Systems · Discrete

Discrete maps

26 iterated maps, subclasses of DiscreteMap. A map is defined by two plain numeric static methods — _step and _jacobian — decorated with @staticjit, which applies Numba's njit (and degrades gracefully to pure Python when Numba is unavailable):

from tsdynamics.utils import staticjit

class Henon(DiscreteMap):
    params = {"a": 1.4, "b": 0.3}
    dim = 2

    @staticjit
    def _step(X, a, b):
        x, y = X[0], X[1]
        return (1.0 - a * x**2 + y, b * x)

    @staticjit
    def _jacobian(X, a, b):
        ...

Parameters arrive positionally, in params-dict order — a mismatched signature raises a TypeError at import time.

Categories

Category	Count	Examples
chaotic_maps	9	Hénon, Ikeda, standard map, ...
exotic_maps	7	Less common chaotic recurrences
geometric_maps	4	Baker, cat-map-style area transformations
polynomial_maps	3	Quadratic and cubic recurrences
population_maps	3	Logistic, Ricker, and relatives

Each map has a generated page in this section with its recurrence, defaults, and an orbit figure.

Iterating

import tsdynamics as ts

h = ts.Henon()
traj = h.iterate(steps=10_000)     # Trajectory; traj.t is arange(steps)

On first call, a Numba loop is compiled with the current parameter values inlined and cached per (class, params); changing a parameter costs one quick re-JIT. If an orbit diverges (random ICs can land outside the attractor basin), iterate retries with fresh random ICs up to max_retries times. Maps whose basin is small declare a class-level default_ic so the first try lands inside.

Lyapunov spectrum

h.lyapunov_spectrum(steps=5000)    # ≈ [0.42, -1.62]

Computed by QR decomposition of the running Jacobian product in a single forward pass — trajectory and tangent dynamics advance together. A reortho_interval larger than 1 trades a little accuracy for speed on high-dimensional maps. The math is laid out in Theory · Lyapunov exponents.

Maps are the analysis workhorse

The protocol-level tools written for maps also run on flows through the derived wrappers (PoincareMap, StroboscopicMap) — that composition is what turns an orbit diagram over a wrapped flow into its bifurcation diagram. Map-specific tools like fixed-point finding use the explicit _jacobian and apply to DiscreteMap subclasses directly.