Volume 35, Number 3 (2026)

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A Reproducible Expository Analysis of the Logistic Map: Stability, Period Doubling, Conjugacy and Symbolic Dynamics
Joshua Owolabi Adeleke

The logistic map is the standard first example in introductions to discrete dynamical systems, but its treatment is scattered across sources, and the numerical practice behind its familiar pictures is rarely documented. A self-contained account is assembled here. Complete derivations are given for the fixed-point and period-two stability windows, including the frequently misstated marginal parameter, where the fixed point is non-hyperbolic yet still attracting. Period-doubling thresholds are determined to 10 significant figures by a simultaneous Newton solve, and ratios formed from four-decimal thresholds are shown incapable of converging to the Feigenbaum constant. A conjugacy with the tent map is proved and combined with the itinerary map to establish that the fully developed case is chaotic in the sense of Devaney, with topological entropy and Lyapunov exponent both equal to the natural logarithm of two. The numerical tables are independently cross-checked against exact formulas, stability criteria or higher-precision computations.

Keywords: logistic map; chaos; period doubling; Feigenbaum constant; Lyapunov exponent; tent map; symbolic dynamics; reproducible numerics

Cite this publication as:
J. O. Adeleke, “A Reproducible Expository Analysis of the Logistic Map: Stability, Period Doubling, Conjugacy and Symbolic Dynamics,” Complex Systems, 35(3), 2026 pp. 203–235.
https://doi.org/10.25088/ComplexSystems.35.3.203

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Personal Space and Collective Behavior in Crowd Disasters
Igor Lugo and Martha G. Alatriste-Contreras

The importance of personal space (PS) in densely crowded situations is commonly underestimated. However, self-awareness of PS can help prevent harm and reduce risky situations. This paper explores the use of PS as the key factor for designing simple computational models to prevent crushing events. An agent-based model is developed to relate transitional rules with Schelling’s spatial proximity model of segregation. An exploratory data analysis and subsequent model validation find that the dynamics of crowd events associated only with the individual perception of PS show significant statistical regularities between density situations. These results suggest that crushing events in densely crowded situations are highly probable and that lower density levels are key for delaying the onset of such disasters.

Keywords: personal space; collective behavior; crowd disasters; complex systems; agent-based model; Schelling’s spatial model; cellular automata; data analysis

Cite this publication as:
I. Lugo and M. G. Alatriste-Contreras, “Personal Space and Collective Behavior in Crowd Disasters,” Complex Systems, 35(3), 2026 pp. 237–265.
https://doi.org/10.25088/ComplexSystems.35.3.237

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Distinct Regimes of Predictive Accessibility in Elementary Cellular Automata
Rodrigo A. Garrido

Predicting the evolution of deterministic dynamical systems is often treated as a problem of model capacity, architecture design or training procedure. However, in local dynamical systems, long-term predictability may be constrained not only by the expressivity of the predictor, but also by the way dynamically relevant information propagates, disperses and remains accessible under finite learned representations. In this paper, we investigate this issue using elementary cellular automata (ECAs) as a minimal setting in which local deterministic rules generate qualitatively different global dynamics. Rather than framing prediction failure as a generic limitation of machine learning models, we study the emergence of distinct regimes of predictive accessibility: the extent to which information relevant for future states can be captured and exploited by finite learned representations under a given observational framework. Using neural predictors with controlled receptive fields, we show that different cellular automaton rules exhibit qualitatively different forms of predictive behavior. Rule 30 displays an abrupt loss of accessible predictive structure beyond a receptive-field-dependent horizon. Rule 45 exhibits a smoother degradation regime. Rule 90, despite being deterministic and linear over ${\mathbb{F}}_2$, remains near chance-level performance under the tested representations, consistent with distributed XOR-like dependencies that are difficult to compress statistically. Rule 110, by contrast, retains substantial predictive structure over longer horizons, suggesting that complex dynamics need not imply immediate loss of predictability. These results indicate that prediction limits in ECAs are not adequately described as a uniform consequence of error accumulation or insufficient model capacity. Instead, they reflect rule-dependent differences in information structure, causal accessibility and statistical compressibility. The proposed framework reframes predictability as an empirical property of the interaction between dynamics and representation, rather than as a purely algorithmic capability.

Keywords: elementary cellular automata; predictive accessibility; dynamical systems; machine learning; information propagation; statistical compressibility

Cite this publication as:
R. A. Garrido, “Distinct Regimes of Predictive Accessibility in Elementary Cellular Automata,” Complex Systems, 35(3), 2026 pp. 267–298.
https://doi.org/10.25088/ComplexSystems.35.3.267