al (2013a), and Guirao et al (2013b).…”

On the Periodic Auto–Oscillations of an Electric Circuit with Periodic Imperfections on Its Variables

“…For doing this we shall use the averaging theory of dynamical systems, see Appendix for more details on it. We have been inspired by other works where these techniques have been used for studying other perturbed dynamics problems, see for instance [2,3,4,5,6,7,8,9,10].…”