Geodesic
Discretization of Hamiltonian systems and intersection theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 3, pp. 475-492

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We discuss the possibility of using the intersection points of the common level surface of integrals of motion with an auxiliary curve to construct finite-difference equations corresponding to different discretizations of the original integrable system. As an example, we consider the generalized one-dimensional oscillator with third- and fifth-degree nonlinearity, for which we show that the intersection divisors of the hyperelliptic curve with straight lines, quadrics, and cubics generate families of integrable discrete maps.
Keywords: finite-dimensional integrable system, discrete integrable map, intersection theory. 
@article{TMF_2018_197_3_a9,
     author = {A. V. Tsiganov},
     title = {Discretization of {Hamiltonian} systems and intersection theory},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {475--492},
     publisher = {mathdoc},
     volume = {197},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2018_197_3_a9/}
}
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A. V. Tsiganov. Discretization of Hamiltonian systems and intersection theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 197 (2018) no. 3, pp. 475-492. http://geodesic.mathdoc.fr/item/TMF_2018_197_3_a9/