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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_introductions/vanderpol.py"
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.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_introductions_vanderpol.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_introductions_vanderpol.py:


The Van der Pol Oscillator
==========================

Here, we will introduce the Van der Pol model, a widely studied non-linear oscillator model [1]_.
In its two-dimensional form, the Van der Pol oscillator is governed by the following non-linear ODEs:

.. math::
        \dot x &= z, 

        \dot z &= \mu (1-x^2) z - x,

with damping constant :math:`\mu`. Depending on the latter parameter, the periodic solutions of the Van der Pol oscillator change,
with simple harmonic oscillations for :math:`\mu = 0` and non-harmonic limit cycle oscillations for :math:`\mu > 0`.
In this gallery, we will simulate and visualize the model dynamics for :math:`\mu = 0` and :math:`\mu = 0`.

**References**

.. [1] T. Kanamaru (2007) *Van der Pol oscillator* Scholarpedia 2(1): 2202. DOI: 10.4249/scholarpedia.2202.

.. GENERATED FROM PYTHON SOURCE LINES 23-33

Step 1: Numerical simulation of the model dynamics for :math:`\mu = 0`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

First, we will load a :code:`CircuitTemplate` instance via the path to the model definition.
We will the use the :code:`CircuitTemplate.run` method to solve the initial value problem of the above defined
differential equations for a time interval from 0 to the given simulation time.
This solution will be calculated numerically by a differential equation solver in the backend, starting with a
defined step-size. Here, we use the default backend and a Runge-Kutta 2(3) solver.
Check out the arguments of the :code:`CircuitTemplate.run()` method for a detailed explanation of the
arguments that you can use to adjust this numerical procedure.

.. GENERATED FROM PYTHON SOURCE LINES 34-56

.. code-block:: Python


    from pyrates import CircuitTemplate, clear

    # define simulation time and input start and stop
    T = 100.0
    step_size = 1e-2

    # load model template
    model = CircuitTemplate.from_yaml("model_templates.oscillators.vanderpol.vdp")

    # set mu to 0
    model.update_var({'p/vdp_op/mu': 0.0})
    results = model.run(step_size=step_size, simulation_time=T, outputs={'x': 'p/vdp_op/x'}, solver='scipy', method='RK23')

    # plot resulting phases
    import matplotlib.pyplot as plt
    plt.plot(results)
    plt.show()

    # clear results
    clear(model)


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Step 2: Numerical simulation of the model dynamics for :math:`\mu = 5`
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

As expected, we found harmonical oscillations for :math:`\mu = 0`.
Now, we will repeat the procedure for :math:`\mu = 5`.

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.. code-block:: Python


    # load model template
    model = CircuitTemplate.from_yaml("model_templates.oscillators.vanderpol.vdp")

    # set mu to 0
    model.update_var({'p/vdp_op/mu': 5.0})
    results = model.run(step_size=step_size, simulation_time=T, outputs={'x': 'p/vdp_op/x'}, solver='scipy', method='RK23')

    # plot resulting phases
    plt.plot(results)
    plt.show()


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We find a much less harmonical waveform of the oscillations. For investigations of the Van der Pol oscillator driven
by a periodic input signal, see the `entrainment example <https://pyrates.readthedocs.io/en/latest/auto_analysis/entrainment.html>`_
in the use examples section `Model analysis`.


.. _sphx_glr_download_auto_introductions_vanderpol.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: vanderpol.ipynb <vanderpol.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: vanderpol.py <vanderpol.py>`


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 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_