Computer Science Faculty Publications

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. A power series expansion in the damping parameter E of the limit cycle U(t; E) of the free van der Pol equation Ul + E( U2 _1) U + U =0 is constructed and analyzed. Coefficients in the expansion are computed up to 0(E24) in exact rational arithmetic using the symbolic manipulation system MACSYMA and up to O(E163) using a FORTRAN program. The series is analyzed using Pade approximants. The convergence of the series for the maximum amplitude of the limit cycle is limited by two pairs of complex conjugate singularities in the complex e-plane. These singularities are the same as those which limit the convergence of the series expansion of the frequency of the limit cycle. A new expansion parameter is introduced which maps these singularities to infinity and leads to a new expansion for the amplitude which converges for all real values of E. Amplitudes computed from this transformed series agree very well with reported numerical and asymptotic results. For the limit cycle itself, convergence of the series expansion is limited by three pairs of complex conjugate branch point singularities. Two pairs remain fixed throughout the cycle and correspond to the singularities found in the maximum amplitude series, while the third pair moves in the E-plane as a function of t from one of the fixed pairs to the other. This moving pair of singularities dominate the fixed singularities for certain ranges of t and hence account for a nonuniformity in the convergence of the series. The limit cycle series is transformed using a new expansion parameter, which leads to a new series that converges for larger values of E.

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SIAM Journal on Applied Mathematics