.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/05_kuramoto_sync.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_auto_examples_05_kuramoto_sync.py: Kuramoto network: synchronization transition =============================================== Sweeps the global coupling strength ``G`` and tracks the top two Lyapunov exponents of a small heterogeneous-frequency Kuramoto network. At ``G=0`` both are exactly 0 (decoupled oscillators: ``dtheta/dt = omega`` is state-independent, so the step map's Jacobian is exactly the identity -- see ``tests/test_network.py::test_kuramoto_zero_coupling_gives_exactly_zero_spectrum``). The Kuramoto model has an exact continuous symmetry (``theta_i -> theta_i + c`` for all ``i`` leaves the dynamics unchanged), which guarantees one Lyapunov exponent stays *exactly* 0 once phase-locked -- it never goes negative. What actually signals synchronization is ``lambda_2``: below threshold both leading exponents are small and positive (weakly unstable/incoherent regime); once the network locks, ``lambda_1`` snaps to machine-precision 0 (the marginal rotation mode) and ``lambda_2`` turns negative, growing more negative as coupling strengthens the lock. This is exactly what the run below shows -- not a hypothesis, it's what lyapax actually produces for this network. Note: this sweep calls ``lyapunov_spectrum`` once per ``G`` value in a plain Python loop; see ``examples/11_vmap_parameter_sweep.py`` for the same sweep done as a single batched ``jax.vmap`` call instead, and ``examples/07_speed_and_accuracy.py`` for what the per-call cost is. **The system.** 6 oscillators, all-to-all coupled (``weights`` is the complete graph: 1 everywhere off the diagonal, 0 on it), with heterogeneous natural frequencies ``omega`` evenly spaced over ``[-1, 1]`` so the population would never sync on its own -- staggered initial phases start it far from the locked state. The classic Kuramoto ODE ``dtheta_i/dt = omega_i + (G/N) sum_j sin(theta_j - theta_i)`` is built here from the same machinery as :ref:`04_linear_network.py `: ``ModelSpec``/``build_jax_dfun`` compile the bare node dynamics ``"omega + c"`` (``c`` is the coupling input) into a JAX function, and ``kuramoto_coupling`` supplies the sine-coupling term as ``c``; ``network_problem`` wires the two together (plus a ``lyapax.Network`` topology and ``state0``/``dt``) into the flat step function the Lyapunov engine steps. Only ``k=2`` exponents are tracked per run (not the full spectrum of 6) since only the top two are needed to read off the transition -- see the ``k`` parameter in ``lyapunov_spectrum``'s docstring for the leading-exponents-only cost tradeoff. .. GENERATED FROM PYTHON SOURCE LINES 47-64 .. code-block:: Python import os os.environ["JAX_PLATFORMS"] = "cpu" import time import jax import jax.numpy as jnp import matplotlib.pyplot as plt import numpy as np jax.config.update("jax_enable_x64", True) from lyapax.core import lyapunov_spectrum from lyapax.coupling import kuramoto_coupling from lyapax.network import Network, network_problem from lyapax.simulator import ModelSpec, Parameter, StateVar, build_jax_dfun .. GENERATED FROM PYTHON SOURCE LINES 65-103 .. code-block:: Python n_nodes = 6 omega = jnp.linspace(-1.0, 1.0, n_nodes) weights = jnp.ones((n_nodes, n_nodes)) - jnp.eye(n_nodes) model = ModelSpec( name="kuramoto", state_variables=(StateVar("theta", default_init=0.0),), parameters=(Parameter("omega", 0.0),), cvar=("theta",), dfun_str={"theta": "omega + c"}, ) dfun = build_jax_dfun(model) network = Network(weights=weights, cvar_indices=model.cvar_indices) dt = 1e-2 state0 = jnp.linspace(0.0, 2 * jnp.pi, n_nodes, endpoint=False) G_values = np.linspace(0.0, 4.0, 13) lambda1, lambda2 = [], [] t0 = time.perf_counter() for G in G_values: params = {"omega": omega, "G": float(G)} problem = network_problem( dfun, network, kuramoto_coupling(alpha=0.0), params=params, state0=state0, dt=dt, ) result = lyapunov_spectrum( problem, n_steps=5_000, renorm_every=10, k=2, t_transient=50.0, ) lambda1.append(float(result.exponents[0])) lambda2.append(float(result.exponents[1])) elapsed = time.perf_counter() - t0 print(f"swept {len(G_values)} values of G in {elapsed:.1f}s " f"({elapsed / len(G_values):.2f}s/run)") for G, l1, l2 in zip(G_values, lambda1, lambda2): print(f" G={G:.2f} lambda1={l1:+.4f} lambda2={l2:+.4f}") .. rst-class:: sphx-glr-script-out .. code-block:: none swept 13 values of G in 5.8s (0.45s/run) G=0.00 lambda1=+0.0000 lambda2=+0.0000 G=0.33 lambda1=+0.0040 lambda2=+0.0034 G=0.67 lambda1=+0.0126 lambda2=+0.0070 G=1.00 lambda1=+0.0289 lambda2=+0.0121 G=1.33 lambda1=+0.0243 lambda2=+0.0116 G=1.67 lambda1=+0.0000 lambda2=-0.7518 G=2.00 lambda1=+0.0000 lambda2=-1.3269 G=2.33 lambda1=+0.0000 lambda2=-1.7867 G=2.67 lambda1=+0.0000 lambda2=-2.2025 G=3.00 lambda1=+0.0000 lambda2=-2.5951 G=3.33 lambda1=+0.0000 lambda2=-2.9734 G=3.67 lambda1=+0.0000 lambda2=-3.3423 G=4.00 lambda1=+0.0000 lambda2=-3.7044 .. GENERATED FROM PYTHON SOURCE LINES 104-114 .. code-block:: Python fig, ax = plt.subplots(figsize=(6, 4)) ax.plot(G_values, lambda1, "o-", label=r"$\lambda_1$ (always -> 0: symmetry mode)") ax.plot(G_values, lambda2, "o-", label=r"$\lambda_2$ (synchronization strength)") ax.axhline(0.0, color="gray", lw=0.5) ax.set_xlabel("coupling strength G") ax.set_ylabel("Lyapunov exponent") ax.set_title("Kuramoto network: synchronization transition") ax.legend() fig.tight_layout() plt.show() .. image-sg:: /auto_examples/images/sphx_glr_05_kuramoto_sync_001.png :alt: Kuramoto network: synchronization transition :srcset: /auto_examples/images/sphx_glr_05_kuramoto_sync_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 6.249 seconds) .. _sphx_glr_download_auto_examples_05_kuramoto_sync.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: 05_kuramoto_sync.ipynb <05_kuramoto_sync.ipynb>` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 05_kuramoto_sync.py <05_kuramoto_sync.py>` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: 05_kuramoto_sync.zip <05_kuramoto_sync.zip>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_