Critical Transition (Fold)

iterated-maps · 37 views

iterated-mapsintermittent

What It Is

Saddle-node (fold) normal form dx/dt = x - x^3 + mu(t) driven by a slowly ramped control parameter across the tipping point at mu_c = 2/(3*sqrt(3)). As the occupied well flattens, the system shows critical slowing down --- rising autocorrelation and variance --- then transitions abruptly to the surviving stable branch (the canonical early-warning-signal model; Scheffer 2009, Dakos 2012). The control-parameter ramp makes the emission monotone-trending BY DESIGN (|Spearman(x,idx)| ~ 0.99): a genuine nonstationary trajectory --- full float cardinality, ~50% local decreases, an abrupt branch jump at ~92% and rising pre-tip variance (critical slowing down) --- NOT a rank-collapse footgun, and it must not be de-trended (that would erase the early-warning signal).

Interpretation

Standard analysis sees: heavy-tailed; right-skewed; smooth / autocorrelated; long-range memory (persistent); time-irreversible (sharp rises, slow decay); volatility-clustering (bursty); nonstationary / drifting. The atlas additionally detects Nonstationarity:adf_pvalue, Higher-Order Statistics:c3_energy. It sits beside Stochastic Resetting Walk in the atlas (standard-bank rank 21) — a neighbor conventional features miss.

What standard analysis sees

tail heaviness0.90

asymmetry0.96

occupancy0.33

short-range corr0.90

long-range memory0.87

spectral colour0.31

periodicity0.78

complexity0.74

time-irreversibility0.86

volatility clustering0.91

multifractality0.63

dimensionality0.21

nonstationarity1.00

What the atlas adds

Nonstationarity:adf_pvalue+6.0z

unit-root nonstationarity (ADF cannot reject random-walk null)

Higher-Order Statistics:c3_energy+5.3z

third-cumulant burst energy (SOC irreducibility signature)

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

`Wasserstein:self_similarity`	-5.7z	bank-miss 1.0σ
`S² × ℝ (Thurston):height_drift`	+5.4z	bank-miss 1.6σ
`Bispectrum:off_diagonal_ratio`	-3.7z	bank-miss 2.1σ
`Ulam Spiral (Square):polynomial_concentration`	-3.1z	bank-miss 1.4σ
`Mostow Rigidity:spectral_rigidity`	-3.0z	bank-miss 1.2σ
`Chladni:domain_ks_exponential`	+2.1z	bank-miss 1.3σ

Composition

dtypefloat64

range[-1.192, 1.21]

unique values16384 / 16384

mean ± std-0.818 ± 0.573

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
BTC Close Price	4.35	cross-origin
Stochastic Resetting Walk	4.69	cross-origin
Geometric Brownian Motion	4.71	cross-origin

Open in Atlas →

Which Geometries Light Up

Bispectrum › `Bispectrum:bicoherence_gini`	rank 5/307	0.9689
Bispectrum › `Bispectrum:off_diagonal_ratio`	rank 306/307	0.1213
Bispectrum › `Bispectrum:coupling_frequency_centroid`	rank 307/307	0.1230
Catch24 › `Catch24:DN_OutlierInclude_p_001_mdrmd`	rank 1/307	0.9291
Catch24 › `Catch24:DN_OutlierInclude_n_001_mdrmd`	rank 306/307	-0.5100
Catch24 › `Catch24:PD_PeriodicityWang_th0_01`	rank 306/307	0.0000
Catch24 › `Catch24:FC_LocalSimple_mean1_tauresrat`	rank 307/307	0.0002
Higher-Order Statistics › `Higher-Order Statistics:c3_energy`	rank 3/307	3.0238
Hodge–Laplacian › `Hodge–Laplacian:poisson_recovery_error`	rank 305/307	0.0196
Inflation (Substitution) › `Inflation (Substitution):discrepancy`	rank 1/307	30.1687
Julia Set › `Julia Set:potential_variance`	rank 303/307	1377.0891
Moiré › `Moiré:moire_dilation_entropy`	rank 1/307	4.2169
Mostow Rigidity › `Mostow Rigidity:distance_rigidity`	rank 306/307	0.0152
Multi-Scale Wasserstein › `Multi-Scale Wasserstein:w_slope`	rank 305/307	-0.5941
Navier-Stokes › `Navier-Stokes:cascade_asymmetry`	rank 2/307	1.0000
Nonstationarity › `Nonstationarity:ac1_trend`	rank 1/307	0.4892
Nonstationarity › `Nonstationarity:adf_pvalue`	rank 1/307	0.9904
Nonstationarity › `Nonstationarity:variance_trend`	rank 3/307	0.4497
SL(2,ℝ) (Thurston) › `SL(2,ℝ) (Thurston):cocycle_growth_burstiness`	rank 1/307	0.7121
Spirograph › `Spirograph:angular_order`	rank 3/307	12.2000
S² × ℝ (Thurston) › `S² × ℝ (Thurston):height_drift`	rank 2/307	0.9718
Ulam Spiral (Square) › `Ulam Spiral (Square):polynomial_concentration`	rank 307/307	0.0007
Wasserstein › `Wasserstein:transport_variability`	rank 2/307	0.1606
Wasserstein › `Wasserstein:distributional_stationarity`	rank 306/307	0.6873
Wavelet Cascade › `Wavelet Cascade:cascade_coupling`	rank 5/307	0.8068

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