Bifurcation Evolution of Duffing Systems on 2-Parameter Planes

ZHANG Yanlong; WANG Li; SHI Jianfei

doi:10.21656/1000-0887.380089

Volume 39 Issue 3

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ZHANG Yanlong, WANG Li, SHI Jianfei. Bifurcation Evolution of Duffing Systems on 2-Parameter Planes[J]. Applied Mathematics and Mechanics, 2018, 39(3): 324-333. doi: 10.21656/1000-0887.380089

Citation:

ZHANG Yanlong, WANG Li, SHI Jianfei. Bifurcation Evolution of Duffing Systems on 2-Parameter Planes[J]. Applied Mathematics and Mechanics, 2018, 39(3): 324-333. doi: 10.21656/1000-0887.380089

Citation:

ZHANG Yanlong, WANG Li, SHI Jianfei. Bifurcation Evolution of Duffing Systems on 2-Parameter Planes[J]. Applied Mathematics and Mechanics, 2018, 39(3): 324-333. doi: 10.21656/1000-0887.380089

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Bifurcation Evolution of Duffing Systems on 2-Parameter Planes

doi: 10.21656/1000-0887.380089

¹School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, P.R.China;²School of Mathematics, Lanzhou City University, Lanzhou 730070, P.R.China

Funds: The National Natural Science Foundation of China（11302092; 11362008）

Received Date: 2017-04-07
Rev Recd Date: 2017-04-26
Publish Date: 2018-03-15

Abstract

Abstract

The calculation method for the top Lyapunov exponents in the parameter space was given. The top Lyapunov exponents of Dufffing systems on 2-parameter planes were calculated with the numerical method. Combined with the single-parameter top Lyapunov exponents, the bifurcation diagrams, the phase diagrams and the time response diagrams, the bifurcation and the bifurcation evolution process of Duffing systems on the 2-parameter planes were discussed in view of the change of system parameters. The results show that 2 different regions with the phenomena of missing edges appear when the pitchfork bifurcation occurs. The system has strong sensitivity to initial values in the regions where 2 attractors coexist. The system vibration amplitude decreases suddenly when the system moves through the period jump curve. The system flutter motion often occurs when the excitation frequency is relatively small. In addition, when the stiffness coefficient increases, the period-doubling bifurcation curve cycles constantly exist and nest each other in the 2 regions with the phenomena of missing edges, which makes the system finally evolve into a chaotic state via the period-doubling bifurcation sequences. The dynamic properties of the system are very complex on 2-parameter planes with the change of control parameters.
- Duffing system,
- Lyapunov exponent,
- 2-parameter characteristic,
- bifurcation,
- periodic jump

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References(17)

References

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