Guide

The algorithms in this package take as input a system of ODEs with polynomial right-hand side and return the new variable expressions and the corresponding system(s).

Creating an ODE and finding reductions

As an example, consider the folliwing system of differential equations

\[\begin{cases} \dot{x}_1 = x_1^2 + 2x_1x_2,\\ \dot{x}_2 = x_2^2 + x_3 + x_4,\\ \dot{x}_3 = x_2 + x_4, \\ \dot{x}_4 = x_1 + x_3 \end{cases}\]

To create a system as above in ExactODEReduction.jl, use the ODEsystem macro. This is the easiest way to do so.

using ExactODEReduction

odes = @ODEsystem(
    x1'(t) = x1^2 + 2x1*x2,
    x2'(t) = x2^2 + x3 + x4,
    x3'(t) = x2 + x4,
    x4'(t) = x1 + x3
)

x1'(t) = x1(t)^2 + 2*x1(t)*x2(t)
x2'(t) = x2(t)^2 + x3(t) + x4(t)
x3'(t) = x2(t) + x4(t)
x4'(t) = x1(t) + x3(t)

After using the macro, we can call find_reductions function. This function accepts the ODE model and outputs possible linear reductions.

reductions = find_reductions(odes)

A chain of 2 reductions of dimensions 2, 3.
==================================
1. Reduction of dimension 2.
New system:
y1'(t) = y1(t)^2 + y2(t)
y2'(t) = y1(t) + y2(t)
New variables:
y1 = x1 + x2
y2 = x3 + x4
==================================
2. Reduction of dimension 3.
New system:
y1'(t) = y1(t)^2 + 2*y1(t)*y2(t)
y2'(t) = y2(t)^2 + y3(t)
y3'(t) = y1(t) + y2(t) + y3(t)
New variables:
y1 = x1
y2 = x2
y3 = x3 + x4

Note that because of the randomization the result may be different between several runs. If you need to fix the result, use the seed keyword argument (see Finding reductions).

The reductions object can be treated as an array of reductions. For example, to access the second reduction from the list, type

reduction2 = reductions[2]

Reduction of dimension 3.
New system:
y1'(t) = y1(t)^2 + 2*y1(t)*y2(t)
y2'(t) = y2(t)^2 + y3(t)
y3'(t) = y1(t) + y2(t) + y3(t)
New variables:
y1 = x1
y2 = x2
y3 = x3 + x4

Further, we can use new_system and new_vars functions to explore this reduction.

new_ode = new_system(reduction2)

y1'(t) = y1(t)^2 + 2*y1(t)*y2(t)
y2'(t) = y2(t)^2 + y3(t)
y3'(t) = y1(t) + y2(t) + y3(t)

Note that new_ode from above is again an ODE object. In particular, this means that all functions from (Section Functions for manipulating ODEs) will also work for it.

new_var = new_vars(reduction2)

Dict{Nemo.fmpq_mpoly, Nemo.fmpq_mpoly} with 3 entries:
  y2 => x2
  y1 => x1
  y3 => x3 + x4

The function find_reductions additionaly provides several useful keyword arguments, which are described in Finding reductions.