Subharmonic response of a Duffing oscillator
--------------------------------------------

MMS example on a Duffing oscillator subject to hard forcing triggering a subharmonic response. 
This configuration was studied by Nayfeh and Mook :cite:`nayfehNonlinearOscillations1995`, sections 4.1.2 and 4.1.4.

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

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

   Illustration of a forced Duffing oscillator. Subharmonic oscillations (of order 3) are triggered provided :math:`F` is large and :math:`\omega_0 \approx 1/3 \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 3 times the oscillator's frequency is sought so the frequency is set to

.. math::
    \omega = 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_subharmonic.py
   :language: python
   :linenos:


Plot outputs
^^^^^^^^^^^^
The plot outputs shown below are generated from the code above. 

Frequency response curve
~~~~~~~~~~~~~~~~~~~~~~~~

The two figures below display the amplitude and phase responses (blue) of the :math:`1/3^{\text{rd}}` harmonic of the Duffing oscillator as a function of the excitation frequency. The backbone curve is shown in grey and the linear frequency multiplied by 3 in black.

.. figure:: /_static/examples/Duffing_subH_plots/FRC_a.svg
   :alt: Frequency response curve - amplitude
   :width: 80%
   :align: center

   Frequency response curve of the Duffing oscillator in subharmonic regime of order 3 (amplitude).

.. figure:: /_static/examples/Duffing_subH_plots/FRC_beta.svg
   :alt: Frequency response curve - phase
   :width: 80%
   :align: center

   Frequency response curve of the Duffing oscillator in subharmonic regime of order 3 (phase).

Amplitude response curve
~~~~~~~~~~~~~~~~~~~~~~~~

The two figures below display the amplitude and phase responses (blue) of the :math:`1/3^{\text{rd}}` harmonic of the Duffing oscillator as a function of the excitation amplitude.

.. figure:: /_static/examples/Duffing_subH_plots/ARC_a.svg
   :alt: Amplitude response curve - amplitude
   :width: 80%
   :align: center

   Amplitude response curve of the Duffing oscillator in subharmonic regime of order 3 (amplitude).

.. figure:: /_static/examples/Duffing_subH_plots/ARC_beta.svg
   :alt: Amplitude response curve - phase
   :width: 80%
   :align: center

   Amplitude-response curve of the Duffing oscillator in subharmonic regime of order 3 (phase).