Example 8: Superharmonic response of a Duffing oscillator
---------------------------------------------------------

MMS example on a Duffing oscillator subject to hard forcing triggering a superharmonic response. 
This configuration was studied by Nayfeh and Mook in *Nonlinear Oscillations* (1995), sections 4.1.2 and 4.1.3.

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

.. figure:: /_static/Duffing_oscillator.svg
   :alt: Nonlinear system.
   :width: 70%
   :align: center

   Illustration of a forced Duffing oscillator. Superharmonic oscillations (of order 3) are triggered provided :math:`F` is large and :math:`\omega_0 \approx 3 \omega`.

The system's equation is

.. math::
    \ddot{x} + c \dot{x} + \gamma \dot{x}^{3} + \omega_{0}^{2} x = F \cos(\omega t),

where 

- :math:`x` is the oscillator's coordinate,
- :math:`t` is the time,
- :math:`\dot{(\bullet)} = \mathrm{d}(\bullet)/\mathrm{d}t` is a time derivative,
- :math:`c` is the linear viscous damping coefficient,
- :math:`\omega_0` is the oscillator's natural frequency,
- :math:`\gamma` is the nonlinear coefficient,
- :math:`F` is the forcing amplitude,
- :math:`\omega` is the forcing frequency.

A parametric response around :math:`1/3` times the oscillator's frequency is sought so the frequency is set to

.. math::
    \omega = \frac{1}{3}\omega_0 + \epsilon \sigma

where

- :math:`\epsilon` is a small parameter involved in the MMS,
- :math:`\sigma` is the detuning.

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

- :math:`c = \epsilon \tilde{c}` indicates that damping is weak,
- :math:`\gamma = \epsilon \tilde{\gamma}` indicates that nonlinear damping is weak.

Note that the forcing is not scaled, i.e. :math:`F` appears at leading order. This is called hard forcing. 

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 forced response and the backbone curve,

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