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https://github.com/x3042/exactodereduction.jl
Exact reduction of ODE models via linear transformations
https://github.com/x3042/exactodereduction.jl
differential-equations dimensionality-reduction julia ode ode-model
Last synced: about 1 month ago
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Exact reduction of ODE models via linear transformations
- Host: GitHub
- URL: https://github.com/x3042/exactodereduction.jl
- Owner: x3042
- License: mit
- Created: 2020-10-04T10:15:51.000Z (about 4 years ago)
- Default Branch: main
- Last Pushed: 2024-02-17T20:32:27.000Z (9 months ago)
- Last Synced: 2024-10-12T23:21:48.376Z (about 1 month ago)
- Topics: differential-equations, dimensionality-reduction, julia, ode, ode-model
- Language: Julia
- Homepage:
- Size: 124 MB
- Stars: 13
- Watchers: 3
- Forks: 3
- Open Issues: 4
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
# ExactODEReduction
[](https://github.com/x3042/ExactODEReduction.jl/actions/workflows/Runtests.yml) [](https://x3042.github.io/ExactODEReduction.jl/dev)
This repository contains a Julia implementation of algorithms for finding exact reductions of ODE systems via a linear change of variables.
Online documentation could be found at [https://x3042.github.io/ExactODEReduction.jl](https://x3042.github.io/ExactODEReduction.jl).
## Installation
To install `ExactODEReduction.jl`, run the following in Julia:
```julia
import Pkg
Pkg.add(url="https://github.com/x3042/ExactODEReduction.jl")
```
For the usage examples, please see examples below in this file, or in the `examples` directory.
## What is exact reduction?
Exact reduction of the system of differential equations is an exact variable substitution which preserves the invariants of the system. In this project we consider reductions obtained with **linear transformations**. We will explain it using a toy example. Consider the system
$$\begin{cases}
\dot{x}_1 = x_1^2 + 2x_1x_2,\\
\dot{x}_2 = x_2^2 + x_3 + x_4,\\
\dot{x}_3 = x_2 + x_4, \\
\dot{x}_4 = x_1 + x_3
\end{cases}$$
An example of an exact reduction in this case would be the following set of new variables
$$y_1 = x_1 + x_2 \quad \text{ and } \quad y_2 = x_3 + x_4$$
The important feature of variables $y_1, y_2$ is that their derivatives can be written in terms of $y_1$ and $y_2$ only:
$$\dot{y}_1 = \dot{x}_1 + \dot{x}_2 = y_1^2 + y_2$$
and
$$\dot{y}_2 = \dot{x}_3 + \dot{x}_4 = y_1 + y_2$$
Therefore, the original system can be **reduced exactly** to the following system:
$$\begin{cases}
\dot{y}_1 = y_1^2 + y_2,\\
\dot{y}_2 = y_1 + y_2
\end{cases}$$
## What does this package do and how to use it?
We implement an algorithm that takes as **input** a system of ODEs with polynomial right-hand side and **returns** a list of possible linear transformations and corresponding systems.
We will demonstrate the usage on the example above. For more details on the package usage, including reading dynamical systems from `*.ode` files, please see the documentation.
1. Import the package
```julia
using ExactODEReduction
```
2. Construct the system (as in the example above)
```julia
odes = @ODEsystem(
x1'(t) = x1^2 + 2x1*x2,
x2'(t) = x2^2 + x3 + x4,
x3'(t) = x2 + x4,
x4'(t) = x1 + x3
)
```
3. Call `find_reductions` providing the system
```julia
reductions = find_reductions(odes)
```
which returns the list of possible reductions. You will get the following result printed
```julia
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
```
Notice that the first reduction is the same as we have seen earlier. We can access it through the `reductions` object
```julia
red1 = reductions[1]
```
```julia
new_system(red1)
## Prints:
y1'(t) = y1(t)^2 + y2(t)
y2'(t) = y1(t) + y2(t)
```
```julia
new_vars(system)
## Prints:
Dict{Nemo.fmpq_mpoly, Nemo.fmpq_mpoly} with 2 entries:
y2 => x3 + x4
y1 => x1 + x2
```
For more examples we refer to the documentation and the `examples` directory.