.. _example_VdP:

Van der Pol oscillator
----------------------

MMS example on a Van der Pol oscillator. 
This configuration is very close from that studied by Nayfeh and Mook :cite:`nayfehNonlinearOscillations1995`, section 3.3.4.


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

.. figure:: /_static/examples/VdP.svg
   :alt: Nonlinear system.
   :width: 50%
   :align: center

   Illustration of a Van der Pol oscillator. 

The system's equation is

.. math::
    \ddot{x} + \omega_{0}^{2} x + \mu \left(x^{2} - 1\right) \dot{x} = 0,

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:`\omega_0` is the oscillator's natural frequency,
- :math:`\mu` is the linear and nonlinear damping coefficient.

A parametric response around the oscillator's frequency 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 reference frequency :math:`\omega_0`.

The parameter :math:`\mu = \epsilon \tilde{\mu}` is then scaled such that 

.. math::
    \mu = \epsilon \tilde{\mu}

to indicate that linear and nonlinear dampings are weak. 

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

- Construct the dynamical system.
- Apply the MMS to the system,
- Solve the modulation equations (yields the transient response),
- Evaluate the MMS results at steady state,
- Solve the steady state modulation equations (yields the limit cycle),
- Evaluate the symbolic expressions for given numerical parameters and plot the limit cycle and two trajectories in the phase portrait.

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

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


Phase portrait
~~~~~~~~~~~~~~
The figure below displays the phase portrait of the Van der Pol oscillator. Three different trajectories are shown: the limit cycle (black), an external one (red) and an internal one (blue). The limit cycle is stable, such that the trajectories are converging towards it.

.. figure:: /_static/examples/VdP_plots/phase_portrait.svg
   :alt: Phase portrait
   :width: 80%
   :align: center

   Phase portrait of the Van der Pol oscillator.

Transient time signal
~~~~~~~~~~~~~~~~~~~~~
The figure below displays the time signal associated to the transient, internal trajectory shown in the phase portrait of the Van der Pol oscillator. The horizontal black lines are associated to the amplitude on the limit cycle.  

.. figure:: /_static/examples/VdP_plots/time_signal.svg
   :alt: Time signal
   :width: 80%
   :align: center

   Transient response of the Van der Pol oscillator.