Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays
In this paper, the dynamics of Duffing-van der Pol oscillators under linear-plus-nonlinear position feedback control with two time delays is studied analytically and numerically. By the averaging method, together with truncation of Taylor expansions for those terms with time delay, the slow-flow equ...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Shock and Vibration |
| Online Access: | http://dx.doi.org/10.3233/SAV-2010-0570 |
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| _version_ | 1849304614352977920 |
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| author | Xinye Li Huabiao Zhang Lijuan Zhang |
| author_facet | Xinye Li Huabiao Zhang Lijuan Zhang |
| author_sort | Xinye Li |
| collection | DOAJ |
| description | In this paper, the dynamics of Duffing-van der Pol oscillators under linear-plus-nonlinear position feedback control with two time delays is studied analytically and numerically. By the averaging method, together with truncation of Taylor expansions for those terms with time delay, the slow-flow equations are obtained from which the trivial and nontrivial solutions can be found. It is shown that the trivial solution can be stabilized by appropriate gain and time delay in linear feedback although it loses its stability via Hopf bifurcation and results in periodic solution for uncontrolled systems. And the stability of the trivial solution is independent of nonlinear feedback. Different from the case of the trivial solution, the stability of nontrivial solutions is also associated with nonlinear feedback besides linear feedback. Non-trivial solutions may lose their stability via saddle-node or Hopf bifurcation and the resulting response of the system may be quasi-periodic or chaotic. The feedback gains and time delays have great effects on the amplitude of the periodic solutions and their bifurcation control. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results. |
| format | Article |
| id | doaj-art-3a37cfeb1cfc4104bea1556c8122f3d0 |
| institution | Kabale University |
| issn | 1070-9622 1875-9203 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Shock and Vibration |
| spelling | doaj-art-3a37cfeb1cfc4104bea1556c8122f3d02025-08-20T03:55:41ZengWileyShock and Vibration1070-96221875-92032011-01-01181-237738610.3233/SAV-2010-0570Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time DelaysXinye Li0Huabiao Zhang1Lijuan Zhang2School of Mechanical Engineering, Hebei University of Technology, Tianjin, ChinaSchool of Astronautics, Harbin Institute of Technology, Harbin, ChinaSchool of Mechanical Engineering, Hebei University of Technology, Tianjin, ChinaIn this paper, the dynamics of Duffing-van der Pol oscillators under linear-plus-nonlinear position feedback control with two time delays is studied analytically and numerically. By the averaging method, together with truncation of Taylor expansions for those terms with time delay, the slow-flow equations are obtained from which the trivial and nontrivial solutions can be found. It is shown that the trivial solution can be stabilized by appropriate gain and time delay in linear feedback although it loses its stability via Hopf bifurcation and results in periodic solution for uncontrolled systems. And the stability of the trivial solution is independent of nonlinear feedback. Different from the case of the trivial solution, the stability of nontrivial solutions is also associated with nonlinear feedback besides linear feedback. Non-trivial solutions may lose their stability via saddle-node or Hopf bifurcation and the resulting response of the system may be quasi-periodic or chaotic. The feedback gains and time delays have great effects on the amplitude of the periodic solutions and their bifurcation control. The simulations, obtained by numerically integrating the original system, are in good agreement with the analytical results.http://dx.doi.org/10.3233/SAV-2010-0570 |
| spellingShingle | Xinye Li Huabiao Zhang Lijuan Zhang Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays Shock and Vibration |
| title | Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays |
| title_full | Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays |
| title_fullStr | Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays |
| title_full_unstemmed | Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays |
| title_short | Response of the Duffing-Van der Pol Oscillator under Position Feedback Control with Two Time Delays |
| title_sort | response of the duffing van der pol oscillator under position feedback control with two time delays |
| url | http://dx.doi.org/10.3233/SAV-2010-0570 |
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