Periodic motions in a double-well Duffing oscillator under periodic excitation through discrete implicit mappings



In this paper, periodic motions of a periodically forced, damped, Duffing oscillator with double-well potential are analytically predicted through discrete implicit mappings. The discrete implicit maps are obtained from the differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions of the Duffing oscillator are predicted analytically, and the corresponding stability and bifurcation analysis of periodic motions are carried out through the eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are performed by the numerical method of the differential equation to verify the semi-analytical prediction, and the corresponding harmonic amplitudes are computed through discrete Fourier series of the analytically predicted node points of periodic motions, and the complexity of periodic motions can be measured by the harmonic amplitude. The frame work presented in this paper can provide a semi-analytical method to find periodic motions and to determine routes of periodic motions to chaos rather than numerical simulation only in nonlinear dynamical systems, and the stable and unstable periodic motions and even chaos can be predicted analytically.


Duffing oscillator Period-1 motions to chaos Bifurcation tree Implicit mapping Mapping structures 


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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.McCoy School of EngineeringMidwestern State UniversityWichita FallsUSA
  2. 2.Department of Mechanical and Industrial EngineeringSouthern Illinois University EdwardsvilleEdwardsvilleUSA

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