.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/08_delayed_coupling.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_08_delayed_coupling.py: Delayed coupling: two-node linear DDE network vs delay =========================================================== The simplest possible "delayed coupling" demo: two identical linear nodes coupled through each other's *delayed* state, ``x1' = gamma*x1 + G*x2(t-tau)``, ``x2' = gamma*x2 + G*x1(t-tau)``. This sweeps the delay ``tau`` itself, to show what delayed coupling does to the spectrum -- how it differs from the zero-delay case in :ref:`04_linear_network.py `. **Why this system.** Its characteristic equation is tractable: the symmetric mode (``x1=x2``) gives ``lambda_sym = gamma + W(G*tau*e^{-gamma*tau})/tau`` and the antisymmetric mode (``x1=-x2``) gives ``lambda_antisym = gamma + W(-G*tau*e^{-gamma*tau})/tau`` (Lambert W, principal branch) -- so every point on the sweep below has an independent closed-form answer, not just the single spot-check the test performs. **Delay range, deliberately narrow.** The antisymmetric branch's Lambert-W argument, ``-G*tau*e^{-gamma*tau}``, crosses the branch point at ``-1/e`` around ``tau ~ 0.46`` for the parameters below -- past that, ``W``'s principal branch stops being real/single-valued and taking ``.real`` of it no longer tracks the true dominant root (verified numerically while writing this demo: the naive formula produces a non-monotonic jump right at the crossing, an artifact of the branch point, not real dynamics). The sweep below stays at ``tau <= 0.4``, safely inside the region where the closed form is valid, so every plotted "exact" point is trustworthy. **The machinery.** Coupling between two *different* nodes' delayed states (as opposed to a single scalar equation delayed against its own past, e.g. Mackey-Glass) needs the general, per-edge ``delay_steps`` path: ``simulator.Connectivity(weights, tract_lengths, speed)`` turns a physical ``tau`` into an integer ``delay_steps`` matrix and the ring-buffer ``horizon`` needed to store enough history, exactly as ``lyapax.dde.resolve_tau_steps`` does for a single scalar delay -- then ``simulator.make_step_fn(..., delay_steps=...)`` and ``lyapax.dde.lyapunov_spectrum_dde`` do the rest: propagate the state and its ring-buffer history together, periodically re-orthonormalizing the tangent directions via QR, the same Benettin's-method idea as the non-delayed engine, just carrying delay history alongside the state. Deliberately staying low-level here: ``network_dde_problem`` (the problem-object front door used by :ref:`09_kuramoto_delayed_network.py ` and :ref:`12_public_api_overview.py `) only covers a single *uniform* delay shared by every edge, and explicitly rejects a per-edge ``delay_steps`` matrix like the one this script needs -- see that function's docstring. A per-edge delay matrix combined with a custom ``coupling_fn`` isn't wired up at all yet (only this hardcoded-linear ``delay_steps`` path is); this script is the one place that still needs the raw ``Connectivity``/``make_step_fn``/``lyapunov_spectrum_dde`` call form for that reason, not because it predates the new API. .. GENERATED FROM PYTHON SOURCE LINES 57-80 .. 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 from scipy.special import lambertw jax.config.update("jax_enable_x64", True) from lyapax.dde import constant_history_buf0, lyapunov_spectrum_dde from lyapax.simulator import ( Connectivity, ModelSpec, Parameter, StateVar, build_jax_dfun, make_step_fn, ) .. GENERATED FROM PYTHON SOURCE LINES 81-134 .. code-block:: Python gamma, G, dt = -1.0, 0.5, 1e-3 def lambda_sym(tau): return gamma + float((lambertw(G * tau * np.exp(-gamma * tau), k=0) / tau).real) def lambda_antisym(tau): return gamma + float((lambertw(-G * tau * np.exp(-gamma * tau), k=0) / tau).real) model = ModelSpec( name="linear_node", state_variables=(StateVar("x", default_init=0.0),), parameters=(Parameter("gamma", gamma),), cvar=("x",), dfun_str={"x": "gamma * x + c"}, ) dfun = build_jax_dfun(model) weights = jnp.array([[0., 1.], [1., 0.]]) params = {"gamma": gamma, "G": G} tau_values = np.linspace(0.05, 0.4, 8) lambda1, lambda2 = [], [] t0 = time.perf_counter() for tau in tau_values: conn = Connectivity( weights=np.array([[0., 1.], [1., 0.]]), tract_lengths=np.array([[0., tau], [tau, 0.]]), speed=1.0, ) delay_steps = jnp.array(conn.delay_steps(dt)) horizon = conn.horizon(dt) step_fn = make_step_fn( dfun=dfun, weights=weights, has_delays=True, horizon=horizon, n_nodes=2, cvar_indices=model.cvar_indices, dt=dt, delay_steps=delay_steps, G_default=G, coup_a=1.0, coup_b=0.0, integrator="heun", ) state0 = jnp.array([[0.3, -0.2]]) buf0 = constant_history_buf0(state0, horizon) result = lyapunov_spectrum_dde( step_fn, state0=state0, buf0=buf0, params=params, dt=dt, n_steps=20_000, k=2, renorm_every=10, t_transient=5.0, ) lambda1.append(float(result.exponents[0])) lambda2.append(float(result.exponents[1])) elapsed = time.perf_counter() - t0 print(f"swept {len(tau_values)} values of tau in {elapsed:.1f}s " f"({elapsed / len(tau_values):.2f}s/run)") for tau, l1, l2 in zip(tau_values, lambda1, lambda2): print(f" tau={tau:.3f} lambda1={l1:+.4f} (sym exact {lambda_sym(tau):+.4f}) " f"lambda2={l2:+.4f} (antisym exact {lambda_antisym(tau):+.4f})") .. rst-class:: sphx-glr-script-out .. code-block:: none swept 8 values of tau in 10.3s (1.28s/run) tau=0.050 lambda1=-0.4875 (sym exact -0.4877) lambda2=-1.5404 (antisym exact -1.5400) tau=0.100 lambda1=-0.4755 (sym exact -0.4756) lambda2=-1.5864 (antisym exact -1.5859) tau=0.150 lambda1=-0.4638 (sym exact -0.4640) lambda2=-1.6400 (antisym exact -1.6394) tau=0.200 lambda1=-0.4525 (sym exact -0.4526) lambda2=-1.7036 (antisym exact -1.7029) tau=0.250 lambda1=-0.4415 (sym exact -0.4416) lambda2=-1.7812 (antisym exact -1.7803) tau=0.300 lambda1=-0.4309 (sym exact -0.4310) lambda2=-1.8796 (antisym exact -1.8784) tau=0.350 lambda1=-0.4206 (sym exact -0.4207) lambda2=-2.0122 (antisym exact -2.0106) tau=0.400 lambda1=-0.4106 (sym exact -0.4107) lambda2=-2.2131 (antisym exact -2.2105) .. GENERATED FROM PYTHON SOURCE LINES 135-151 .. code-block:: Python tau_dense = np.linspace(0.02, 0.42, 200) sym_dense = [lambda_sym(t) for t in tau_dense] antisym_dense = [lambda_antisym(t) for t in tau_dense] fig, ax = plt.subplots(figsize=(6, 4)) ax.plot(tau_dense, sym_dense, "-", color="C0", label=r"$\lambda_{sym}$ (exact, Lambert W)") ax.plot(tau_dense, antisym_dense, "-", color="C1", label=r"$\lambda_{antisym}$ (exact, Lambert W)") ax.plot(tau_values, lambda1, "o", color="C0", label=r"lyapax $\lambda_1$") ax.plot(tau_values, lambda2, "o", color="C1", label=r"lyapax $\lambda_2$") ax.axhline(0.0, color="gray", lw=0.5) ax.set_xlabel(r"delay $\tau$") ax.set_ylabel("Lyapunov exponent") ax.set_title("2-node delayed linear network: spectrum vs delay") ax.legend() fig.tight_layout() plt.show() .. image-sg:: /auto_examples/images/sphx_glr_08_delayed_coupling_001.png :alt: 2-node delayed linear network: spectrum vs delay :srcset: /auto_examples/images/sphx_glr_08_delayed_coupling_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 10.465 seconds) .. _sphx_glr_download_auto_examples_08_delayed_coupling.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: 08_delayed_coupling.ipynb <08_delayed_coupling.ipynb>` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: 08_delayed_coupling.py <08_delayed_coupling.py>` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: 08_delayed_coupling.zip <08_delayed_coupling.zip>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_