Presentation

The OSCILATE project allows the application of the Method of Multiple Scales (MMS) to a nonlinear equation or systems of \(N\) coupled nonlinear equations.

The source codes of OSCILATE (Oscillators’ nonlinear analysis through SymboliC ImpLementATion of the mEthod of multiple scales) are hosted on GitHub. A full documentation is available.

Nonlinear systems considered

The nonlinear systems tackled are of the form

The \(x_i(t)\) (\(i=0,...,N-1\)) are the oscillators’ coordinates,

is the vector containing all the oscillators’ coordinates (the \(^\intercal\) denotes the transpose), \(\omega_i\) are their natural frequencies, \(t\) is the time and \(\dot{(\bullet)} = \textrm{d}(\bullet)/\textrm{d}t\) denotes a time-derivative.

The \(f_i\) are functions which can contain:

• Weak linear terms in \(x_i\), \(\dot{x}_i\), or \(\ddot{x}_i\).

• Weak linear coupling terms involving \(x_j\), \(\dot{x}_j\), or \(\ddot{x}_j\), \(j \neq i\).

• Weak nonlinear terms. Taylor expansions are performed to approximate nonlinear terms as polynomial nonlinearities.

• Forcing terms:

  • Can be hard (appearing at leading order) or weak (small).

  • Primarily harmonic, e.g., \(F \cos(\omega t)\), where \(F\) and \(\omega\) are the forcing amplitude and frequency, respectively.

  • Modulated by any function (constant, linear, or nonlinear), for instance to model parametric forcing (e.g., \(x_i(t) F \cos(\omega t)\)).

Internal resonance relations among oscillators can be specified in a second step by expressing the \(\omega_i\) as a function of a reference frequency. Detuning can also be introduced during this step.

Overview

The package associated with the OSCILATE project is called oscilate. The package organisation is depicted below

oscilate
│   sympy_functions.py
│   __init__.py
│   __version__.py
│
└───MMS
        dyn_sys.py
        mms.py
        mms_oscillator.py
        mms_complex.py
        steady_state.py
        visualisation.py
        __init__.py

It contains two modules:

• The MMS module is the MMS solver. . It is divided into six sub-modules:

  • The oscilate.MMS.dyn_sys sub-module defines the dynamical system of interest,

  • The oscilate.MMS.mms, oscilate.MMS.mms_oscillator and oscilate.MMS.mms_complex sub-modules apply the MMS to the dynamical system,

  • The oscilate.MMS.steady_state sub-module allows for a steady state analysis,

  • The oscilate.MMS.visualisation sub-module contains numerical evaluation and plotting functions,

• The sympy_functions module contains additional functions that are not directly related to the MMS but which are used in MMS.

print(vlatex(the_expr))

Documentation

A full documentation is available.

Citation

Please cite this package when using it. See the Citation section of the documentation for details. A regular entry and a LaTeX/BibTeX users entry are given.

Installation guide

To install the oscilate package, refer to the Installation guide section of the documentation.

Disclaimer

This code is provided as-is and has been tested on a limited number of nonlinear systems. Other test cases might trigger bugs or unexpected behavior that I am not yet aware of. If you encounter any issues, find a bug, or have suggestions for improvements, please feel free to:

• Open an issue on the GitHub repository (if applicable).

• Propose a solution.

• Contact me directly at vincent.mahe@ec-nantes.fr.

Your feedback is highly appreciated.

Vincent Mahé

License

This project is licensed under the Apache License 2.0 – see the LICENSE file for details.