Stable Manifolds

Choose a discrete linear saddle map or Hénon map, then Calculate. Blue is stable (forward iterates approach the fixed point), orange unstable (backward iterates approach it), pink the fixed point. For discrete maps stability uses eigenvalue magnitude, |λₛ|<1<|λᵤ|, not the sign criterion for continuous flows. Negative eigenvalues reverse the side on each iterate.

Linear example: A=R(θ)diag(λₛ,λᵤ)R(θ)ᵀ. The eigen-directions are orthonormal; half-length is the Euclidean distance from the origin to each endpoint. The lines are exact invariant subspaces of this map. Rotation and eigenvalues are editable; the actual matrix is in details.

Hénon: F(x,y)=(1−ax²+y,bx), inverse F⁻¹(x,y)=(y/b,x−1+a(y/b)²). Choose either real fixed point. Only a saddle with one stable and one unstable direction is accepted; an attracting or marginal point is an error, not a fabricated curve.

A degree-20 local Taylor chart K(t) solves F(K(t))=K(λt) coefficient by coefficient. Its tangent is the normalized eigenvector. Local radius is reduced until the sampled invariance defect and polynomial tail are small. These are numerical indicators, not rigorous error enclosures. Fundamental segments of both signs extend forward for the unstable chart and backward for the stable chart. They are adaptively resolved by segment length and midpoint curvature, clipped to the selected square, and never connected across separate branches.

Extension steps are finite, with limits on evaluations and refinement depth; unresolved visible curves produce an error. Very distant/escaped portions are omitted. Finite sampling can miss thin returning branches, and inverse iteration amplifies rounding errors. Neither the whole global manifold nor intersections/chaos are certified. Compare shorter extensions and smaller views; local defect does not bound every extended point.

Calculation runs in a cancellable worker; resize redraws cached data. PNG exports actual curves, equal-scale axes, parameters and color meaning. Raw and inactive fields, panels and completed curves are retained locally. Reload evaluates the saved subdivision plan, without adaptive refinement or rounding stored coordinates; Calculate runs the adaptive calculation again. The plan describes this finite approximation, not a proof of complete coverage. Blue is stable, dashed orange is unstable, pink marks the fixed point when visible.

Gonzalez & Mireles James · Parameterization

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