Space of initial conditions for a cubic Hamiltonian system

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In this paper we perform the analysis that leads to the space of initial conditions for the Hamiltonian system q′ = p2 + zq + α, p′ = −q2 zpβ, studied by the author in a previous article. By compactifying the phase space of the system from C2 to CP2 three base points arise in the standard coordinate charts covering the complex projective space. Each of these is removed by a sequence of three blow-ups, a construction to regularise the system at these points. The resulting space, where the exceptional curves introduced after the first and second blow-up are removed, is the so-called Okamoto’s space of initial conditions for this system which, at every point, defines a regular initial value problem in some coordinate chart of the space. The solutions in these coordinates will be compared to the solutions in the original variables.
Original languageEnglish
Number of pages12
JournalComplex Variables and Elliptic Equations
Early online date17 Jan 2018
Publication statusEarly online - 17 Jan 2018


  • Painleve equations
  • space of inital conditions
  • Hamiltonian systems


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