Example 3: Coupled Duffings in 1:1 internal resonance
-----------------------------------------------------

MMS example on coupled Duffing oscillators in 1:1 internal resonance subject to harmonic forcing. 

System description
^^^^^^^^^^^^^^^^^^

.. figure:: /_static/Duffing_oscillators_11.svg
   :alt: Nonlinear system.
   :width: 90%
   :align: center

   Illustration of two forced, nonlinearly coupled Duffing oscillators in 1:1 internal resonance.

The system's equations are

.. math::
    
    \begin{cases}
        \ddot{x}_{0} + \omega_{0}^{2} x_{0} + c_{0} \dot{x}_{0} + \gamma_{0} x_{0}^{3} + \gamma_{01} x_{0} x_{1}^{2} & = F \cos(\omega t), \\
        \ddot{x}_{1} + \omega_{0}^{2} x_{1} + c_{1} \dot{x}_{1} + \gamma_{1} x_{1}^{3} + \gamma_{01} x_{0}^{2} x_{1} & = F \cos(\omega t),
    \end{cases}
    

where 

- :math:`x_0,\; x_1` are the oscillators' coordinates,
- :math:`t` is the time,
- :math:`\dot{(\bullet)} = \mathrm{d}(\bullet)/\mathrm{d}t` is a time derivative,
- :math:`c_0,\; c_1` are the linear viscous damping coefficients,
- :math:`\omega_0` is the oscillators' natural frequency,
- :math:`\gamma_0,\; \gamma_1,\; \gamma_{01}` are nonlinear coefficients,
- :math:`F` is the forcing amplitude,
- :math:`\omega` is the forcing frequency.

A response around :math:`\omega_0` is sought so the frequency is set to

.. math::
    \omega = \omega_0 + \epsilon \sigma

where

- :math:`\epsilon` is a small parameter involved in the MMS,
- :math:`\sigma` is the detuning wrt the oscillators' frequency :math:`\omega_0`.

The parameters are then scaled to indicate how weak they are:

- :math:`c_i = \epsilon \tilde{c}_i` indicates that damping is weak,
- :math:`F = \epsilon \tilde{F}` indicates that forcing is weak,
- :math:`\gamma_i = \epsilon \tilde{\gamma}_i, \; i\in\{0, 1, 01\}` indicates that nonlinearities are weak.

Code description
^^^^^^^^^^^^^^^^
The script below allows to

- Construct the dynamical system.
- Apply the MMS to the system,
- Evaluate the MMS results at steady state,
- Compute the backbone curve and forced response when only oscillator 1 responds,
- Evaluate the stability of the forced solution,
- Compute the coupled-mode backbone curve. 

.. literalinclude:: ../../../examples/Duffing_11_direct.py
   :language: python
   :linenos: