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Analysis · 05

Fixed points

fixed_points finds the solutions of \(f(x) = x\) for a discrete map and classifies their stability from the Jacobian eigenvalues.

import tsdynamics as ts

fps = ts.fixed_points(ts.Henon())
fps
# [FixedPoint([-1.131354  -0.339406], unstable, |λ|max=3.2598),
#  FixedPoint([ 0.631354  0.189406], unstable, |λ|max=1.9237)]

How it works

  1. Seeding — n_seeds random points are drawn from a search box (default: the orbit's bounding box padded by 50 %), plus points sampled from a short orbit of the map itself, which biases the search toward the attractor region.
  2. Newton — each seed runs Newton's method on \(g(x) = f(x) - x\), using the map's exact _jacobian (no finite differences), up to max_iter iterations or residual tol.
  3. Dedup & classify — converged roots closer than dedup_tol are merged; each survivor is classified stable iff every \(|\lambda_i| < 1\) for the eigenvalues \(\lambda_i\) of \(J(x^*)\).

Pass seed= for reproducible seeding, and a box=((lo,...), (hi,...)) to search a specific region (roots outside the box are discarded).

Checking against the analytic answer: Hénon

For the Hénon map \(x' = 1 - a x^2 + y\), \(y' = b x\), a fixed point satisfies \(a x^2 + (1 - b)x - 1 = 0\), so

\[ x^* = \frac{-(1 - b) \pm \sqrt{(1 - b)^2 + 4a}}{2a}, \qquad y^* = b\,x^*. \]

With the classic \(a = 1.4\), \(b = 0.3\): \(x^* \approx 0.6314\) and \(x^* \approx -1.1314\) — matching the numeric output above. Both are saddles (one eigenvalue inside the unit circle, one outside), which is exactly what the strange attractor needs: orbits are attracted along the stable direction and stretched along the unstable one.

import numpy as np

a, b = 1.4, 0.3
x_exact = (-(1 - b) + np.sqrt((1 - b) ** 2 + 4 * a)) / (2 * a)
assert abs(fps[1].x[0] - x_exact) < 1e-10

The FixedPoint record

fp = fps[0]
fp.x              # the fixed point, shape (dim,)
fp.eigenvalues    # eigenvalues of the Jacobian at fp.x
fp.stable         # True iff all |λ| < 1

Maps only, for now

fixed_points requires a DiscreteMap — it relies on the explicit _jacobian. For equilibria of flows, the analogous problem is \(f(x) = 0\) on the right-hand side; a flow-aware version is a natural extension and not yet implemented.

See also