Baker¶

Systems / Maps / Geometric maps

The baker's map — stretch, cut, and stack; the cleanest model of mixing and the horseshoe.

discrete · map2 dimensionsmixing

Definition¶

@staticmethod
def _step(X, alpha):
    """
    Right-hand side of the Baker map.

    x, y: Current state variables
    alpha: Contraction ratio of the fold (0 < alpha <= 0.5)

    The canonical ``x`` fold is ``(2 * x) % 1``, but the pure doubling map
    drains a float mantissa one bit per step (``2 * x`` is an exact binary
    shift), so every orbit collapses onto the ``x = 0`` fixed point in ~52
    iterations and the chaos dies. Wrapping just below 1 injects a ~5e-8
    offset at each fold that keeps the orbit non-degenerate; the dynamics
    (and the exponents ``ln 2`` / ``ln alpha``) are unchanged to that
    tolerance. This is the same guard :class:`KaplanYorke` carries.

    The ``y`` branch is written with ``np.where`` (rather than a Python
    ``if`` on the state) so the step traces to a straight-line tape and runs
    on the Rust engine.
    """
    x, y = X
    xp = (2 * x) % 0.99999995  # stretch in x (wrapped just below 1)
    yp = np.where(x < 0.5, alpha * y, alpha * y + (1 - alpha))  # contract + stack
    return xp, yp

Parameters¶

Symbol	Default	Role
`alpha`	`0.5`	fold fraction splitting the height (0.5 = area-preserving)

State variables: x, y

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

1 fixed points
0 stable · 1 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Baker()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Hopf (1937), Ergodentheorie (Springer, Berlin)
doi:10.1007/978-3-642-86630-2

BibTeX

@misc{baker,
  title = {Baker system},
  note = {Hopf (1937), Ergodentheorie (Springer, Berlin)},
  doi = {10.1007/978-3-642-86630-2}
}