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Learning Chaos Without Seeing Chaos: Extrapolation of Global Dynamics in Autoregressive Transformers

Yilun Liu, Yi Zhang, Ganyu Wu, Sikuan Yan, Mengyue Wang, Alois Knoll, Volker Tresp, Yunpu Ma

arXiv:2609.38814v1cs.LGnlin.CD

TL;DR

Can local observations teach a model to preserve global dynamical organization when its recursively generated trajectories reach unseen parameter regimes? The paper trains autoregressive transformers on restricted regimes and finds they recover unseen structures, including verified period-doubling cascades and long-time behavior in other systems.

  • Problem

    It is uncertain whether locally learned dynamics remain faithful when recursively applied to model-generated states and extrapolated beyond observed parameter conditions.

    Source: §1 · p. 1

  • Method

    The paper studies parameter-conditioned autoregressive predictors as autonomous dynamical systems under recursive application beyond their training regimes.

    Source: §2.2 · p. 3, §3.4 · p. 6

  • Results

    Transformers recover unseen global dynamics across systems; the logistic map reaches period 128 with a finite-order ratio of 4.668686 versus 4.669132 for the reference cascade at the same order.

    Source: §4.1 · p. 6, §6 · p. 9

  • Takeaways & Limitations

    Local next-state supervision can produce parameter-conditioned dynamics that preserve nontrivial global organization beyond the observed regime.

    Source: §6 · p. 9

Abstract

from arXiv · show

Autoregressive models are trained to predict a system's behavior one step at a time, and recursive generation allows the learned dynamics to unfold over long horizons. To what extent can such dynamics learned from local observations recover broader organization of an underlying system that was only partially observed during training? Here we study small autoregressive transformers trained from scratch on trajectories sampled from restricted parameter regimes of several non-linear dynamical systems, including logistic and sine maps, the Lorenz system, and the generalized Hopf system, with control parameters and state trajectories represented as sequences of continuous tokens. Under closed-loop evaluation at parameters far outside the training distribution, the models can recover self-similar period-doubling cascades, chaotic dynamics, and attractor structures with remarkable visual and numerical fidelity. For the logistic map, a transformer reproduces successive period doublings up to period 128, yielding a finite-order scaling ratio of 4.6687, matching the Feigenbaum constant to within $5\times10^{-4}$. We further investigate how these structures emerge over the course of training, and reveal with causal interventions how control-parameter information is processed through attention into state prediction and shapes the resulting closed-loop dynamics. These results suggest that a surprisingly narrow window into a system's local behavior may suffice for autoregressive transformers to generalize to its unseen global dynamical organization.

1 INTRODUCTION

The section asks whether local observations can teach a parameter-conditioned predictor to preserve a system’s global organization when recursively run beyond its training regime. Experiments find that small autoregressive transformers can recover global structures, including period-doubling cascades and chaotic behavior, from restricted training data.

Research question

  • The study tests whether locally learned, parameter-conditioned transitions preserve the underlying system’s global organization under closed-loop extrapolation.Recursive generation shifts inputs from observed states to model-generated states, while parameter extrapolation extends transitions beyond training conditions.Source: §1 · p. 1

Motivation

  • Bifurcations provide a stringent test because extrapolation must capture qualitative changes from stable fixed points to periodic or chaotic behavior.The study assesses not only whether period doublings occur, but whether transitions are placed correctly and their scaling follows the same universality structure.Source: §1 · p. 2

Approach

  • Small transformers are trained from scratch on local next-state prediction using continuous parameter and state tokens, then evaluated recursively far outside training regimes.The systems include logistic and sine maps, Lorenz systems, and generalized Hopf systems, without stability labels, governing equations, or long-time dynamical summaries.Source: §1 · p. 2

Main findings

  • Across nonlinear systems, transformers extrapolate global dynamics beyond training, including period-doubling cascades, chaos, and unseen attractor behavior.The results also connect these dynamics to learned pathways through which control-parameter information shapes state computation.Source: §1 · p. 2

2 BACKGROUND AND PRELIMINARIES

The background distinguishes learned predictive models from approaches that seek explicit equations or other representations, and frames the paper’s target as preserving structural consequences of dynamics beyond observed conditions. It defines the parameterized transitions and bifurcations used to assess that target, while reviewing prior work on regime extrapolation.

2.1 FROM SYSTEM IDENTIFICATION TO LEARNED SCIENTIFIC MODELS

  • System-identification and scientific-discovery methods target transition laws or explicit symbolic descriptions, while predictive models may instead recover structural consequences beyond training conditions.The surveyed alternatives include Koopman-inspired models, neural differential equations, reservoirs, scientific surrogates, and world models.Source: §2.1 · pp. 2–3

2.2 PARAMETERIZED DYNAMICAL SYSTEMS AND BIFURCATIONS

  • A learned predictor induces a parameterized autonomous system whose iterated behavior can be assessed for attractors, periodicity, bifurcations, scaling, and chaotic instability.For scalar maps, a flip occurs when the cycle multiplier passes through −1; successive flip intervals in the quadratic universality class approach the Feigenbaum scaling ratio.Source: §2.2 · p. 3

2.3 LEARNING DYNAMICS ACROSS PARAMETER-CONDITIONED REGIMES

  • Parameterized scientific surrogates can generalize solution maps, but that does not necessarily establish recovery of qualitative regime transitions.Reservoir and recurrent models have also reproduced long-run or attractor-level behavior of chaotic systems [Vlachas et al., 2018; Pathak et al., 2018; Lu et al., 2018; Gauthier et al., 2021].Source: §2.3 · p. 4
  • Prior parameter-conditioned models have extrapolated bifurcations, but transformer results include both emergent behavior and failures, so regime extrapolation is an empirical property.Examples include recurrent networks, parameter-aware reservoirs [Köglmayr & Räth, 2024], and parameterized Neural ODEs; transformer studies include autonomous behavior and reported failures [Zhai et al., 2026].Source: §2.3 · p. 4

3 METHODOLOGY AND EXPERIMENTS

The methodology trains parameter-conditioned autoregressive predictors on local trajectories, then studies the autonomous dynamics induced by recursive prediction. It extends the setup to discretely observed flows, probes parameter-to-state influence with interventions, and evaluates structural properties beyond local prediction accuracy.

3.1 SYSTEMS, DATA, AND TRAINING OBJECTIVE

  • Logistic and sine map models use teacher-forced next-state training on restricted parameter ranges, with model-generated states introduced only during closed-loop evaluation.Training parameters stop at or before the first period-doubling bifurcation in each map family.Source: §3.1 · p. 4

3.2 PARAMETERIZED TRANSFORMER AND INDUCED DYNAMICS

  • Separate continuous parameter and state tokens let attention transmit parameter information into state-token computation, where nonlinear transformations can combine it with state information.The scalar-map decoder-only Transformer uses two residual layers, two attention heads, width 16, and a 64-wide feed-forward block.Source: §3.2 · p. 5
  • Recursive prediction defines the learned autonomous update; for contexts longer than one state, the generated process is not a one-dimensional map.Scalar predictions are clipped to the allowed state interval, while local bifurcation calculations use the smooth transition where clipping is inactive.Source: §3.2 · p. 5

3.3 EXTENSION TO CONTINUOUS-TIME FLOWS

  • For discretely observed continuous-time systems, the model predicts a finite-time state increment rather than receiving or reconstructing the continuous vector field.Experiments apply the construction to Lorenz and generalized Hopf systems, with parameter and state variables represented in tokens.Source: §3.3 · p. 5

3.4 INTERVENTION ON PARAMETER CONDITIONING

  • Activation patching holds state history fixed while replacing a layer’s final-state attention output with one from a different parameter condition.Complementary closed-loop interventions block the selected parameter-to-state attention edge, with sham and equal-norm random controls.Source: §3.4 · p. 6

3.5 MEASURING STRUCTURAL EXTRAPOLATION

  • Structural evaluation separately measures long-run distributions, periodic organization, bifurcation placement, and cascade scaling, alongside local transition accuracy over training.The logistic cascade figure averages results across seeds and marks whether each scaling segment is supported by all seeds or only some.Source: §3.5 · p. 6, Figure 2 · p. 6

4 RESULTS AND ANALYSES

Across scalar maps and continuous flows, restricted local training yielded closed-loop dynamics with unseen global structure. The analyses trace how that structure emerges, how parameter information shapes it, and how it extends across systems.

4.1 EMERGENCE OF GLOBAL STRUCTURE BEYOND THE OBSERVED REGIME

  • 4.668686 is the deepest shared finite-order ratio across logistic seeds, versus 4.669132 for the reference cascade; the learned cascade reaches period 128.The sine model also shows qualitative period doubling, with a reference finite-order ratio of 4.668805.Source: §4.1 · pp. 6–7, Figure 3 · p. 7
  • The logistic cascade appears progressively during training, while later one-step OOD MSE improvement does not correspond to monotonic improvement in closed-loop OOD W1.No doubling is found by 2,000 updates; one appears by 3,000 and three by 4,000. Between 6,000 and 10,000 updates, OOD MSE falls from 1.10×10−3 to 2.41×10−4 while closed-loop OOD W1 rises from 0.0296 to 0.0867.Source: §4.1 · p. 7
  • At least six flips are verified under each of four training windows, and a coarse detector finds a cascade in 11 of 12 window–seed combinations.The windows include [2, 3], [2.5, 3], [2.9, 3], and [2, 2.5], so training observations need not be immediately adjacent to the transition.Source: §4.1 · p. 7
  • The logistic and sine Transformers have fitted critical exponents z = 2.00 and z ≈2.10, respectively, consistent with a quadratic critical point.The paper connects the recovered Feigenbaum scaling and these exponents to organization expected of the quadratic universality class.Source: §4.1 · p. 7

4.2 PARAMETER CONDITIONING CAUSALLY SHAPES THE LEARNED DYNAMICS

  • Attention interventions and parameter-label permutation show that extrapolated organization depends on learned parameter-to-trajectory correspondence.For logistic seed 42, blocking relevant second-layer attention raises OOD W1 from 0.03497 to 0.60605 and eliminates two consecutive flips; permuting training labels removes coarse-grid flips and periodic agreement.Source: §4.2 · pp. 7–8

4.3 STRUCTURAL EXTRAPOLATION IN CONTINUOUS FLOWS

  • Across 593 of 625 Lorenz evaluation cells, fixed/nonfixed classification matches the reference; repeated wing switching appears in 362 of 375 reference cells exhibiting it.The model also reconstructs the transition boundary, with median absolute displacement of 0.95 in ρ and a maximum of 10.7 at the steep low-σ end.Source: §4.3 · p. 8, Figure 4 · p. 8
  • The generalized Hopf model matches point-versus-cycle organization at all nine tested parameter pairs, including coexistence from different initial radii.Across 36 trajectories, 14 approach fixed points and 22 approach cycles; at (µ1, µ2) = (−0.1, 1), the predicted outer-cycle radius is 0.948 versus 0.942 for the reference.Source: §4.3 · p. 8, Figure 5 · p. 8
  • Together, scalar maps, Lorenz, and generalized Hopf yield unseen chaotic, periodic, and multistable long-time organization beyond their training regimes.The continuous-flow examples extend the reported structural extrapolation beyond scalar maps.Source: §4.3 · pp. 8–9

5 DISCUSSION

The discussion frames recursively applied predictors as autonomous systems whose organization can exceed what their local training regimes represented. It links this global behavior to parameter routing and to scientific analysis of learned dynamics.

5 DISCUSSION

  • Across systems, local next-state learning yields autonomous dynamics with global structure absent from the training regime.Scalar maps develop extended period-doubling hierarchies, while Lorenz and generalized Hopf models produce unseen chaotic and multistable behavior.Source: §5 · p. 9
  • Structural organization can emerge early and evolve differently from one-step prediction accuracy, including when closed-loop fidelity does not improve monotonically.The discussion also describes Lorenz transitions to sustained double-wing motion and generalized Hopf coexistence between point and cycle attractors.Source: §5 · p. 9
  • Counterfactual patches and closed-loop blocking connect learned routes for parameter information to the autonomous dynamics produced under recursive feedback.The discussion distinguishes these parameter-routing mechanisms from structural phenomena shared across transition representations.Source: §5 · p. 9
  • Bifurcations, universality scaling, chaotic instability, and attractor selection are proposed as objects for probing what models learn about physical systems.The discussion presents learned autonomous dynamics as a potentially useful substrate for scientific discovery from local observations.Source: §5 · p. 9

6 CONCLUSION

The conclusion reports that restricted local observations can yield unseen global organization when autoregressive Transformers are recursively applied. It highlights both verified scalar-map cascades and chaotic or multistable behavior in continuous systems.

6 CONCLUSION

  • Transformers trained on restricted local observations generate unseen global dynamical organization under recursive execution.Source: §6 · p. 9
  • Scalar-map closed-loop systems develop numerically verified high-order period-doubling cascades with Feigenbaum scaling.Source: §6 · p. 9
  • The parameter-conditioned formulation produces unseen chaotic and multistable long-time behavior in Lorenz and generalized Hopf systems.Source: §6 · p. 9

AI USE STATEMENT

Generative AI assisted with language editing and literature retrieval, but not with the scientific work described. The authors state that they checked references and reviewed and approved the scientific content and final wording.

AI USE STATEMENT

  • Generative AI supported language editing and discovery of potentially relevant literature; authors independently checked retrieved references.Source: AI USE STATEMENT · p. 10
  • Generative AI was not used to design methods or experiments, analyze or interpret results, generate synthetic datasets, or prove mathematical claims.Source: AI USE STATEMENT · p. 10
  • The authors reviewed and approved all scientific content, analyses, results, and final wording.Source: AI USE STATEMENT · p. 10

CONTENTS OF THE APPENDICES

The scalar comparisons use a fixed neural experiment set, while flow experiments use separate supports and budgets.

A EXPERIMENTAL SETUP AND TRAINING PROTOCOLS

A.1 DATA GENERATION AND SAMPLING

The appendix specifies short-trajectory data generation, a compact Transformer, and a fixed training protocol; it then introduces evaluation and verification procedures.

A.1 DATA GENERATION AND SAMPLING

  • Each batch samples 512 parameters and 512 initial states, then generates float32 physical-map trajectories without discarding burn-in.Sequence lengths have seed-specific means near 5.5 and maxima of 22–28.Source: §A.1 · p. 15

A.2 TRANSFORMER ARCHITECTURE

  • The Transformer uses width 16, two residual layers, two attention heads, feed-forward width 64, and learned parameter/state type embeddings.Its maximum total context is 256.Source: §A.2 · p. 15

A.3 OPTIMIZATION AND CHECKPOINT SELECTION

  • Both architectures use teacher forcing and AdamW for 20,000 updates, with warm-up and cosine learning-rate decay.The final checkpoint supplies fixed-grid comparisons, high-order verification, and interventions; selected figures also show intermediate checkpoints.Source: §A.3 · p. 15

B EVALUATION AND NUMERICAL VERIFICATION

B.1 DISTRIBUTIONAL AND LOCAL PREDICTION METRICS

The appendix defines scalar and flow diagnostics and verifies the learned cascade with independent numerical checks. It also tracks how cascade structure, physical alignment, and autonomous fidelity change during training.

B.1 DISTRIBUTIONAL AND LOCAL PREDICTION METRICS

  • W1 distributional agreement and period matching measure complementary properties; unresolved sequences include long transients or insufficient resolution and are not classified as chaos.Cascade detection also requires agreement across all three initial states and consecutive grid support on both sides of a transition.Source: §B.1 · p. 16, §B.2 · p. 16

B.3 ADAPTIVE REFINEMENT AND FLIP VERIFICATION

  • Adaptive verification refines blind trajectory-detected brackets, checks persistence across burn-in budgets, and confirms flips through orbit closure, multiplier, and two-sided stability tests.Physical flip locations and finite-order interval ratios are reported separately because they measure parameter alignment and cascade geometry, respectively.Source: §B.3 · pp. 16–17

B.4 EVALUATION OF CONTINUOUS FLOWS

  • Continuous-flow evaluation uses fixed/nonfixed classification, wing switching and Lyapunov estimates for Lorenz, and radius, angular motion, and multiple initial conditions for generalized Hopf.Repeated wing switching alone does not establish chaos; multiple initial conditions test for coexisting Hopf attractors.Source: §B.4 · p. 17

C.1 VERIFIED CASCADES AND SCALING

  • The sine reference ladder reaches a final finite-order ratio of 4.668805, and seven logistic Transformer flip parameters are numerically verified.The reported roots lie inside independently detected trajectory brackets, and the bracket widths represent spatial/recognition resolution rather than statistical confidence intervals.Source: §C.1 · p. 18, Table 2 · p. 18

C.1.1 PRECISION OF THE VERIFIED CASCADE

  • Float32 evaluation preserves the verified cascade, but the period-64 and period-128 windows are not resolvable at deployment precision.One-sided period probes recover the expected period in six of seven windows in both float32 and float64.Source: §C.1.1 · p. 18

C.1.2 DETAILED CASCADE GEOMETRY

  • Local views verify successive logistic doublings from parent period 4 to emerging period 128 and show physical-coordinate splitting and displacement.The magnifications use separate local roots and crops; dense clouds can include slow transients, and dots are shown only at evaluated parameter values.Source: §C.1.3 · p. 18, Figure 7 · p. 19, Figure 8 · p. 20

C.2.1 STRUCTURAL EMERGENCE AND PHYSICAL ALIGNMENT

  • The logistic fixed-grid detector confirms no doubling through 2,000 updates, one at 3,000, and three at 4,000, when the first transition remains near r = 3.32 rather than the physical r = 3.The sine replay has two detected doublings at 1,000 updates and three at 2,000; counts and locations later vary.Source: §C.2.1 · p. 22
  • From 6,000 to 10,000 updates, logistic fixed-probe OOD one-step MSE falls from 1.10 × 10−3 to 2.41 × 10−4 while closed-loop OOD W1 rises from 0.0296 to 0.0867.The attractor-evolution figures show all 23 closed-loop stages alongside the ground-truth reference for logistic and sine maps.Source: §C.2.1 · p. 22, Figure 12 · p. 22, Figure 16 · p. 24

C.2.4 MATCHED TRAJECTORIES AND COBWEBS ACROSS TRAINING

The appendix tests learned closed-loop dynamics across training stages, observation windows, seeds, contexts, and map families, then compares transition representations. It reports cascades that persist across windows, context-dependent fidelity, and failure to reproduce the quartic cascade.

C.2.4 MATCHED TRAJECTORIES AND COBWEBS ACROSS TRAINING

  • At the 10,000-update checkpoint, after 2,048 burn-in steps and 2,048 iterates, logistic r = 3.8 yields Lyapunov estimates of 0.422 for ground truth and 0.431 for Transformer; at r = 3.9, the model converges to period 3 despite a chaotic reference.Source: §C.2.4 · p. 26

C.3 DEPENDENCE ON THE OBSERVATION WINDOW

  • Across training windows, the cascade persists and finite-order ratios approach the same-definition reference as more levels are resolved, while displaced first flips alter early ratios.The ablation changes only the parameter-sampling interval; verified-flip counts for its new arms are refinement-budget-limited.Source: §C.3 · p. 31, Table 5 · p. 31

C.4 VARIATION ACROSS SEEDS AND CONTEXTS

  • Period agreement is 73.7% versus 24.5% on logistic and 78.1% versus 50.4% on sine for C = 4 versus C = 255; the latter is a length-extrapolation stress test, not a clean context-length probe.The atlas pairs ground truth with seeds and contexts; C excludes the parameter token.Source: §C.4 · p. 32, §C.4.1 · p. 32

C.5 CRITICAL GEOMETRY AND UNIVERSALITY

  • The critical-point analysis fits the local exponent around each maximum and evaluates the Schwarzian where |F'| > 0.05, with a negative Schwarzian and one critical point given as a sufficient condition for a full cascade.The class diagnostics report the fitted order of the maximum.Source: §C.5 · p. 34, Table 6 · p. 34
  • 1 flip was verified for the learned map trained on the quartic family, whose fitted critical exponent is quadratic, so no learned finite-order scaling ratio could be estimated.The result suggests the quadratic-family scaling match reflects the learned map’s universality class rather than identification of the underlying analytic family.Source: §C.5 · pp. 32, 34, Table 7 · p. 34

C.6 TRANSITION ACCURACY AND PARAMETER ALIGNMENT

  • Across seeds, the final checkpoint displaces the reference ladder by −0.060 to −0.031 for seed 42, +0.001 to +0.036 for seed 43, and −0.020 to −0.005 for seed 44, a spread of 0.096 in r.The learned map has parameter-derivative differences and non-zero parameter curvature, consistent with a smooth surrogate rather than a recovered equation.Source: §C.6 · pp. 34–35

D COMPARISONS ACROSS TRANSITION REPRESENTATIONS

  • The representation comparison evaluates recursive dynamical structure and distributional agreement; its direct parameter–state MLP uses three width-64 SiLU hidden layers and 8,577 parameters.The MLP takes (r, x_t) directly, without token communication.Source: D COMPARISONS ACROSS TRANSITION… · p. 35, §D.1 · p. 35

D.2 POLYNOMIAL TRANSITION MODEL

  • NVAR fits current-state polynomial features consisting of monomials r̃^i x̃^j with i + j ≤ d, selecting degree and ridge strength by minimizing ID validation MSE.Each logistic seed selects (d, λ) = (3, 10^-10), and each sine seed selects (7, 10^-10).Source: §D.2 · p. 35

D.3 DISTRIBUTIONAL COMPARISONS

This section compares distributional fidelity, cascade recovery, and parameter-dependent causal effects, including a Lorenz extension that reproduces regime structure and positive Lyapunov exponents beyond fixed-point training support. It also examines how data and model width affect regime-map recovery, validation error, and full-grid stability.

D.3 DISTRIBUTIONAL COMPARISONS

  • The scalar W1 ordering differs from cascade fidelity: the MLP leads on logistic, the Transformer on sine, and the degree-3 polynomial reaches the sampling floor for an exactly expressible update.Reference-versus-reference mean OOD W1 is 0.002104 (SD 0.000092) for logistic and 0.002239 (SD 0.000188) for sine; this quantifies sampling variability, not a strict lower bound on model error.Source: §D.3 · pp. 35–36

D.4 STRUCTURAL COMPARISONS

  • The MLP and Transformer both recover the quadratic cascade, with final finite-order ratios of 4.668296 and 4.668686, respectively, versus 4.669132 for the reference map.Their seventh flips lie at 3.586416 for the MLP and 3.510275 for the Transformer, compared with 3.569891 for the reference, showing opposite-direction seed-specific displacement.Source: §D.4 · p. 36

E.1 REPRESENTATIONAL ANALYSIS

  • For logistic and sine maps, the separable form Fr(x) = rg(x) offers a representational route: encode the state, approximate g, and multiply by r, without implying that training uses this decomposition.A parameter token can be transported through attention or supplied as a channel; these constructions establish possible access routes, not an expressivity advantage or an identified internal circuit. Function-level diagnostics include parameter-Jacobian, parameter-curvature, and state-Jacobian errors; improved one-step loss need not improve orbit stability or long-run distributions.Source: §E.1.1 · p. 38, §E.1.2 · p. 39, §E.1.3 · p. 39

E.2 ACTIVATION PATCHING

  • Parameter mediation varies by seed: logistic seed 44 has patch reductions R0 = 0.522 and R1 = 0.130, while sine seed 44 has R1 = 0.591.Sine seeds 42 and 43 yield approximately 1.000 and 0.969; full attention-output patches test layer-level mediation, not a unique head or feature circuit.Source: §E.2 · p. 40

E.3 CLOSED-LOOP INTERVENTIONS

  • Blocking parameter-to-state attention can markedly change distributions even when blocked four-state models retain two detected flips, so coarse-grid flip counts alone are insufficient measures of fidelity.The blocking scan uses 36 runs across six Transformers and contexts C ∈ {1, 4}; its coarse-grid counts are not comparable to the separate adaptive seven-flip analysis.Source: §E.3 · pp. 40–41

E.4 PARAMETER–TRAJECTORY CORRESPONDENCE

  • Permuting parameter labels makes the cascade disappear and leaves no systematic parameter dependence: mean parameter-derivative mismatch is 0.164 against a reference scale of 0.173, while the state critical exponent remains quadratic (z = 2.03).This contrasts with post-training blocking, which degrades a structure already built during training.Source: §E.4 · p. 41

F.2 REGIME STRUCTURE NEAR CONVENTIONAL PARAMETERS

  • The displayed Lorenz model matches reference fixed/nonfixed labels in 593/625 cells and switching in 362/375 reference switching cells, while the larger-data model scores 600/625 and 369/375, respectively.The displayed model has lower distributional and amplitude errors: mean/median W1 0.08947/0.06861 and normalized standard-deviation error 0.02270, versus 0.14725/0.14473 and 0.04183 for the larger-data model.Source: §F.2 · p. 42

F.3 LYAPUNOV DIAGNOSTICS

  • The frozen Lorenz model and reference agree on Lyapunov-exponent sign at all ten parameter pairs and on order of magnitude, with discrepancies up to a few tenths; fixed-state cases are correctly non-chaotic.The positive model exponents at conventional chaotic parameters support calling the extension chaotic, without establishing matching attractor geometry or dimension.Source: §F.3 · p. 43

F.4 REGIME RECOVERY OVER THE EXTENDED PARAMETER DOMAIN

  • Increasing data raised P2 agreement from 1 to 149 cells at width 64, but none of the four conditions recovered the full P2/P4/P8 structure.These comparisons distinguish fitting training transitions from recovering the joint regime map.Source: §F.4 · p. 44
  • Width 128 attained lower within-condition validation MSE but less reliable full-grid stability in the larger-data condition.The comparisons do not identify a unique cause of extrapolation errors.Source: §F.4 · p. 44
  • With one model seed per condition, the comparisons do not estimate variability across training seeds.Source: §F.4 · p. 44

F.5 SINGLE-PARAMETER TRAINING-REGION COMPARISONS

This section tests whether Lorenz models trained on different parameter regions reproduce sustained double-wing motion at unseen parameters. It specifies the training supports and matched evaluation protocol, while distinguishing transient atlas displays from long-time tests.

F.5 SINGLE-PARAMETER TRAINING-REGION COMPARISONS

  • The mixed condition samples I0 = (0, 1), I1 = (1, 13.9265), and I2 = (13.9265, 24.0579), with 171, 171, and 170 training trajectories, respectively.Initial states are sampled without basin filtering or convergence-based rejection, and short observations may include rotations and cross-wing motion.Source: §F.5 · p. 44
  • The two-layer Transformer uses one state token and one parameter token, predicts normalized state increments, and receives no physical vector field.It has width 64, four attention heads, and is trained for 10,000 updates; the final checkpoint is used without OOD weight selection.Source: §F.5 · p. 44
  • The fixed-checkpoint comparison varies parameter support while holding architecture, model seed, trajectory count, initial-state draws, and optimization budget fixed.Wasserstein distances are scaled using state standard deviations from the original broad training dataset, which supplies no training samples to these conditions.Source: §F.5 · pp. 44–45, Table 16 · p. 45
  • At ρ = 28 and 32, mixed and I2 conditions reproduce sustained double-wing motion, whereas I0 and I1 do not at the fixed checkpoint.This result is specific to the stated protocol and does not establish an architectural impossibility.Source: §F.5 · pp. 44–45
  • The supplementary atlas separates short transients in gray panels from long-time behavior in white panels, but its differing windows are not comparable as convergence tests.The fixed 20-parameter grid spans several Lorenz regimes; for mixed training, its first 15 values are in support and its last five are OOD.Source: §F.5 · p. 45

G.1 EXPERIMENTAL SETUP AND TRAINING SUPPORT

This block defines a generalized Hopf training region and evaluates whether a model trained only on trajectories approaching the origin recovers point-versus-cycle organization. Tests find matching outcomes at nine parameter pairs, while parameter slices show displaced stability boundaries.

G.1 EXPERIMENTAL SETUP AND TRAINING SUPPORT

  • Training parameters are sampled only where μ1 < −max(μ2, 0)^2/4, so all 512 training trajectories have strictly decreasing radii and approach the origin.The model uses Cartesian state and parameter tokens, with a final checkpoint fixed without OOD selection.Source: §G.1 · pp. 45–47

G.2 ATTRACTOR COEXISTENCE AND BASIN TESTS

  • Across nine parameter pairs, all 36 rollouts remain finite through t = 300, and predicted origin-versus-cycle outcomes match the reference: 14 trajectories approach points and 22 approach cycles.These are tests from multiple initial conditions, not a quasistatic hysteresis experiment.Source: §G.2 · p. 47

G.3 PARAMETER SLICES AND STABILITY BOUNDARIES

  • For μ2 = −1 and 1, mean absolute radius errors are 0.00894 and 0.04975, respectively; the model recovers coexistence but displaces its boundaries.At μ2 = 1, the large initial state incorrectly collapses at μ1 = −0.225, while small initial states incorrectly approach cycles at μ1 = −0.05 and −0.025; full unstable-cycle continuation and dense two-dimensional asymptotic classification are not claimed.Source: §G.3 · p. 47

G.4 TRAJECTORY VISUALIZATIONS

  • Figure 5 displays four initial states at each of 64 grid centers over [−0.5, 0.5] × [−1, 1], without burn-in; slow convergence near bifurcations may exceed its display window.Gray marks the exact curved training-support intersection, not a classification inferred from trajectory appearance.Source: §G.4 · p. 47

H.1 SOFTWARE AND HARDWARE

  • Ten neural runs used an NVIDIA RTX 5060 Ti GPU, while two seed-44 MLP runs used CPU execution; mixed-device wall-clock times are not treated as an efficiency comparison.The implementation records Python 3.13.9, PyTorch 2.9.1 with CUDA 12.8, NumPy 2.3.4, and SciPy 1.16.3.Source: §H.1 · p. 47

H.2 EXPERIMENT RECORDS AND INTEGRITY CHECKS

  • The experiment archive retains rollout arrays, diagnostics, checkpoints, configuration records, and verification results, with integrity checks covering saved evaluations and figure provenance.Checks include recovery of generalized Hopf outcomes and recomputation of mixed-Lorenz switching and distributional distances.Source: §H.2 · p. 47

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