Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation
The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and su...
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Wiley
2018-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.1155/2018/5207910 |
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| author | Xindong Ma Shuqian Cao |
| author_facet | Xindong Ma Shuqian Cao |
| author_sort | Xindong Ma |
| collection | DOAJ |
| description | The coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameter b by means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency. |
| format | Article |
| id | doaj-art-475ecd66918042e984a9cb1ce4e14901 |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-475ecd66918042e984a9cb1ce4e149012025-02-03T00:59:46ZengWileyShock and Vibration1070-96221875-92032018-01-01201810.1155/2018/52079105207910Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic ExcitationXindong Ma0Shuqian Cao1Department of Mechanics, Tianjin University, Tianjin 300354, ChinaDepartment of Mechanics, Tianjin University, Tianjin 300354, ChinaThe coupling effect of two different frequency scales between the exciting frequency and the natural frequency of the Shimizu-Morioka system with slow-varying periodic excitation is investigated. First, based on the analysis of the equilibrium states, homoclinic bifurcation, fold bifurcation, and supercritical Hopf bifurcation are observed in the system under a certain parameter condition. When the exciting frequency is much smaller than the natural frequency, we can regard the periodic excitation as a slow-varying parameter. Second, complicated dynamic behaviors are analyzed when the slow-varying parameter passes through different bifurcation points, of which the mechanisms of four different bursting patterns, namely, symmetric “homoclinic/homoclinic” bursting oscillation, symmetric “fold/Hopf” bursting oscillation, symmetric “fold/fold” bursting oscillation, and symmetric “Hopf/Hopf” bursting oscillation via “fold/fold” hysteresis loop, are revealed with different values of the parameter b by means of the transformed phase portrait. Finally, we can find that the time interval between two symmetric adjacent spikes of bursting oscillations exhibits dependency on the periodic excitation frequency.http://dx.doi.org/10.1155/2018/5207910 |
| spellingShingle | Xindong Ma Shuqian Cao Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation Shock and Vibration |
| title | Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation |
| title_full | Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation |
| title_fullStr | Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation |
| title_full_unstemmed | Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation |
| title_short | Bursting Oscillations in Shimizu-Morioka System with Slow-Varying Periodic Excitation |
| title_sort | bursting oscillations in shimizu morioka system with slow varying periodic excitation |
| url | http://dx.doi.org/10.1155/2018/5207910 |
| work_keys_str_mv | AT xindongma burstingoscillationsinshimizumoriokasystemwithslowvaryingperiodicexcitation AT shuqiancao burstingoscillationsinshimizumoriokasystemwithslowvaryingperiodicexcitation |