Introduction:
Laboratory and numerical studies show that fault complexity, quantified by the Kaplan–Yorke dimension (effective number of dynamical degrees of freedom), can decrease as stress and load increase, but the reason remains uncertain.
Materials and methods:
We analyze a synthetic seismic cycle from a previously validated heterogeneous-rupture model: moment magnitude
M
W
(macroscopic earthquake size) and its perturbation response
Δ
M
W
(magnitude change after a small energy increase). A thermodynamic estimate gives
Λ
, the sum of all Lyapunov exponents, which measure whether nearby states diverge (expansion) or converge (contraction). In sliding windows of
W
= 100 samples,
k
-nearest neighbors (KNN) forecasts yield a finite-horizon stability descriptor
λ
^
M
W
, the slope of log forecast error versus forecast horizon.
λ
^
M
W
is positive, negative or zero when errors grow (expansion), shrink (contraction) and are neutral, respectively. As
M
W
is macroscopic, subtracting
λ
^
M
W
from
Λ
leaves a residual
λ
R
, the microscopic part. Slopes require straight-line fits (coefficient of determination
R
2
≥ 0.98, normalized mean absolute error ≤ 0.08) over four or more early horizons; robustness is tested with
W
= 80 and
W
= 120.
Results:
In
M
W
, 334 of 460 windows (72.61%) show contraction, but the mean slope is near zero (−0.005159), so errors shrink slightly: near-neutral stability. Of these, 230 (68.86%) fit well;
W
= 80 and 120 give similar contraction fractions (69.58%, 72.27%). In
Δ
M
W
, 406 of 460 windows (88.26%) show expansion, so perturbations grow. Only 6 of 54 contracting windows fit well.
Λ
becomes more negative through the cycle (stronger contraction) while
λ
^
M
W
stays near neutral, so
λ
R
becomes increasingly negative: contraction concentrates in the microscopic part. As accumulated energy scales with squared stress, the Kaplan–Yorke dimension decreases approximately as
1
/
stress
2
, matching laboratory data.
Conclusions:
As load accumulates, stability is redistributed across scales: the macroscopic part stays near neutral, whereas microscopic contraction strengthens, lowering effective dimensionality. Small-scale activity thus fades as the simulated system approaches rupture.