Chaotic systems - Nixtla

Chaotic systems are deterministic yet unpredictable: fully specified equations whose sensitivity to initial conditions produces series that look random but have rich hidden structure. They test whether a model captures nonlinear dynamics rather than just fitting noise.

The model $x_{n+1} = r\, x_n (1 - x_n) \quad \text{(logistic map; \texttt{lorenz} and \texttt{mackey\_glass} differ)}$ Choose a classic system via system — lorenz (a strange attractor), logistic (the period-doubling map), or mackey_glass (a delay-differential equation). Each is generated by iterating its exact dynamics, so the same seed and parameters reproduce the trajectory precisely.

import polars as pl
import matplotlib.pyplot as plt

from synforecast.generators import ChaoticSystemGenerator

1. The three systems

Each system is fully deterministic given its initial condition, yet none looks periodic. lorenz is a continuous attractor sampled once per time unit, logistic is a period-doubling map, and mackey_glass is a delay-differential equation.

lorenz_df = ChaoticSystemGenerator(
    engine="polars",
    min_length=500,
    max_length=500,
    freq="D",
    system="lorenz",
    observation_noise=0.1,
    seed=42,
).generate(n_series=1)

logistic_df = ChaoticSystemGenerator(
    engine="polars",
    min_length=500,
    max_length=500,
    freq="D",
    system="logistic",
    logistic_r=3.9,
    observation_noise=0.01,
    seed=42,
).generate(n_series=1)

mackey_glass_df = ChaoticSystemGenerator(
    engine="polars",
    min_length=500,
    max_length=500,
    freq="D",
    system="mackey_glass",
    mg_tau=17,
    observation_noise=0.01,
    seed=42,
).generate(n_series=1)

panels = [
    ("lorenz (x-component)", lorenz_df),
    ("logistic map (r=3.9)", logistic_df),
    ("mackey_glass (tau=17)", mackey_glass_df),
]
fig, axes = plt.subplots(3, 1, figsize=(12, 7.5), sharex=False)
for ax, (label, df) in zip(axes, panels):
    ax.plot(df["ds"].to_list(), df["y"].to_list(), alpha=0.85, linewidth=1)
    ax.set(ylabel="Value", title=label)
axes[-1].set_xlabel("Timestamp")
plt.tight_layout()
plt.show()

2. Sensitivity to initial conditions

Multiple series from the same chaotic system diverge due to tiny differences in initial perturbations.

multi_gen = ChaoticSystemGenerator(engine="polars", 
    min_length=200, max_length=200, freq="D",
    system="lorenz", seed=42,
)
multi_df = multi_gen.generate(n_series=3)

fig, ax = plt.subplots(figsize=(12, 4))
for uid in multi_df["unique_id"].unique().to_list():
    series = multi_df.filter(pl.col("unique_id") == uid)
    ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8, linewidth=0.8)
ax.set_xlabel("Timestamp")
ax.set_ylabel("Value")
ax.set_title("Multiple Lorenz series (sensitive to initial conditions)")
ax.legend()
plt.tight_layout()
plt.show()

Related generators

Cyclic — irregular but stochastic oscillation.

State space — custom deterministic-plus-noise dynamics.

Available systems and parameters are in the generator reference.

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