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.. "auto_introductions/stuartlandau.py"
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.. only:: html

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

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

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

.. _sphx_glr_auto_introductions_stuartlandau.py:


The Stuart-Landau Oscillator
============================

Here, we will introduce the Stuart Landau equations, a model of non-linear oscillating systems near a Hopf bifurcation [1]_.
In cartesian coordinates, the evolution equations of the Stuart Landau oscillator can be written as:

.. math::
        \dot x &= \omega*z + x*(1-z^2-x^2), 

        \dot z &= -\omega*x + z*(1-z^2-x^2),

with intrinsic angular frequency :math:`\omega`. It can be used as the generating model of a harmonic oscillation with
frequency :math:`\omega` and is thus useful to incorporate periodic forcing into dynamical systems models.
Below, we provide an example where we apply periodic forcing to a Van der Pol oscillator (for a detailed description of
the latter, see the corresponding
`use example <https://pyrates.readthedocs.io/en/latest/auto_introductions/vanderpol.html>`_ in the use example section
`Model introductions`).

**References**

.. [1] L.D. Landau (1944) *On the problem of turbulence* Conference paper: Dokl. Akad. Nauk UDSSR 44: 311.

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Step 1: The Stuart-Landau oscillator dynamics
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Let us first have a look at the signal that the Stuart-Landau oscillator generates.
To this end, we will load a :code:`CircuitTemplate` instance via the path to the Stuart-Landau 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.

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.. code-block:: Python


    from pyrates import CircuitTemplate, clear
    import numpy as np

    # 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.stuartlandau.sl")

    # define omega
    omega = 2*np.pi/12.0
    model.update_var({'p/sl_op/omega': omega})
    results = model.run(step_size=step_size, simulation_time=T, outputs={'x': 'p/sl_op/x'}, solver='scipy', method='RK23')

    # visualize model dynamics
    import matplotlib.pyplot as plt
    plt.plot(results)
    plt.show()

    # clear results
    clear(model)


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Step 2: Dynamics of the Van der Pol oscillator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

In a second step, lets examine the dynamics of the autonomous Van der Pol oscillator system.

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.. code-block:: Python


    # simulate model dynamics
    from pyrates import integrate
    results = integrate("model_templates.oscillators.vanderpol.vdp",
                        step_size=step_size, simulation_time=T, outputs={'x': 'p/vdp_op/x'},
                        solver='scipy', method='RK23', clear=True)

    # visualize model dynamics
    plt.plot(results)
    plt.show()


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Step 3: Periodic forcing of the Van der Pol oscillator
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Now in a final step, lets see how the Van der Pol model dynamics change in response to periodic forcing
as generated by the Start Landau equations.

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.. code-block:: Python


    from pyrates import NodeTemplate

    # define nodes
    vpo = NodeTemplate.from_yaml("model_templates.oscillators.vanderpol.vdp_pop")
    sl = NodeTemplate.from_yaml("model_templates.oscillators.stuartlandau.sl_pop")

    # define circuit
    model = CircuitTemplate(name='vpo_forced', nodes={'vpo': vpo, 'sl': sl},
                            edges=[('sl/sl_op/x', 'vpo/vdp_op/inp', None, {'weight': 10.0})])

    # set omega to the value defined above
    model.update_var({'sl/sl_op/omega': omega})
    results = model.run(step_size=step_size, simulation_time=T, outputs={'x': 'vpo/vdp_op/x'}, solver='scipy', method='RK23')

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


.. _sphx_glr_download_auto_introductions_stuartlandau.py:

.. only:: html

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

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

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

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

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


.. only:: html

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

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