r"""
The Van der Pol limit cycle
================================

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).
"""

# %%
import matplotlib.pyplot as plt

from mathematicskit.ode_dynamics.systems.limit_cycles import VanDerPolOscillator, estimate_limit_cycle_amplitude
from mathematicskit.ode_dynamics.visualizers.plots import plot_phase_portrait

# %%
# Trajectories from very different initial conditions converge
# -----------------------------------------------------------------------

fig, ax = plt.subplots(figsize=(6, 6))
for state0 in ([0.1, 0.0], [4.0, 0.0], [0.0, 3.0]):
    system = VanDerPolOscillator(state0, mu=1.0)
    traj = system.integrate((0.0, 30.0), dt=1e-3, method="rk4")
    plot_phase_portrait(traj, ax=ax, lw=0.8, label=f"start={state0}")
ax.legend()
ax.set_title("Van der Pol (mu=1): all orbits approach the same limit cycle")
fig.tight_layout()

# %%
# Amplitude is independent of mu's transient details
# --------------------------------------------------------

for mu in (0.5, 1.0, 2.0, 5.0):
    amp = estimate_limit_cycle_amplitude(mu=mu, t_transient=150.0, t_observe=40.0)
    print(f"mu={mu}: limit-cycle amplitude ~ {amp:.4f}")

plt.show()