"""
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 <sphx_glr_auto_examples_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.
"""
# %%
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

# %%
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}")

# %%
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()