Parametrically excited Rayleigh oscillator
------------------------------------------

MMS example on a Rayleigh oscillator subject to parametric forcing. 
This configuration was studied by Nayfeh and Mook :cite:`nayfehNonlinearOscillations1995`, section 5.7.2. Note that the Rayleigh oscillator is quite similar to the Van der Pol oscillator as they are both associated to damping nonlinearities.

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

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

   Illustration of a parametrically forced Rayleigh oscillator through the time-varying stiffness :math:`\omega_0^2 + 2 F \cos(\omega t)`.

The system's equation is

.. math::
    \ddot{x} + c \dot{x} + \gamma \dot{x}^{3} + \omega_{0}^{2} x = -2x 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 damping coefficient,
- :math:`F` is the forcing amplitude,
- :math:`\omega` is the forcing frequency.

A parametric response around twice the oscillator's frequency is sought so the frequency is set to

.. math::
    \omega = 2\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:`F = \epsilon \tilde{F}` indicates that forcing is weak,
- :math:`\gamma = \epsilon \tilde{\gamma}` indicates that nonlinear damping is 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 forced response and the backbone curve,
- Evaluate the stability of the computed forced solution.

.. literalinclude:: ../../../examples/Rayleigh_parametric.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^{\text{st}}` harmonic of the Duffing oscillator as a function of the excitation frequency.

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

   Frequency response curve of the Rayleigh oscillator (amplitude).

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

   Frequency response curve of the Rayleigh oscillator (phase).

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

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

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

   Amplitude response curve of the Rayleigh oscillator (amplitude).

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

   Amplitude-response curve of the Rayleigh oscillator (phase).