Critical Transition (Fold)
chaos · 36 views
chaos
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).
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 critical slowing down. It sits beside Stochastic Resetting Walk in the atlas (standard-bank rank 24) — 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
complexity0.75
time-irreversibility0.86
volatility clustering0.91
multifractality0.64
dimensionality0.21
nonstationarity1.00
What the atlas adds
critical slowing down+6.6z
lag-1 autocorrelation rises across the series — the early-warning signature of an approaching tipping point / regime shift (also fired by frequency chirps)
also fires on nonstationary arithmetic sequences whose autocorrelation drifts for non-dynamical reasonsAtlas-extreme metrics the standard bank can’t predict for this source
Wasserstein:self_similarity | -6.0z | bank-miss 1.5σ |
S² × ℝ (Thurston):height_drift | +5.5z | bank-miss 2.0σ |
Mostow Rigidity:distance_rigidity | -5.0z | bank-miss 1.2σ |
Bispectrum:off_diagonal_ratio | -3.7z | bank-miss 1.9σ |
Ulam Spiral (Square):polynomial_concentration | -3.2z | bank-miss 1.3σ |
Ulam Spiral (Square):arm_density_variance | +2.8z | bank-miss 1.5σ |
Zipf–Mandelbrot (8-bit):hapax_ratio | +2.8z | bank-miss 1.6σ |
Hodge–Laplacian:poisson_recovery_error | -2.2z | bank-miss 1.8σ |
Composition
dtypefloat64
range[-1.192, 1.21]
unique values16384 / 16384
mean ± std-0.818 ± 0.573
Render Gallery
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Native > phase_plotraw
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Native > timeseriesraw
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Native > triple_latticeraw
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Boltzmann > lag_map
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Boltzmann > spin_grid
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Cayley > default
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Chladni > plate_bottom
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Chladni > plate_top
_(centered)/signed_log_z/Critical_Transition_(Fold).png)
Heisenberg (Nil) (centered) > signed_log_z
_(centered)/xy_path/Critical_Transition_(Fold).png)
Heisenberg (Nil) (centered) > xy_path
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Hodge–Laplacian > default
/barcode/Critical_Transition_(Fold).png)
Inflation (Substitution) > barcode
/d_curve/Critical_Transition_(Fold).png)
Inflation (Substitution) > d_curve
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Information Theory > default
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Julia Set > escape_histogram
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Klein Bottle > default
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Laplacian > default
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Logarithmic Spiral > default
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Möbius-S³ > mobius_cylinder
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Möbius-S³ > s2_base_map
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Ordinal Partition > default
/phi_spectrum/Critical_Transition_(Fold).png)
Penrose (Quasicrystal) > phi_spectrum
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Persistent Homology > barcode
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Persistent Homology > h1_diagram
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Predictability > default
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Recurrence Quantification > default
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Spectral Analysis > default
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Spectral Graph > default
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Spirograph > mode_spectrum
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Spirograph > phasor_trajectory
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Symplectic > flux_grid
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Symplectic > phase_space
/default/Critical_Transition_(Fold).png)
Ulam Spiral (Sacks) > default
/default/Critical_Transition_(Fold).png)
Ulam Spiral (Square) > default
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Visibility Graph > default
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Wavelet Cascade > default
Atlas Position
| Nearest neighbor | Distance | |
|---|---|---|
| BTC Close Price | 4.33 | cross-domain |
| Stochastic Resetting Walk | 4.77 | cross-domain |
| Geometric Brownian Motion | 4.80 | cross-domain |
Which Geometries Light Up
Bispectrum › Bispectrum:bicoherence_concentration | rank 4/298 | 0.9689 |
Bispectrum › Bispectrum:off_diagonal_ratio | rank 297/298 | 0.1213 |
Bispectrum › Bispectrum:coupling_frequency_centroid | rank 298/298 | 0.1230 |
Catch24 › Catch24:DN_OutlierInclude_p_001_mdrmd | rank 1/298 | 0.9291 |
Catch24 › Catch24:DN_OutlierInclude_n_001_mdrmd | rank 297/298 | -0.5100 |
Catch24 › Catch24:FC_LocalSimple_mean1_tauresrat | rank 298/298 | 0.0002 |
Catch24 › Catch24:PD_PeriodicityWang_th0_01 | rank 298/298 | 0.0000 |
Higher-Order Statistics › Higher-Order Statistics:c3_energy | rank 3/298 | 3.0238 |
Hodge–Laplacian › Hodge–Laplacian:poisson_recovery_error | rank 296/298 | 0.0196 |
Inflation (Substitution) › Inflation (Substitution):discrepancy | rank 1/298 | 30.1687 |
Julia Set › Julia Set:potential_smoothness | rank 294/298 | 29.3720 |
Julia Set › Julia Set:potential_variance | rank 294/298 | 1377.0891 |
Moiré › Moiré:moire_dilation_entropy | rank 1/298 | 4.2169 |
Mostow Rigidity › Mostow Rigidity:distance_rigidity | rank 297/298 | 0.0152 |
Multi-Scale Wasserstein › Multi-Scale Wasserstein:w_slope | rank 295/298 | -0.5941 |
Navier-Stokes › Navier-Stokes:cascade_asymmetry | rank 2/298 | 1.0000 |
Nonstationarity › Nonstationarity:ac1_trend | rank 1/298 | 0.4892 |
Nonstationarity › Nonstationarity:adf_pvalue | rank 1/298 | 0.9904 |
Nonstationarity › Nonstationarity:variance_trend | rank 4/298 | 0.4497 |
SL(2,ℝ) (Thurston) › SL(2,ℝ) (Thurston):lyapunov_std | rank 1/298 | 0.7121 |
Septagonal (Danzer) › Septagonal (Danzer):z_conjugate | rank 294/298 | -0.5370 |
Spirograph › Spirograph:angular_order | rank 3/298 | 12.2000 |
S² × ℝ (Thurston) › S² × ℝ (Thurston):height_drift | rank 2/298 | 0.9718 |
Ulam Spiral (Square) › Ulam Spiral (Square):arm_density_variance | rank 3/298 | 2.2270 |
Ulam Spiral (Square) › Ulam Spiral (Square):polynomial_concentration | rank 298/298 | 0.0007 |
Wasserstein › Wasserstein:transport_variability | rank 2/298 | 0.1606 |
Wasserstein › Wasserstein:self_similarity | rank 297/298 | 0.6873 |
Wavelet Cascade › Wavelet Cascade:cascade_coupling | rank 5/298 | 0.8068 |
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