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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/09_tds_control_manual/example_2_06.py"
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.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_09_tds_control_manual_example_2_06.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_09_tds_control_manual_example_2_06.py:


Example 2.6 - Stability analysis of neutral DDE with multiple delays
====================================================================

Example 2.6 from :cite:`appeltans2023analysis` Section 2.3.2 demonstrates that
the exponential stability of neutral systems might be fragile with respect to
small delay perturbations.

Consider the following neutral delay differential equation (NDDE):

.. math::

    \dot{x}(t) = \frac{1}{4} x(t) - \frac{1}{3} x(t-\tau_1) + 
        \frac{3}{4} \dot{x}(t-\tau_1) - \frac{1}{2} \dot{x}(t-\tau_2),

with :math:`\tau_1 = 1` and :math:`\tau_2 = 2`. The associated delay
differential equation is

.. math::

    x(t) - \frac{3}{4}x(t-\tau_1) + \frac{1}{2}x(t-\tau_2) = 0.

.. GENERATED FROM PYTHON SOURCE LINES 27-50



.. image-sg:: /auto_examples/09_tds_control_manual/images/sphx_glr_example_2_06_001.png
   :alt: $\tau_2=2$, $\tau_2=2.05$, $\tau_2=2.005$
   :srcset: /auto_examples/09_tds_control_manual/images/sphx_glr_example_2_06_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np
    import tdcpy
    import tdcpy.plot

    A0 = np.array([[0.25]])
    A1 = np.array([[-1./3]])

    H1 = np.array([[-0.75]])
    H2 = np.array([[0.5]])

    r = [-0.9, 0.2, -500, 500]

    fig, axes = plt.subplots(1, 3, sharex=True, sharey=True)

    for delay, ax in zip([2, 2.05, 2.005], axes):
        ndde = tdcpy.NDDE(A=[A0, A1], hA=[0, 1], H=[H1, H2], hH=[1, delay])
        cr, info = tdcpy.roots(ndde, r=r, max_size_evp=1500)
        tdcpy.plot.eigen_plot(cr, ax=ax)
        ax.set_title(rf"$\tau_2={delay}$")

    plt.show()


.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 36.668 seconds)


.. _sphx_glr_download_auto_examples_09_tds_control_manual_example_2_06.py:

.. only:: html

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      :download:`Download Python source code: example_2_06.py <example_2_06.py>`

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.. only:: html

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

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