<abstract xmlns="http://www.w3.org/1999/xhtml"><p>In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing Rayleigh damping and generalized Duffing functions is considered. General conditions for the stability and periodicity of solutions are deduced via fixed point theorems and the Lyapunov function method. A gyro dynamic application represented by the motion of axi-symmetric gyro mounted on a sinusoidal vibrating base is analyzed by the interpretation of its dynamical motion in terms of Euler’s angles via the deduced theoretical results. Moreover, the existence of homoclinic bifurcation and the transition to chaotic behaviour of the gyro motion in terms of main gyro parameters are proved. Numerical verifications of theoretical results are also considered.</p></abstract>

In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing Rayleigh damping and generalized Duffing functions is considered. General conditions for the stability and periodicity of solutions are deduced via fixed point theorems and the Lyapunov function method. A gyro dynamic application represented by the motion of axi-symmetric gyro mounted on a sinusoidal vibrating base is analyzed by the interpretation of its dynamical motion in terms of Euler’s angles via the deduced theoretical results. Moreover, the existence of homoclinic bifurcation and the transition to chaotic behaviour of the gyro motion in terms of main gyro parameters are proved. Numerical verifications of theoretical results are also considered.

On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic

Department of Engineering Mathematics and Physics, Faculty of Engineering

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

In this article, the study of qualitative properties of a special type of non-autonomous nonlinear second order ordinary differential equations containing...

On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic

Published Online: Mar 30, 2020

Page range: 93 - 108

Received: Aug 06, 2019

Accepted: Nov 09, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00010

Keywords
Nonlinear Ordinary Differential Equations, Stability Theory, Periodic Solutions, Bifurcation, Chaotic Dynamic, Gyroscope

© 2020 Mohamed El-Borhamy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

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On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic

Published Online: Mar 30, 2020

Page range: 93 - 108

Received: Aug 06, 2019

Accepted: Nov 09, 2019

DOI: https://doi.org/10.2478/amns.2020.1.00010

KeywordsNonlinear Ordinary Differential Equations, Stability Theory, Periodic Solutions, Bifurcation, Chaotic Dynamic, Gyroscope

© 2020 Mohamed El-Borhamy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Keywords
Nonlinear Ordinary Differential Equations, Stability Theory, Periodic Solutions, Bifurcation, Chaotic Dynamic, Gyroscope