Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators
By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limi...
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/5/772 |
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| author | Chun Zhang Qiaoxia Tang Zhixiang Wang |
| author_facet | Chun Zhang Qiaoxia Tang Zhixiang Wang |
| author_sort | Chun Zhang |
| collection | DOAJ |
| description | By introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented. |
| first_indexed | 2024-03-09T20:31:07Z |
| format | Article |
| id | doaj.art-6cceeb7aa4e144f586a59caac75c0759 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T20:31:07Z |
| publishDate | 2022-02-01 |
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| spelling | doaj.art-6cceeb7aa4e144f586a59caac75c07592023-11-23T23:23:25ZengMDPI AGMathematics2227-73902022-02-0110577210.3390/math10050772Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol OscillatorsChun Zhang0Qiaoxia Tang1Zhixiang Wang2School of Mathematical Science, Huaiyin Normal University, Huai’an 223300, ChinaSchool of Mathematical Science, Huaiyin Normal University, Huai’an 223300, ChinaSchool of Mathematical Science, Huaiyin Normal University, Huai’an 223300, ChinaBy introducing a switching scheme related to the state and time, a typical switched model alternating between a Duffing oscillator and van der Pol oscillator is established to explore the typical dynamical behaviors as well as the mechanism of the switched system. Shooting methods to locate the limit cycle and specify bifurcation sets are described by defining an appropriate Poincaré map. Different types of multiple-Focus/Cycle and single-Focus/Cycle period oscillations in the system can be observed. Symmetry-breaking, period-doubling, and grazing bifurcation curves are obtained in the plane of bifurcation parameters, dividing the parameters plane into several regions corresponding to different kinds of oscillations. Meanwhile, based on the numerical simulation and bifurcation analysis, the mechanisms of several typical dynamical behaviors observed in different regions are presented.https://www.mdpi.com/2227-7390/10/5/772switched systemmixed-mode oscillationnonsmooth evolutionbifurcation mechanismtransformed phase portrait |
| spellingShingle | Chun Zhang Qiaoxia Tang Zhixiang Wang Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators Mathematics switched system mixed-mode oscillation nonsmooth evolution bifurcation mechanism transformed phase portrait |
| title | Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators |
| title_full | Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators |
| title_fullStr | Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators |
| title_full_unstemmed | Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators |
| title_short | Grazing and Symmetry-Breaking Bifurcations Induced Oscillations in a Switched System Composed of Duffing and van der Pol Oscillators |
| title_sort | grazing and symmetry breaking bifurcations induced oscillations in a switched system composed of duffing and van der pol oscillators |
| topic | switched system mixed-mode oscillation nonsmooth evolution bifurcation mechanism transformed phase portrait |
| url | https://www.mdpi.com/2227-7390/10/5/772 |
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