.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/chaos/chaos_metrics/plot_phase_volume_expansion.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_chaos_chaos_metrics_plot_phase_volume_expansion.py: Phase-Space Volume Contraction (Liouville's Theorem) ======================================================= The flow studied here is the Lorenz system, .. math:: \dot{x} = \sigma (y - x), \qquad \dot{y} = x (\rho - z) - y, \qquad \dot{z} = x y - \beta z. Liouville's theorem says a Hamiltonian flow preserves phase-space volume :math:`V`; a *dissipative* flow like Lorenz's instead contracts it, according to .. math:: \frac{d(\log V)}{dt} = \operatorname{tr} J(x(t)), where :math:`J` is the flow's Jacobian. For Lorenz, :math:`\operatorname{tr} J = -\sigma - 1 - \beta` everywhere (a constant, independent of the state), so the contraction rate is exactly :math:`-(\sigma + 1 + \beta)`. Rather than taking that on faith from a formula, watch it happen: seed a small ball of nearby initial conditions and co-evolve every point independently -- the ball visibly flattens from a sphere into a thin, wispy sheet draped over the attractor, which is both what "volume contracts" looks like and, not coincidentally, why the attractor it collapses onto is a lower-dimensional (here fractal) object rather than filling space. .. GENERATED FROM PYTHON SOURCE LINES 30-39 .. code-block:: Python import matplotlib.pyplot as plt from physicskit.chaos.systems.continuous import Lorenz from physicskit.chaos.utils.metrics import phase_volume_expansion from physicskit.chaos.visualizers import animate_phase_volume_contraction system = Lorenz(sigma=10.0, rho=28.0, beta=8.0 / 3.0) .. GENERATED FROM PYTHON SOURCE LINES 40-46 Animation: a ball of initial conditions, collapsing ------------------------------------------------------------ 800 points, seeded in a small ball around a point already on the attractor, each integrated forward independently: watch the ball stretch into a comet-like streak within the first couple of time units, then thin out into a sparse sheet tracing both lobes of the Lorenz butterfly. .. GENERATED FROM PYTHON SOURCE LINES 46-50 .. code-block:: Python anim = animate_phase_volume_contraction(system, n_points=800, t_max=8.0, n_frames=150) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 51-55 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("phase_volume_contraction_animation.gif", writer="pillow", fps=20) .. GENERATED FROM PYTHON SOURCE LINES 57-65 Quantifying it: the exact contraction rate ----------------------------------------------- :func:`physicskit.chaos.utils.metrics.phase_volume_expansion` tracks the cumulative log phase-space volume ``log(V(t) / V(0))`` of an infinitesimal ball co-evolved with a *single* reference trajectory (rather than the many finite-sized points animated above). Because the Lorenz system's divergence is the same constant everywhere, this curve is exactly a straight line, at exactly the analytically predicted slope. .. GENERATED FROM PYTHON SOURCE LINES 65-79 .. code-block:: Python t, states = system.trajectory(n_steps=2000, dt=0.01) log_volume = phase_volume_expansion(system, t, states) expected_rate = -(system.sigma + 1.0 + system.beta) fig, ax = plt.subplots(figsize=(7, 5)) ax.plot(t, log_volume, label="measured") ax.plot(t, expected_rate * t, "--", color="gray", label=f"expected slope = {expected_rate:.3f}") ax.set_xlabel("t") ax.set_ylabel(r"$\log(V(t)/V(0))$") ax.set_title("Lorenz system: phase-space volume contracts at a constant rate") ax.legend() plt.show() .. image-sg:: /api/gallery/chaos/chaos_metrics/images/sphx_glr_plot_phase_volume_expansion_002.png :alt: Lorenz system: phase-space volume contracts at a constant rate :srcset: /api/gallery/chaos/chaos_metrics/images/sphx_glr_plot_phase_volume_expansion_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 10.172 seconds) .. _sphx_glr_download_api_gallery_chaos_chaos_metrics_plot_phase_volume_expansion.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_phase_volume_expansion.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_phase_volume_expansion.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_phase_volume_expansion.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_