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Research Papers

# An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

[+] Author and Article Information
Albert C. J. Luo

Department of Mechanical and
Industrial Engineering,
Southern Illinois University Edwardsville,
Edwardsville, IL 62026-1805
e-mail: aluo@siue.edu

Arash Baghaei Lakeh

Department of Mechanical and
Industrial Engineering,
Southern Illinois University Edwardsville,
Edwardsville, IL 62026-1805

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received January 27, 2013; final manuscript received December 31, 2013; published online February 13, 2014. Assoc. Editor: Claude-Henri Lamarque.

J. Comput. Nonlinear Dynam 9(3), 031001 (Feb 13, 2014) (7 pages) Paper No: CND-13-1021; doi: 10.1115/1.4026425 History: Received January 27, 2013; Revised December 31, 2013

## Abstract

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms ($N$) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., $AN≤ɛ$ and $ɛ=10-8$). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

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## Figures

Fig. 1

Frequency–amplitude responses of period-1 motion for van der Pol oscillator from the 22 harmonic terms (HB22): (a)–(d) A1-A11 (α1 = 0.5,α2 = 5,α3 = 10,Q0 = 0.5,1.0,2.0,4.0,8.0). Acronym “HB” represents Hopf bifurcation.

Fig. 3

Analytical and numerical solutions of limit cycle with 22 harmonic terms (HB22) with Ω≈3.1574 and Q0 = 0: (a) phase plane and (b) analytical harmonic amplitude spectrum. Solid and symbol curves are for numerical and analytical solutions. The initial condition is x0 = 0.0844137, and y0 = 2.02906 (α1 = 0.5,α2 = 5,α3 = 10).

Fig. 4

Analytical and numerical solutions of stable period-1 motion with 22 harmonic terms (HB22) at Ω = 1.048 and Q0 = 4.0: (a) phase plane and (b) analytical harmonic amplitude spectrum. Solid and symbol curves are for numerical and analytical solutions. The initial condition is x0 = 0.391628, and y0 =  -0.556119 (α1 = 0.5,α2 = 5,α3 = 10).

Fig. 5

Analytical and numerical solutions of unstable period-1 motion with 22 harmonic terms (HB22) at Ω = 0.5 and Q0 = 4: (a) phase plane, (b) displacement, (c) velocity, and (d) analytical harmonic amplitude spectrum. Solid and symbol curves are for numerical and analytical solutions. The initial condition is x0 = 0.409654, and y0 = 6.472250×10-3 (α1 = 0.5,α2 = 5,α3 = 10).

Fig. 2

Parameter map for the period-1 motion of the periodically forced van der Pol oscillator. The gray and white areas are for stable and unstable period-1 motions, respectively (α1 = 0.5,α2 = 5,α3 = 10).

Fig. 6

Analytical and numerical solutions of unstable period-1 motion with 22 harmonic terms (HB22) at Ω = 8 and Q0 = 4: (a) phase plane and (b) analytical harmonic amplitude spectrum. Solid and symbol curves are for numerical and analytical solutions. The initial condition is x0 = -0.073682, and y0 =  -0.042901 (α1 = 0.5,α2 = 5,α3 = 10).

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