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Equivalence of \(n\)-point Gauss–Chebyshev rule and \(4n\)-point midpoint rule in computing the period of a Lotka–Volterra system

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Abstract

Two integrals (3.6), (4.7) for the period of a periodic solution of the Lotka–Volterra system are presented in terms of two inverse functions of \(x\exp(x)\) restricted on \(x \geq -1\), \( x \leq -1\), respectively. In computing this period numerically, the integral (3.6), which possesses a weak singularity of the square root type at each endpoint of the integration, is an excellent example of using the Gauss–Chebyshev integration rule of the first kind; while the integral (4.7), which is an integral of a smooth periodic function over its period \([0, 2\pi]\), is an excellent example of using the midpoint rule, but not the trapezoidal rule, suggested by Waldvogel [39, 40], due to a removable singularity of the integrand at \(0\), \(\pi/2\), \(\pi\), \(3\pi/2\), and \(2\pi\), respectively. This paper shows, in computing the period of a periodic solution of the Lotka–Volterra system, the \(n\)-point Gauss–Chebyshev integration rule of the first kind applied to the integral (3.6) becomes the \(4n\)-point midpoint rule to the integral (4.7).

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Authors and Affiliations

  1. Department of Mathematics, University of Wyoming, Laramie, WY, 82071-3036, USA

    Shagi-Di Shih

  2. Department of Mathematics, Brigham Young University, Provo, UT, 84602-6539, USA

    Shue-Sum Chow

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  1. Shagi-Di Shih
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  2. Shue-Sum Chow
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Correspondence to Shagi-Di Shih.

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Dedicated to R. Bruce Kellogg on the occasion of his 75th birthday.

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Shih, SD., Chow, SS. Equivalence of \(n\)-point Gauss–Chebyshev rule and \(4n\)-point midpoint rule in computing the period of a Lotka–Volterra system. Adv Comput Math 28, 63–79 (2008). https://doi.org/10.1007/s10444-006-9013-4

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