How Recurrence Condensation Uncovers Critical Transitions in Complex Systems

Background

Critical transitions—small parameter changes that cause abrupt shifts between qualitatively different dynamical regimes—appear in climate, ecology, finance, and engineering. Detecting such transitions in short, noisy, high‑dimensional time series is difficult with conventional statistical tools.

Methodology

1. Phase‑space reconstruction : For each scalar time series (e.g., pressure p(t)) a Takens delay embedding is built x_i = [p(t_i), p(t_i+τ), ..., p(t_i+(m-1)τ)] where the embedding dimension m and delay τ are chosen by false‑nearest‑neighbors and mutual‑information criteria. 2. Recurrence matrix : A binary matrix is defined as R_{ij}=Θ(ε-‖x_i-x_j‖) with Euclidean distance ‖·‖, Heaviside step Θ, and a threshold ε set to 20 % of the attractor’s diameter (or 0.2 × standard deviation of pairwise distances). The matrix is visualised as a black‑white Recurrence Plot (RP). 3. Recurrence Quantification Analysis (RQA) : From the RP the following metrics are extracted (minimum diagonal/vertical length = 2):

Determinism (DET) : proportion of recurrence points forming diagonal lines, indicating trajectory predictability.

DET = \frac{\sum_{l\ge l_{min}} l\,P(l)}{\sum_{l\ge1} l\,P(l)}

Entropy (ENTR) : Shannon entropy of the diagonal‑line length distribution, measuring complexity. ENTR = -\sum_{l\ge l_{min}} p(l)\log p(l) Laminarity (LAM) and Trapping Time (TT) : analogous quantities for vertical line structures, reflecting the duration the system stays in a given state.

LAM = \frac{\sum_{v\ge v_{min}} v\,P(v)}{\sum_{v\ge1} v\,P(v)}\quad\text{TT}=\frac{\sum_{v\ge v_{min}} v\,P(v)}{\sum_{v\ge v_{min}} P(v)}

Recurrence Time (RT) and its entropy (RTE): statistics of the time between successive recurrences.

RT = \langle\Delta t\rangle,\qquad RTE = -\sum_{k} p_k\log p_k

4. Scaling analysis : Each metric M is plotted against the distance to the control parameter (e.g., Reynolds number Re) from an assumed critical value Re_c. A power‑law fit M ∝ |Re‑Re_c|^{-α} provides both the exponent α and an estimate of Re_c where the fit error is minimal.

Experimental System: Thermoacoustic Combustor

A laboratory combustor equipped with a high‑frequency pressure sensor is operated at three mean flow rates corresponding to Reynolds numbers Re = 2.13×10⁴ , 2.88×10⁴ , and 3.09×10⁴ . Increasing Re drives the system from low‑amplitude chaotic combustion noise to high‑amplitude self‑sustained thermoacoustic oscillations. Recurrence plots constructed from pressure data at the three Re values show a clear evolution:

Fragmented short diagonals (chaotic regime).

Emergence of ordered short diagonal blocks (intermittent regime).

Long continuous diagonals (periodic regime).

Results: Recurrence Condensation and Early‑Warning Signals

All RQA metrics exhibit systematic trends as Re approaches the transition:

DET rises sharply, indicating increasing predictability.

ENTR shows a rise‑then‑fall pattern, reflecting a temporary increase in complexity before ordering.

LAM and TT decrease, meaning the system spends less time in laminar‑like states.

RT and RTE converge toward a single value, signalling a collapse of multiple time scales into one dominant recurrence time.

Power‑law scaling of each metric with |Re‑Re_c| yields a consistent critical Reynolds number Re_c = 2.96×10⁴ ± 0.07×10⁴ . The scaling exponent differs among metrics but the fitted Re_c aligns with the visual onset of long diagonal structures in the RP.

Model Validation with Noisy Hopf Bifurcation

To test universality, synthetic time series are generated from a noisy Hopf normal‑form model: <code>\dot{z} = (μ + iω)z - (1 + iβ)|z|^2z + ξ(t)</code> where z∈ℂ , μ is the bifurcation parameter, ω the linear frequency, β the nonlinear frequency shift, and ξ(t) Gaussian white noise. Both sub‑critical (μ<0) and super‑critical (μ>0) cases are examined. RPs and RQA metrics display the same recurrence condensation pattern as the experimental combustor. Power‑law fits recover the theoretical bifurcation point (μ=0) within numerical tolerance, confirming that the phenomenon is not specific to thermoacoustic combustion.

Conclusion and Outlook

The study defines recurrence condensation as the collapse of multiple dynamical time scales into a single dominant scale during a critical transition. Quantitative RQA metrics provide early‑warning signals and enable precise localisation of the critical point through power‑law scaling. Although demonstrated on a thermoacoustic combustor, the methodology is applicable to any noisy, high‑dimensional system—e.g., climate ice‑sheet collapse, ecological regime shifts, or financial market crashes—offering a robust tool for anticipating abrupt changes.