On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic
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 v...
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| Format: | Article |
| Language: | English |
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Sciendo
2020-03-01
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| Series: | Applied Mathematics and Nonlinear Sciences |
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| Online Access: | https://doi.org/10.2478/amns.2020.1.00010 |
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| author | El-Borhamy Mohamed Mosalam Nahla |
| author_facet | El-Borhamy Mohamed Mosalam Nahla |
| author_sort | El-Borhamy Mohamed |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-13T07:19:20Z |
| format | Article |
| id | doaj.art-e23b2613b52c43909c3961bd00ee01ea |
| institution | Directory Open Access Journal |
| issn | 2444-8656 |
| language | English |
| last_indexed | 2024-12-13T07:19:20Z |
| publishDate | 2020-03-01 |
| publisher | Sciendo |
| record_format | Article |
| series | Applied Mathematics and Nonlinear Sciences |
| spelling | doaj.art-e23b2613b52c43909c3961bd00ee01ea2022-12-21T23:55:27ZengSciendoApplied Mathematics and Nonlinear Sciences2444-86562020-03-01519310810.2478/amns.2020.1.00010On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamicEl-Borhamy Mohamed0Mosalam Nahla1Department of Engineering Mathematics and Physics, Faculty of Engineering, University of TantaEgyptDepartment of Engineering Mathematics and Physics, Faculty of Engineering, University of TantaEgyptIn 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.https://doi.org/10.2478/amns.2020.1.00010nonlinear ordinary differential equationsstability theoryperiodic solutionsbifurcationchaotic dynamicgyroscope34l3037c7534c2534f1037d4570e05 |
| spellingShingle | El-Borhamy Mohamed Mosalam Nahla On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic Applied Mathematics and Nonlinear Sciences nonlinear ordinary differential equations stability theory periodic solutions bifurcation chaotic dynamic gyroscope 34l30 37c75 34c25 34f10 37d45 70e05 |
| title | On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic |
| title_full | On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic |
| title_fullStr | On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic |
| title_full_unstemmed | On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic |
| title_short | On the existence of periodic solution and the transition to chaos of Rayleigh-Duffing equation with application of gyro dynamic |
| title_sort | on the existence of periodic solution and the transition to chaos of rayleigh duffing equation with application of gyro dynamic |
| topic | nonlinear ordinary differential equations stability theory periodic solutions bifurcation chaotic dynamic gyroscope 34l30 37c75 34c25 34f10 37d45 70e05 |
| url | https://doi.org/10.2478/amns.2020.1.00010 |
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