## Project Details

### Description

In this project we aim to understand the nature of so-called spaces of initial values for various classes of non-linear differential equations in the complex plane and related Hamiltonian systems the solutions of which have movable singularities more general than poles, for which the concept of the space of initial values was originally devised by K. Okamoto (i.e. for equations with the Painlevé property).

The objectives of this project are three-fold:

1) Understand and classify the geometries of spaces of initial values for certain classes of differential equations, in particular equations for which the solutions admit movable algebraic singularities, but no others (known as the quasi-Painlevé property).

2) Develop a general software / computer algebra tool which allows, by automating the computations of blow-ups for a given input equation from within some class (e.g. second-order rational ordinary differential equations, or polynomial Hamiltonian systems in two variables, etc.), to obtain the space of initial values for the equation, from which the possible types of movable singularities in the complex plane that can occur in the solutions of the equations can be determined.

3) To apply the method of blowing up phase spaces of higher-order differential equations and higher-dimensional Hamiltonian systems, and determine classes of equations for which a space of initial conditions can be constructed in a finite number of steps. The main difficulty to be overcome here is that the indeterminacies in the equations to be blown up are no longer points, but higher-dimensional sub-varieties.

This research project is conducted under an EPSRC New Investigator Award held by Dr Thomas Kecker.

The objectives of this project are three-fold:

1) Understand and classify the geometries of spaces of initial values for certain classes of differential equations, in particular equations for which the solutions admit movable algebraic singularities, but no others (known as the quasi-Painlevé property).

2) Develop a general software / computer algebra tool which allows, by automating the computations of blow-ups for a given input equation from within some class (e.g. second-order rational ordinary differential equations, or polynomial Hamiltonian systems in two variables, etc.), to obtain the space of initial values for the equation, from which the possible types of movable singularities in the complex plane that can occur in the solutions of the equations can be determined.

3) To apply the method of blowing up phase spaces of higher-order differential equations and higher-dimensional Hamiltonian systems, and determine classes of equations for which a space of initial conditions can be constructed in a finite number of steps. The main difficulty to be overcome here is that the indeterminacies in the equations to be blown up are no longer points, but higher-dimensional sub-varieties.

This research project is conducted under an EPSRC New Investigator Award held by Dr Thomas Kecker.

Status | Active |
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Effective start/end date | 1/06/22 → 31/05/24 |

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