:orphan:

Examples
========

This gallery walks through every public feature of ``mathematicskit.ode_dynamics``:
fixed-point stability classification, phase portraits, the logistic
map's route to chaos, canonical bifurcation normal forms, the Van der
Pol limit cycle, Poincare sections of the driven Duffing oscillator,
Lyapunov functions and Bendixson's criterion, population, epidemic,
neuron, and chemical models, Kuramoto synchronization, and the Lorenz and
Rossler attractors.

See also the narrative tutorials:

- :doc:`/tutorials/chaos_and_fractals`
- :doc:`/tutorials/eigenvalues_everywhere`

Each script in this gallery is self-contained and can be run directly with
``python examples/ode_dynamics/<section>/<script>.py``.

Sections
--------

- **stability** -- Jacobian linearization, the trace-determinant
  classification of node/saddle/spiral/center fixed points, and
  quadratic Lyapunov functions.
- **phase_portrait** -- linear and nonlinear planar flows, and vector-field
  quiver plots.
- **logistic_map** -- the period-doubling bifurcation cascade and an
  estimate of the Feigenbaum constant.
- **bifurcations** -- saddle-node, pitchfork, and Hopf normal forms.
- **limit_cycles** -- the Van der Pol oscillator's amplitude-independent
  limit cycle, Bendixson's negative criterion, and the Brusselator's Hopf
  bifurcation.
- **poincare** -- stroboscopic Poincare sections of the periodically
  driven Duffing oscillator.
- **population** -- Verhulst's logistic growth and Lotka-Volterra
  predator-prey cycles.
- **epidemics** -- the Kermack-McKendrick SIR model's threshold, peak, and
  final size.
- **excitable** -- the FitzHugh-Nagumo neuron's excitability and spiking.
- **synchronization** -- the Kuramoto model's synchronization transition.
- **chaotic_flows** -- the Lorenz attractor and its butterfly effect, and
  the Rossler attractor and its return map.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. thumbnail-parent-div-close

.. raw:: html

    </div>

Bifurcation normal forms
------------------------

Saddle-node, pitchfork, and Hopf bifurcations.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Plots the closed-form fixed points (or limit-cycle radius) of the three canonical bifurcation normal forms as a function of the control parameter r, showing the characteristic collision-and-annihilation, splitting, and birth-of-a-limit-cycle shapes.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/bifurcations/images/thumb/sphx_glr_plot_01_normal_forms_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/bifurcations/plot_01_normal_forms`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Saddle-node, pitchfork, and Hopf bifurcation diagrams</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Chaotic flows
-------------

Lorenz's butterfly attractor and its sensitive dependence on initial
conditions, and the Rossler attractor, a minimal three-dimensional
chaotic flow.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Lorenz&#x27;s 1963 convection model \dot x = \sigma(y - x), \dot y = x(\rho - z) - y, \dot z = xy - \beta z is fully deterministic, yet for \sigma = 10, \rho = 28, \beta = 8/3 its orbits wind forever around a butterfly-shaped strange attractor. This script repeats Lorenz&#x27;s accidental experiment: restart a run from a state rounded to three decimal places, and watch the two forecasts agree for a while, then diverge completely.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/chaotic_flows/images/thumb/sphx_glr_plot_01_lorenz_butterfly_effect_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/chaotic_flows/plot_01_lorenz_butterfly_effect`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lorenz's butterfly effect: deterministic chaos</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Rossler&#x27;s flow has a single nonlinear term, yet for a = b = 0.2, c = 5.7 it settles onto a chaotic attractor: orbits spiral outward near the plane z \approx 0, are lifted and folded back when x exceeds c, and never exactly repeat. Plotting each loop&#x27;s maximum of x against the previous one gives a Lorenz-style return map, a thin single-humped curve.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/chaotic_flows/images/thumb/sphx_glr_plot_02_rossler_attractor_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/chaotic_flows/plot_02_rossler_attractor`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Rossler attractor</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Epidemic models
---------------

The Kermack-McKendrick SIR model, its threshold, and its final size.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="An epidemic grows only if R_0 S_0 &gt; 1, peaks when the susceptible fraction falls to 1/R_0, and burns out before infecting everyone. The simulated peak and final size agree with the closed-form Kermack-McKendrick relations.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/epidemics/images/thumb/sphx_glr_plot_01_sir_model_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/epidemics/plot_01_sir_model`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Kermack-McKendrick SIR model</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Excitable systems
-----------------

The FitzHugh-Nagumo neuron: excitability and periodic spiking.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="With no injected current the model neuron rests at a stable fixed point. Between the two Hopf currents that fixed point is unstable, and the neuron fires repetitively on a relaxation limit cycle that runs along the outer branches of the cubic v-nullcline.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/excitable/images/thumb/sphx_glr_plot_01_fitzhugh_nagumo_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/excitable/plot_01_fitzhugh_nagumo`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">FitzHugh-Nagumo: excitability and spiking</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Limit cycles
------------

The Van der Pol oscillator's amplitude-independent limit cycle.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Trajectories starting both inside and outside the eventual limit cycle converge onto the same closed orbit -- the defining property that distinguishes a limit cycle from a linear center (whose orbit amplitude is set entirely by the initial condition).">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/limit_cycles/images/thumb/sphx_glr_plot_01_van_der_pol_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/limit_cycles/plot_01_van_der_pol`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Van der Pol limit cycle</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="If the divergence of a planar vector field keeps one sign on a simply connected region, no closed orbit fits inside it. A damped oscillator passes the test everywhere; the Van der Pol field fails it because its divergence \mu(1 - x^2) changes sign at |x| = 1 -- and indeed its limit cycle must cross those lines.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/limit_cycles/images/thumb/sphx_glr_plot_02_bendixson_criterion_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/limit_cycles/plot_02_bendixson_criterion`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Bendixson's negative criterion</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The Brusselator&#x27;s fixed point (a, b/a) is stable until b crosses b_c = 1 + a^2, where the trace of its Jacobian changes sign and a limit cycle is born: the model chemical reaction starts to oscillate.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/limit_cycles/images/thumb/sphx_glr_plot_03_brusselator_hopf_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/limit_cycles/plot_03_brusselator_hopf`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Brusselator's Hopf bifurcation</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

The logistic map
----------------

The period-doubling route to chaos and the Feigenbaum constant.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="As the parameter r grows, the logistic map x \mapsto rx(1-x) doubles its period again and again -- 1, 2, 4, 8, ... -- at parameter values whose spacing shrinks by a constant factor, Feigenbaum&#x27;s \delta \approx 4.6692. Universality means the same \delta appears for any smooth one-humped map. This script measures \delta for the logistic map and for the unrelated sine map x \mapsto r\sin(\pi x) and shows both sequences of ratios converging to the one constant.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/logistic_map/images/thumb/sphx_glr_plot_01_bifurcation_cascade_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/logistic_map/plot_01_bifurcation_cascade`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Feigenbaum's universal constant in the period-doubling cascade</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Phase portraits
---------------

Linear and nonlinear planar flows and their vector fields.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="The nonlinear pendulum \ddot\theta = -\sin\theta has no elementary closed-form solution, but Poincaré&#x27;s qualitative approach does not need one. Locate the fixed points, classify each from its linearization, and sketch the vector field: the global picture of librating orbits around the centers, rotating orbits above and below, and separatrices joining the saddles follows without solving anything. Integrated trajectories are overlaid only to confirm the prediction.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/phase_portrait/images/thumb/sphx_glr_plot_01_vector_fields_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/phase_portrait/plot_01_vector_fields`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Poincaré's qualitative method: the pendulum's phase portrait</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Poincare sections
-----------------

Stroboscopic Poincare sections of the periodically driven Duffing
oscillator.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Sampling the Duffing oscillator&#x27;s state once per drive period (stroboscopically) collapses its effectively-3D state space (position, velocity, and drive phase) back to a 2D picture: a periodic response shows up as one (or a few) fixed points, while chaotic parameters produce a scattered, fractal-looking set of points.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/poincare/images/thumb/sphx_glr_plot_01_duffing_poincare_section_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/poincare/plot_01_duffing_poincare_section`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Poincare sections of the driven Duffing oscillator</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Population models
-----------------

Verhulst's logistic growth and the Lotka-Volterra predator-prey cycles.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Integrating \dot N = rN(1 - N/K) numerically reproduces Verhulst&#x27;s closed-form sigmoid exactly: growth looks exponential at first, is fastest at N = K/2, and saturates at the carrying capacity K from any positive starting population.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/population/images/thumb/sphx_glr_plot_01_verhulst_logistic_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/population/plot_01_verhulst_logistic`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Verhulst's logistic growth</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Predator and prey populations rise and fall periodically, out of phase. Every orbit is closed because the Lotka-Volterra invariant V = \delta x - \gamma\ln x + \beta y - \alpha\ln y is conserved: orbits are its level sets around the coexistence fixed point.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/population/images/thumb/sphx_glr_plot_02_lotka_volterra_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/population/plot_02_lotka_volterra`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lotka-Volterra predator-prey cycles</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Fixed-point stability
----------------------

Jacobian linearization and the trace-determinant classification of
node/saddle/spiral/center fixed points.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Poincaré&#x27;s qualitative theory sorts a planar fixed point into one of four types -- node, saddle, spiral (focus), or center -- using only the eigenvalues of the linearized flow, without solving the equations. The eigenvalues depend only on the Jacobian&#x27;s trace \tau and determinant \Delta, so every type occupies its own region of the (\tau, \Delta) plane. This script classifies six canonical matrices, draws their phase portraits, and places each one on Poincaré&#x27;s trace-determinant diagram.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/stability/images/thumb/sphx_glr_plot_01_classification_zoo_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/stability/plot_01_classification_zoo`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Poincaré's classification of fixed points: nodes, saddles, spirals, centers</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="For a stable linear system \dot x = Ax, solving the Lyapunov equation A^T P + PA = -Q gives an energy-like function V(x) = x^T P x whose level sets are ellipses that every trajectory crosses inward -- a proof of stability that never solves the ODE itself.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/stability/images/thumb/sphx_glr_plot_02_lyapunov_function_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/stability/plot_02_lyapunov_function`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lyapunov's direct method</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>

Synchronization
---------------

The Kuramoto model's transition from incoherence to collective synchrony.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="A thousand oscillators with Lorentzian-distributed natural frequencies stay incoherent for weak coupling. Above the critical coupling K_c = 2\gamma a synchronized cluster forms, and the steady coherence follows Kuramoto&#x27;s exact result r = \sqrt{1 - K_c/K}.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/synchronization/images/thumb/sphx_glr_plot_01_kuramoto_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/synchronization/plot_01_kuramoto`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Kuramoto synchronization transition</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:
   :includehidden:


   /api/gallery/ode_dynamics/bifurcations/index.rst
   /api/gallery/ode_dynamics/chaotic_flows/index.rst
   /api/gallery/ode_dynamics/epidemics/index.rst
   /api/gallery/ode_dynamics/excitable/index.rst
   /api/gallery/ode_dynamics/limit_cycles/index.rst
   /api/gallery/ode_dynamics/logistic_map/index.rst
   /api/gallery/ode_dynamics/phase_portrait/index.rst
   /api/gallery/ode_dynamics/poincare/index.rst
   /api/gallery/ode_dynamics/population/index.rst
   /api/gallery/ode_dynamics/stability/index.rst
   /api/gallery/ode_dynamics/synchronization/index.rst



.. only:: html

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

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