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Research Papers

# Nonlinear Dynamics of Duffing System With Fractional Order Damping

[+] Author and Article Information
Junyi Cao1

State Key Laboratory for Manufacturing Systems Engineering, Research Institute of Diagnostics and Cybernetics, Xi’an Jiaotong University, Xi’an 710049, Chinacaojy@mail.xjtu.edu.cn

Chengbin Ma

University of Michigan–Shanghai Jiaotong University Joint Institute, Shanghai Jiaotong University, Shanghai 200240, China

Hang Xie, Zhuangde Jiang

State Key Laboratory for Manufacturing Systems Engineering, Research Institute of Diagnostics and Cybernetics, Xi’an Jiaotong University, Xi’an 710049, China

1

Corresponding author.

J. Comput. Nonlinear Dynam 5(4), 041012 (Aug 12, 2010) (6 pages) doi:10.1115/1.4002092 History: Received June 12, 2009; Revised October 19, 2009; Published August 12, 2010; Online August 12, 2010

## Abstract

In this paper, nonlinear dynamics of Duffing system with fractional order damping is investigated. The fourth-order Runge–Kutta method and tenth-order CFE-Euler method are introduced to simulate the fractional order Duffing equations. The effect of taking fractional order on system dynamics is investigated using phase diagram, bifurcation diagram and Poincaré map. The bifurcation diagram is introduced to exam the effect of excitation amplitude, frequency, and damping coefficient on the Duffing system with fractional order damping. The analysis results show that the fractional order damped Duffing system exhibits periodic motion, chaos, periodic motion, chaos, and periodic motion in turn when the fractional order varies from 0.1 to 2.0. The period doubling bifurcation route to chaos and inverse period doubling bifurcation out of chaos are clearly observed in the bifurcation diagrams with various excitation amplitude, frequency, and damping coefficient.

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## Figures

Figure 4

Bifurcation diagrams with various ω and A

Figure 1

Phase trajectory and Poincaré map at α=1.0

Figure 7

Bifurcation diagrams of x versus c

Figure 8

Phase trajectory and Poincaré map with various c

Figure 2

Bifurcation diagrams of x versus α

Figure 3

Phase trajectory and Poincaré map with various α

Figure 6

Phase trajectory and Poincaré map with various A

Figure 5

Phase trajectory and Poincaré map with various ω

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