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The Quarterly Journal of Mechanics and Applied Mathematics 1954 7(2):152-167; doi:10.1093/qjmam/7.2.152
© 1954 by Oxford University Press
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ON THE TRANSFORMATIONS OF SINGULARITIES AND LIMIT CYCLES OF THE VARIATIONAL EQUATIONS OF VAN DER POL

A. W. GILLIES

( Northampton Polytechnic London )

The investigation of the solution of van der Pol's equation with forcing term leads to equations for the amplitude, b, and phase of the oscillation, {phi}, of the form

b = b(1–b2)–F cos{phi},

b{phi} = –bx+F sin{phi}.

The solution of this autonomous system of first-order equations has been discussed by Cartwright. By considering the isoclines on the plane with (b,{phi}) as polar coordinates, it is shown that Cartwright's solution is incorrect in one range of the parameters. The corrected solution is given. In consequence of this, it is shown that the hysteresis effects to be expected for a van der Pol oscillator with increasing and decreasing frequency are confined to a narrower frequency interval and are less varied in character than was suggested by Cartwright's solution.


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