Celebratio Mathematica — Cartwright

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Works connected to Solomon Lefschetz

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M. L. Cartwright: “Forced oscillations in nonlinear systems,” pp. 149–241 in Contributions to the theory of nonlinear oscillations, vol. 1. Edited by S. Lefschetz. Annals of Mathematics Studies 20. Princeton University Press, 1950. MR 0035355 Zbl 0039.09901 incollection

M. L. Cartwright: “Van der Pol’s equation for relaxation oscillations,” pp. 3–18 in Contributions to the theory of nonlinear oscillations, vol. 2. Edited by S. Lefschetz. Annals of Mathematics Studies 29. Princeton University Press, 1952. MR 0052617 Zbl 0048.06902 incollection

The equation \[ \ddot{x} - k(1-x^2)\dot{x}+x=0 \] with \( k \) large and positive has only one periodic solution, other than \( x=0 \), and this is of a type usually described as a relaxation oscillation (as opposed to a sinusoidal oscillation). It was discussed by van der Pol [1926] who obtained a graphical solution for \( k=10 \) and by le Corbeiller [1936] who, using Liénard’s method, showed that the period \[ 2T = 2k(3/2 - \log_e 2) + O(k) ,\] and the greatest height \( h = 2 + O(1) \) as \( k\to\infty \). Other authors [Flanders and Stoker 1946; Haag 1943, 1944; LaSalle 1949] have also discussed the equation, in particular Dorodnitsin [1947] has obtained an asympotic formula for \( T \) with smaller error terms but his analysis is difficult to follow.

This paper is based on the joint work of Professor J. E. Littlewood and myself, largely on work which was done before that contained in our other published papers on nonlinear differential equations. We shall show that as \( k\to\infty \) \begin{align*} T &= k(3/2 - \log_e 2) + \frac{3(\alpha+\beta)}{2k^{1/3}} + O\Bigl(\frac{1}{k^{1/3}}\Bigr)\\ h &= 2 + \frac{\alpha+\beta}{3k^{4/3}} + O\Bigl(\frac{1}{k^{4/3}}\Bigr), \end{align*} where \( \alpha \) and \( \beta \) are constants determined as follows: The equation \[ \eta_0\frac{d\eta_0}{d\xi} = 2\xi\eta_0 + 1 \] has one and only one solution \( \eta_0^*(\xi) \) such that \( \eta_0^*(\xi) \to 0 \) as \( \xi\to -\infty \). \[ \alpha = \eta_0^*(0),\qquad \beta = \int_0^{\infty}\frac{d\xi}{\eta_0^*(\xi)}. \]