This is a companion repository to the paper ODE trajectories as abnormal curves in Carnot groups. It contains an implementation to compute abnormal covectors for lifts of trajectories of ODE systems in Carnot groups. The computations for the examples of the paper are also included.

A working installation of SageMath is needed. The code has been tested to work on version 9.0, released 2020-01-01.

Clone the repo with

$ git clone https://github.com/ehaka/ode-abnormals

Test any of the examples in the examples subfolder, for instance

$ cd ode-abnormals
$ cd examples
$ sage logarithmic_spiral.sage

There is no separate installation, everything assumes that the package ode_abnormals is found in sys.path. E.g. in examples/logarithmic_spiral.sage this is guaranteed with the preamble

import sys
import pathlib
path = pathlib.Path().absolute().parent
sys.path.append(str(path))

Here the assumption is that pathlib.Path().absolute() refers to the folder ode-abnormals/examples, so the folder added to path is ode-abnormals.

Note that sys.path when running Sage is different from $PATH.

The easiest way to use the package within Sage is to directly launch Sage in the main folder of the repo

$ cd ode-abnormals
$ sage

Similarly for the Sage notebook

$ cd ode-abnormals
$ sage --notebook=jupyter

Then the functions in the ode_abnormals package can be imported with

sage: from ode_abnormals import *

Otherwise, the repo folder can be added to sys.path for the duration of a Sage session with sys.path.append(PATH-TO-ODE-ABNORMALS-DIRECTORY) command.

Copying any of the scripts in the examples subfolder is the easiest way to compute examples. The barebones structure is the following.

Define variables in a polynomial ring with coefficients in some ring, e.g. the rationals QQ

sage: R.<x,y> = QQ[]
sage: variables = [x, y]

Define the ODE system as a list of polynomials in the order of the variables.

sage: equations = [1-y, x] # dx/dt = 1-y, dy/dt = x

Run the main solver.

sage: from ode_abnormals.ode_search import ode_search
sage: covec, auxdata = ode_search(equations, variables)

The first element of the output is a Python dictionary describing the covector as a dual element of a free Lie algebra.

sage: covec
{[X_2, [X_1, X_2]]: 1,
 [X_1, [X_1, [X_1, X_2]]]: -1,
 [X_2, [X_2, [X_1, X_2]]]: -1}

See the various example scripts for further features. Info on the input and output of the various functions can be accessed with the syntax

sage: ode_search?

The computations for the examples section of the paper can be found in the examplessubfolder. Precomputed outputs are available in examples/output, showing the resulting covectors and abnormal polynomials.

The precomputed examples are

1. A logarithmic spiral.
2. A generic homogeneous linear ODE in rank 2.
3. The Hawaiian earring.
4. The Lorenz butterfly.
5. Linear ODEs with randomly chosen integer coefficients in ranks 3 and 4.

This project is licensed under the MIT License - see the LICENSE.txt file for details.