Abstract
If an ordinary differential equation is discretizised near an asymptotically stable stationary solution with a pair of imaginary eigenvalues by Euler's method with constant step lengthh, small invariant attracting cycles of radiusO(h 1/2) will appear. This Hopf bifurcation theorem is applied to prove the existence of limit cycles in certain difference equations occurring in biomathematics (hypercycle, two loci-two alleles) and is also extended to general Runge—Kutta methods.
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Hofbauer, J., Iooss, G. A Hopf bifurcation theorem for difference equations approximating a differential equation. Monatshefte für Mathematik 98, 99–113 (1984). https://doi.org/10.1007/BF01637279
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DOI: https://doi.org/10.1007/BF01637279