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Rational Involutions and an Application to Planar Systems of ODE

Author

Listed:
  • Ivan Mastev

    (Center for Applied Mathematics and Theoretical Physics, University of Maribor, SI-2000 Maribor, Slovenia)

  • Valery G. Romanovski

    (Center for Applied Mathematics and Theoretical Physics, University of Maribor, SI-2000 Maribor, Slovenia
    Faculty of Natural Science and Mathematics, University of Maribor, SI-2000 Maribor, Slovenia
    Faculty of Electrical Engineering and Computer Science, University of Maribor, SI-2000 Maribor, Slovenia)

  • Yun Tian

    (Department of Mathematics, Shanghai Normal University, 100 Guilin Rd., Shanghai 200234, China)

Abstract

An involution refers to a function that acts as its own inverse. In this paper, our focus lies on exploring two-dimensional involutive maps defined by rational functions. These functions have denominators represented by polynomials of degree one and numerators by polynomials of a degree of, at most, two, depending on parameters. We identify the sets in the parameter space of the maps that correspond to involutions. The investigation relies on leveraging algorithms from computational commutative algebra based on the Groebner basis theory. To expedite the computations, we employ modular arithmetic. Furthermore, we showcase how involution can serve as a valuable tool for identifying reversible and integrable systems within families of planar polynomial ordinary differential equations.

Suggested Citation

  • Ivan Mastev & Valery G. Romanovski & Yun Tian, 2024. "Rational Involutions and an Application to Planar Systems of ODE," Mathematics, MDPI, vol. 12(3), pages 1-16, February.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:3:p:486-:d:1332409
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