Logistic r=3.65 (Banded Chaos)

iterated-maps · 37 views

iterated-mapschaotic

What It Is

Logistic map at r=3.65 --- banded (2-band) chaos: chaotic within two disjoint bands the orbit alternates between, mixing local unpredictability with global period-2 band cycling

Interpretation

Standard analysis sees: bounded / light-tailed; left-skewed; anti-correlated (alternating); anti-persistent; blue spectrum (high-frequency power); strongly periodic; stationary. The atlas additionally detects discrete-map sensitive dependence, Time Reversibility:ordinal_reversal_distance.

What standard analysis sees

tail heaviness0.12

asymmetry0.12

occupancy0.69

short-range corr0.01

long-range memory0.10

spectral colour0.96

periodicity0.96

complexity0.24

time-irreversibility0.15

volatility clustering0.44

multifractality0.27

dimensionality0.43

nonstationarity0.11

What the atlas adds

discrete-map sensitive dependence+2.5z

deterministic chaos (positive λ_max, sensitive dependence)

names a discrete-map-scoped estimate, NOT chaos in general — continuous-flow chaos (Lorenz, Rössler) reads weak/neutral here despite being genuinely chaotic; spiky arithmetic sources can false-positive on the finite-time estimate

Time Reversibility:ordinal_reversal_distance+2.4z

time-reversal symmetry breaking via Bandt-Pompe ordinal patterns

Atlas-extreme metrics the standard bank can’t predict for this source

`E8 Lattice:coset_transition`	+11.6z	bank-miss 1.9σ
`Heisenberg (Nil) (centered):area_length_ratio`	+5.4z	bank-miss 1.3σ
`Heisenberg (Nil) (centered):xy_spread`	+5.0z	bank-miss 1.7σ
`Julia Set:potential_variance`	+4.5z	bank-miss 1.5σ
`Julia Set:potential_smoothness`	+4.5z	bank-miss 1.5σ
`Julia Set:escape_entropy`	-3.0z	bank-miss 1.6σ
`H4 600-Cell:edge_walk_fraction`	+2.3z	bank-miss 2.5σ
`Inflation (Substitution):entropy_rate`	-2.1z	bank-miss 1.3σ

Composition

dtypefloat64

range[0.2914, 0.9125]

unique values16384 / 16384

mean ± std0.655 ± 0.216

Render Gallery

Native > phase_plotraw

Native > timeseriesraw

Native > triple_latticeraw

Boltzmann > lag_map

Boltzmann > spin_grid

Cayley > default

Chladni > free_plate_nodal

Chladni > plate_bottom

Chladni > plate_top

Heisenberg (Nil) (centered) > signed_log_z

Heisenberg (Nil) (centered) > xy_path

Hodge–Laplacian > default

Inflation (Substitution) > barcode

Inflation (Substitution) > d_curve

Information Theory > default

Julia Set > escape_histogram

Klein Bottle > default

Laplacian > default

Logarithmic Spiral > default

Möbius-S³ > mobius_cylinder

Möbius-S³ > s2_base_map

Ordinal Partition > default

Penrose (Quasicrystal) > phi_spectrum

Persistent Homology > barcode

Persistent Homology > h1_diagram

Predictability > default

Recurrence Quantification > default

Spectral Analysis > default

Spectral Graph > default

Spirograph > mode_spectrum

Spirograph > phasor_trajectory

Symplectic > flux_grid

Symplectic > phase_space

Ulam Spiral (Sacks) > default

Ulam Spiral (Square) > default

Visibility Graph > default

Wavelet Cascade > default

Atlas Position

Nearest neighbor	Distance
Quartic Map (Feigenbaum)	4.18
Logistic Edge-of-Chaos	4.34
Logistic r=3.9 (Near-Full Chaos)	4.47

Open in Atlas →

Which Geometries Light Up

E8 Lattice › `E8 Lattice:coset_transition`	rank 1/306	0.6094
Fisher Information › `Fisher Information:effective_dimension`	rank 306/306	1.0381
Lorentzian › `Lorentzian:spacelike_fraction`	rank 4/306	0.1768

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