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Geometry of cycles in quadratic systems

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Abstract

This paper is a study of the affine and euclidean differential geometry of cycles of quadratic systems. While the euclidean curvature must always be strictly positive, the affine curvature can take either sign, although with certain restrictions. Every quadratic cycle has exactly six affine vertices, but the number of euclidean vertices can vary, not just from cycle to cycle, but for the same cycle under a linear coordinate transformation. We prove that an upper bound on the number of euclidean vertices over all non-circular quadratic cycles is twelve, and provide evidence that a sharp upper bound is six.

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Correspondence to Douglas S. Shafer.

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Shafer, D.S., Zegeling, A. Geometry of cycles in quadratic systems. Qual. Th. Dyn. Syst. 3, 251–274 (2002). https://doi.org/10.1007/BF02969340

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Key words

  • affine
  • curvature
  • cycle
  • quadratic system
  • sextactic
  • vertex