Initial-State Precision as a Predictive Resource: From Tori to Strange Nonchaotic Attractors and Chaos
Abstract
How much initial-state precision is required to predict a nonlinear system to a prescribed accuracy over a finite horizon? We formulate this inverse prediction problem through a one-shot resource BN, defined as the number of binary refinement bits required in the initial state by a specified local sensing architecture. Within a common scale-separable error-growth regime, changing a fixed tolerance changes BN only by an N-independent offset. An exactly solvable circle/Chebyshev map provides the chaotic calibration BN=Nlog2 m+O(1). We then apply the framework to the quasiperiodically forced logistic map under phase-only initial uncertainty. Over the resolved horizons, representative smooth-torus, strange-nonchaotic-attractor (SNA), and chaotic regimes exhibit a three-level precision-resource hierarchy: bounded, logarithmic-like, and linear-like growth, respectively. For the SNA, algebraic phase sensitivity GNop∼ Nμ implies an admissible initial uncertainty of order N−μ, and therefore a precision cost proportional to log N. Phase-grid refinement shows that the sampled worst-case maxima are numerically stable with increasing phase resolution. Near the torus--SNA fractalization point, derivative-based phase sensitivity can already grow while finite perturbations remain below a fixed operational tolerance for more than 106 iterations, revealing exceptionally long activation horizons. Thus an SNA can occupy an intermediate finite-precision resource class between a smooth torus and ordinary chaos, while its operational manifestation can occur on a much longer timescale than its derivative-based sensitivity.
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BibTeX
@misc{kim2026initialstate,
title = {{Initial-State Precision as a Predictive Resource: From Tori to Strange Nonchaotic Attractors and Chaos}},
author = {Song-Ju Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38674},
eprint = {2609.38674},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38674},
}
RIS
TY - UNPB
TI - Initial-State Precision as a Predictive Resource: From Tori to Strange Nonchaotic Attractors and Chaos
AU - Song-Ju Kim
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38674
UR - https://arxiv.org/abs/2609.38674
AB - How much initial-state precision is required to predict a nonlinear system to a prescribed accuracy over a finite horizon? We formulate this inverse prediction problem through a one-shot resource B_N, defined as the number of binary refinement bits required in the initial state by a specified local sensing architecture. Within a common scale-separable error-growth regime, changing a fixed tolerance changes B_N only by an N-independent offset. An exactly solvable circle/Chebyshev map provides the chaotic calibration B_N=Nlog_2 m+O(1). We then apply the framework to the quasiperiodically forced logistic map under phase-only initial uncertainty. Over the resolved horizons, representative smooth-torus, strange-nonchaotic-attractor (SNA), and chaotic regimes exhibit a three-level precision-resource hierarchy: bounded, logarithmic-like, and linear-like growth, respectively. For the SNA, algebraic phase sensitivity G_N^op∼ N^μ implies an admissible initial uncertainty of order N^−μ, and therefore a precision cost proportional to log N. Phase-grid refinement shows that the sampled worst-case maxima are numerically stable with increasing phase resolution. Near the torus--SNA fractalization point, derivative-based phase sensitivity can already grow while finite perturbations remain below a fixed operational tolerance for more than 10^6 iterations, revealing exceptionally long activation horizons. Thus an SNA can occupy an intermediate finite-precision resource class between a smooth torus and ordinary chaos, while its operational manifestation can occur on a much longer timescale than its derivative-based sensitivity.
ER -