Symbolic dynamics and topological entropy in two-parameter tent maps—necessary and sufficient conditions for periodic orbits via kneading sequences and renormalisation

Early stopping is a standard form of implicit regularization in neural sequence models, but criteria based solely on validation loss can become unstable or weakly informative in noisy, non-stationary, or weakly separated regimes. We propose Symbolic Early Stopping (SES), a representation-aware hybrid stopping criterion that monitors the evolution of validation hidden-state organization during training. At each epoch, SES constructs a Mapper-based symbolic abstraction of hidden representations extracted from a fixed monitored layer, transforms latent trajectories into symbol sequences, and summarizes them through a compact set of symbolic–dynamic descriptors capturing sequential complexity, transition uncertainty, and geometric dispersion. These descriptors are aggregated into a single symbolic stability score, which is combined with validation-loss monitoring to detect convergence of the learned representation. We evaluate SES on recurrent, bidirectional recurrent, and encoder-only Transformer architectures across multiple time-series regimes with different levels of structural regularity and noise. The results indicate that SES frequently terminates training substantially earlier than conservative loss-based baselines while preserving a competitive quality–efficiency trade-off relative to oracle validation-based stopping. Robustness experiments under additive input noise show that the symbolic monitoring signal remains informative under moderate perturbations, although its advantage is not uniform across all datasets and model classes. A layer-wise analysis further suggests that useful stopping signals may emerge before the final validation curve fully stabilizes, reflecting earlier organization of latent representations. Overall, SES provides an interpretable and computationally tractable framework for representation-level early stopping in neural sequence modeling.

Aesthetic Anhedonia and Psychotic Hyper-Meaning: Dysregulated Symbolic Closure Dynamics.

Cardiorespiratory coupling (CRC) reflects coordination between heart and lung function, but how it changes with increasing intensity during graded exercise remains unclear. We investigated CRC as real-time covariation between cardiac timing and breath-by-breath pulmonary oxygen uptake (VO2) to test whether coupling strengthens with workload in adolescent athletes. We conducted an observational, within-subject analysis of the ACTES cycling dataset. Eighteen adolescents cycled at 50, 110, and 140 W. Beat-to-beat RR intervals and pulmonary VO2 time series were examined in ultra-short 60 s segments. Fluctuations were summarized using SDRR/RMSSD and SDVO2/RMSSDVO2. CRC was measured using joint symbolic dynamics (JSD; Shannon entropy, SE, and Miller-Madow-corrected entropy, CSE) and the highest normalized cross-correlation (X-Corr). Mean RR decreased and mean pulmonary VO2 increased with workload (both p < 0.0001). SDRR and RMSSD were lower at 110 and 140 W versus 50 W; SDVO2 declined from 50 to 110/140 W, whereas RMSSDVO2 was unchanged. X-Corr increased from 50 to 110/140 W (p ≤ 0.0014). JSD indices decreased as workload increased (SE: global p = 0.139; CSE: global p = 0.029), suggesting tighter CRC. CRC becomes more pronounced with increased workload, aligning with reduced heart rate variability and reflecting vagal withdrawal and reflex responses that improve heart-lung integration in adolescent athletes.

This article investigates how temporary clusters forge, stabilize, and sometimes erode symbolic associations between places and industries, focusing on the lifecycle of Italy’s first international fashion shows, the Pitti shows in Florence (1951–84). Bridging research on geographic (dis)associations and temporary and cyclical clusters, we develop a framework linking knowledge spillovers to image spillovers. We argue that knowledge generated within temporary clusters translates into durable geographic associations only when selectively mediated, narratively anchored, and sustained by high-status participants. Drawing on longitudinal archival research combining participation data and systematic trade press coverage, we reconstruct how the Florence shows initially consolidated Italian fashion as an exportable category, yet later experienced symbolic hollowing. We demonstrate, first, that spillover capacity within temporary clusters is structurally asymmetric: A small number of elite actors disproportionately shape both knowledge release and symbolic value. Second, we show that these actors perform a metonymic function, standing in for broader geographic imaginaries. Third, we theorize temporary clusters as status arenas embedded in hierarchical cyclical event circuits, where centrality is dynamically renegotiated and participant mobility unevenly distributed. Fourth, we highlight the role of symbolic governance and curatorial selectivity, showing how growth without coherence can dilute credibility and accelerate reputational decline. Conceptually, the study advances cultural economic geography by theorizing temporary clusters as historically contingent sites of knowledge and symbolic value production, central to geographic associations yet reliant on fragile alignment among organizers, cultural intermediaries, and high-status participants. Methodologically, we show how historical-geographic reconstruction reveals slow symbolic dynamics obscured in cross-sectional analyses.

Reduced-order models are central to motion planning and control of quadruped robots, yet existing templates are often hand-crafted for a specific locomotion modality. This motivates the need for automatic methods that extract task-specific, interpretable low-dimensional dynamics directly from data. We propose a methodology that combines a linear autoencoder with symbolic regression to derive such models. The linear autoencoder provides a consistent latent embedding for configurations, velocities, accelerations, and inputs, enabling the sparse identification of nonlinear dynamics (SINDy) to operate in a compact, physics-aligned space. A multi-phase, hybrid-aware training scheme ensures coherent latent coordinates across contact transitions. We focus our validation on quadruped jumping—a representative, challenging, yet contained scenario in which a principled template model is especially valuable. The resulting symbolic dynamics outperform the state-of-the-art handcrafted actuated spring-loaded inverted pendulum (aSLIP) baseline in simulation and hardware across multiple robots and jumping modalities.

Controlled breathing is a hemodynamic maneuver known to influence both baroreflex and chemoreflex sensitivity. This study investigated the impact of respiratory-driven oscillations on the relationship between cerebral autoregulation and autonomic nervous system (ANS) activity. Sixty-one volunteers (median age: 23years) underwent noninvasive measurements of arterial blood pressure (ABP), cerebral blood velocity (CBv), end-tidal CO2 (EtCO2), and respiratory rate during spontaneous breathing and during three 5-min sessions of controlled breathing at 6, 10, and 15bpm. Cerebral autoregulation was assessed using transfer function analysis by calculating phase shift (PS) and gain between ABP and CBv in the very low frequency (VLF; 0.02-0.07Hz) and breathing frequency (BF; [0.1, 0.17, 0.25] ± 0.02Hz) ranges. ANS activity was assessed using baroreflex sensitivity (xBRS), heart rate variability (HRV) metrics in time and frequency domains, and entropy-based parameters. Cardiovascular coupling was assessed using the joint symbolic dynamics of beat-to-beat pulse interval and systolic blood pressure. Increasing respiratory rate led to decreased EtCO2 (p < 0.001), diminished cardiovascular coupling (p < 0.01), and reduced systemic ABP control,as indicated by lower normalized low-frequency HRV and xBRS (both p < 0.001). A linear mixed-effects model, adjusted for EtCO2 and respiratory rate, showed that PS at VLF and BF was modulated byANS metrics, whereas gain was mainly affected by respiratory parameters, with a nonsignificant contribution from ANS. Higher respiratory rates reduced cardiovascular coupling, diminished ANS activity, and modified its interaction with cerebral autoregulation. Respiratory parameters should be considered when assessing ANS-cerebral autoregulation relationship.

This article maps and investigates the network of intertextual connections activated in Agatha Christie's standalone psychological thriller Endless Night (1967), ranging from poems by William Blake and T. S. Eliot to the tarot. While most of these literary and cultural referents participate in the symbolic dynamics of this text through explicit quotes, others are evoked more subtly; yet their presence can be traced and their role analysed. The resulting literary inquiry aims to shed light onto the ways in which Endless Night conjures up an ambience of superstition while exploring the theme of evil in close connection with its unreliable narrator's lack of self-awareness. Probing the contact zone between psychological realism and spiritual symbolism, the novel even gestures back towards the realm of Greek tragedy, in which the psychological thriller is firmly rooted, as shown by the archetypal story of Oedipus.

This study analyzes the feasibility of tourism projects from economic, social, and cultural perspectives, based on This article aims to analyze the Salinópolis Lighthouse, located on the northeastern coast of Pará, highlighting its historical trajectory, architectural structure, and cultural and tourist relevance for the municipality of Salinópolis. The study was based on bibliographic and documentary research, complemented by field observation, to understand the lighthouse’s role in the urban, economic, and symbolic dynamics of the region. The findings indicate that, beyond its original function as a navigational aid, the lighthouse has become one of the main identity symbols of the municipality, exerting a strong influence on local tourism and the collective memory of its residents. It is concluded that its preservation is essential not only as a historical heritage but also as a structural element of Pará’s coastal cultural landscape.

Although synchronization has been extensively studied, key processes underlying its emergence in complex networks have remained hidden by the use of global order parameters. Here, we uncover how the route to synchronization unfolds through a sequential transition between two well-known but previously unconnected phenomena: (CI) and (IS). Using a symbolic dynamics approach, we show that CI develops as a collective yet unsynchronized exploration of different domains of the high-dimensional attractor of the network, whose dimension is reduced as the coupling increases, ultimately collapsing into the reference chaotic attractor of an individual unit. At this stage, the IS emerges as irregular alternations between synchronous and asynchronous phases. These phenomena are mutually exclusive, each dominating a distinct coupling interval and governed by different mechanisms. Furthermore, network structural heterogeneity enhances itinerant behavior since access to different domains of the attractor depends on the nodes’ topological roles. The CI-IS crossover occurs within a consistent coupling interval across models and topologies. Experiments on electronic oscillator networks confirm this two-step process, establishing a unified framework for understanding the route to synchronization in complex systems.

Abstract We consider a family of piecewise contractions admitting a rotation number and defined for every x ∈ [ 0 , 1 ) by f ( x ) = λ x + δ + d θ a ( x ) ( mod 1 ) , where λ ∈ ( 0 , 1 ) , d ∈ ( 0 , 1 − λ ) , δ ∈ [ 0 , 1 ] , a ∈ [ 0 , 1 ] and θ a ( x ) = 1 if x ⩾ a and θ a ( x ) = 0 otherwise. In the special case where a = 1, the family reduces to the well studied ‘contracted rotations’ x ↦ λ x + δ ( mod 1 ) , which are 2-interval piecewise λ -affine contractions when δ ∈ ( 1 − λ , 1 ) . Considering a ∈ ( 0 , 1 ) allows maps with an additional discontinuity, that is, 3-interval piecewise λ -affine contractions. Supposing λ and d fixed, for any ρ ∈ ( 0 , 1 ) and α ∈ [ 0 , 1 ] , we provide the values of the parameters δ and a for which the corresponding map has rotation number ρ , and a symbolic dynamics containing that of the rotation R ρ : [ 0 , 1 ) → [ 0 , 1 ) of angle ρ with respect to the partition given by the positions of 1 − ρ and α in [ 0 , 1 ) . This enables in particular to determine the maps that have a given number of periodic orbits of an arbitrary period, or a Cantor set attractor supporting a dynamics of a given complexity.

The Collatz map is investigated from a nonlinear-dynamics perspective with emphasis on the structure of its iterative orbits. By embedding integers within Sharkovsky’s ordering, odd initial values are shown to be sufficient for a complete characterization of dynamics. A “direction-phase” decomposition is introduced to separate iterative orbits into upward and downward phases, yielding a family of recursive functions parameterized by the number of upward phases. This formulation reveals a logarithmic scaling relation between the total iteration count and the initial value, confirming finite-time convergence to the period-three orbit. The Collatz dynamics is further shown to be dynamically equivalent to a binary shift map, whose ergodicity implies inevitable evolution toward attractors, thereby reinforcing convergence. Numerical analysis indicates that attraction basins follow a power-law distribution and display pronounced self-similarity. Moreover, odd integers grouped by upward-phase counts are found to follow Gamma statistics. Beyond its research implications, the framework provides a concise pedagogical case study illustrating how nonlinear dynamics, symbolic dynamics, and statistical characterization can be integrated to analyze a classical discrete problem.

In our previous work, we introduced symbolic Extended Dynamic Mode Decomposition (EDMD), a method combining EDMD with symbolic dynamics to estimate the Koopman operator using dictionaries constructed from cylinder sets of a generating partition. That work required knowledge of an invariant measure and was validated on systems with known partitions. In this paper, we make two significant advances. First, we prove that our method can be performed with no knowledge of any invariant measures for the system, allowing any sampling method that can see all open sets. Second, we prove that a byproduct of our method is a sequence of measures μm, which approximate the measure of maximal entropy (MME) so long as the dynamical system is equivalent to a Markov or Sofic symbolic shift. We present numerical results on three newly tested dynamical systems: the Liverani-Saussol-Vaienti map, a non-Markov piecewise linear map, and the Arnold Cat Map. We obtain good results for the estimation of spectral data and the MME for each system. In addition, we present some new data-driven methods for estimation of symbolic cylinder sets for the Cat Map.

In countries with intense solar irradiation such as Brazil, the skin operates simultaneously as a biological organ, a phenotypic marker of aging, and a visible marker of social belonging [1]. Marks produced by chronic photoexposure acquire meanings that extend beyond cutaneous physiology. They become embedded in social, economic, and symbolic dynamics that influence behavior, aesthetic expectations, and subjective experiences of aging. Amid ongoing ethical debates on whether aging should be viewed as pathological or socially constructed, it is relevant to examine how photoaged skin in Brazil may function as a visible stigma linked to social class and structural inequality [2, 3]. This phenomenon is deeply rooted in the structural hierarchy of labor and racialized class stratification, where cutaneous health outcomes are inextricably linked to systemic inequalities that determine environmental exposure and access to protective resources [4]. A significant clinical paradox emerges: while these patients pursue idealized digital textures, they frequently present with Fear of Overfilling (FOF) [9], a clinically meaningful anticipatory anxiety frequently observed in aesthetic practice centered on iatrogenic identity loss that acts as a psychological barrier to treatment. The coexistence of these phenotypes reveals an ethical dilemma for cosmetic dermatology: while one population seeks to reduce stigmata linked to structural inequality, the other navigates a tension between unrealistic digital ideals and the fear of clinical distortion [9]. Sun exposure in Brazil is marked by ambivalence. While culturally associated with leisure, it often reflects occupation and unequal access to care [1, 2]. Conversely, smooth skin signals aesthetic capital. Treating photoaging may partially reduce perceived inequalities, while lack of access tends to reinforce them [2, 8, 10]. At the same time, the hypervisibility of distorted results in digital environments has reshaped public perception. The widespread use of filters modifies subjective perceptions of normality [3], while “Zoom dysmorphia” has intensified self-scrutiny [4, 5]. Among clinicians, repeated exposure to filtered demands may induce “perception drift,” where normal variations are reinterpreted as defects [8]. Furthermore, the normalization of Overfilled Face Syndrome (OFS) in social media reinforces FOF, challenging the therapeutic alliance as patients overestimate the risk of unnatural outcomes [9, 11]. Beyond the aesthetic and psychological dimensions, it is imperative to acknowledge that photoaging, regardless of social class, is the primary clinical precursor to skin cancer. Both phenotypes are subject to the cumulative and deleterious effects of ultraviolet radiation, which remains a critical public health concern in Brazil due to the high incidence of basal cell carcinoma, squamous cell carcinoma, and melanoma [6]. Photoaging in Brazil is largely a socially stratified dermatologic phenomenon. Clinical management thus has a dual mission: preventing malignancy while reducing the symbolic burden of occupational exposure. Within this scenario, the dermatologist functions as a steward of facial identity. By integrating the D.I.F.A. framework and addressing FOF, the clinician can mitigate defensive attitudes and support patient confidence in safe rejuvenation, promoting an aesthetic grounded in health, diversity, and authenticity. Naturally, class-related stratification of photoaging does not imply determinism, and individual variance remains substantial. Identity preservation refers not to resisting rejuvenation, but to preserving facial recognizability, proportional coherence, and cultural self-perception within socially realistic expectations. By framing photoaging as both a biological condition and a socially stratified phenomenon, this commentary invites clinicians to move beyond purely technical intervention and engage with the psychosocial and oncological realities that shape aesthetic demand. The author confirms that the ethical policies of the journal, as noted on the journal's author guidelines page, have been adhered to. No ethical approval was required as this is a review article with no original research data. The author declares no conflicts of interest. The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

Developing the analytical model of a multibody system is often the initial step in control and optimization. The analytical model (equations of motion) describes a system’s time evolution under specified forcing conditions. Although developing these equations is easy for simple systems, this process becomes more complex for systems composed of multiple bodies. Deriving equations of motion for complex multibody systems requires specialized expertise in multibody dynamics, is time-consuming, and is susceptible to error. To address this issue, this paper presents an open-source, easy-to-use, systematic framework to derive symbolic equations of motion in both Python and MATLAB using the joint coordinate formulation. This formulation results in a set of ordinary differential equations that use the minimum set of coordinates needed to model a system. The symbolic representation provides better insight into the influence of design parameters on system performance, facilitates sensitivity analysis and parameter studies, and supports direct implementation of control and optimization routines. The tool enables numerical simulation for specified parameter sets, is modular for straightforward integration with other tools and libraries, and allows incorporation of hydrodynamics, mooring, and other external forces. The result is a reproducible, extensible pipeline for modeling, simulation, and design of complex multibody systems. The proposed tool is versatile and can be applied to domains such as robotics, control, and design. In addition, we integrated external libraries that provide capabilities for modeling offshore systems such as underwater robots and marine energy converters.

It is a well-known fact that studying entropies in physiological systems, like the cardiovascular system, to quantify dynamic complexity is a well-established field that aids in autonomous health condition evaluation, reducing the need for continuous expert monitoring. Pathological cardiac conditions such as Congestive Heart Failure (CHF) patients and Atrial Fibrillation (AFib) often pose a risk of co-occurrence, which can be lethal. The primary goal of this study is to collectively analyse the symbolic dynamics of human heart rhythms obtained from normal (healthy), CHF, and AFib subjects, using Normalised Corrected Shannon Entropy (NCSE). This study performs joint differentiation of CHF and AFib conditions, instead of performing individual detection as done in previous studies. Initially, the RR time series data, both filtered and raw, from ECGs of 15 subjects are symbolised and converted into series of codes. These code series are further evaluated to compute absolute and mean NCSE values across different thresholds and word lengths. In all the scenarios, the NCSE outcomes showed significant differences between normal and diseased cardiovascular systems, with AFib exhibiting the highest entropy, followed by normal and CHF systems. The results of NCSE and Shannon entropy (SE) were compared and it was observed that NCSE demonstrated superior performance in separating the three conditions, as well as in executing joint detection. The NCSE results have also been statistically analysed and validated using surrogate data analysis, one-way ANOVA and pairwise t-tests. Furthermore, this approach did not require any classification algorithms or noise cancellation methods, indicating robustness to noise.

Anomaly detection in system logs has attracted considerable interest with the development of deep learning and big data. Many methods and approaches have achieved good results in log anomaly detection, but exploiting the rich dependencies hidden in log data remains an extremely challenging task. Recently, Transformer-based models, such as LogBERT, represent a strong and promising foundation thanks to their ability to capture rich contextual information within log sequences. The limitation of these models is that they mainly treat logs as a sequence of discrete tokens, neglecting the crucial and continuous information embedded in the temporal and numerical dynamics between events. This limitation can lead to a high false positive rate, thus reducing the practical utility of the model. In this work, we propose DualBERT, a hybrid framework designed to address this gap. Our approach leverages a dual-model architecture: 1) a primary Transformer (LogBERT) to analyze the log’s symbolic key sequence; 2) a parallel and specialized parameter model represented by an LSTM-based regressor that analyzes inter-event features such as temporal deltas and cyclic patterns. The anomaly scores of both models are then fused using an XGBoost classifier to produce a final robust decision. We conduct a rigorous and fully supervised evaluation on the HDFS benchmark dataset. Our results demonstrate that DualBERT achieves an F1 score of 98.75%, significantly surpassing the satisfactorily evaluated LogBERT benchmark (98.29%). This improvement is mainly due to a substantial increase in accuracy (from 96.64% to 97.70%) and a 43% reduction in the false positive rate. Our results confirm that merging symbolic sequence analysis with dedicated inter-event dynamic modeling is an effective strategy to create a more reliable and practical system for real-world logarithmic anomaly detection.

This article offers a critical reading of the material and symbolic dynamics of power relations, based on Geography and the framework of intersectionality. It analyzes how geographic space is shaped by social and cultural practices that reproduce structural inequalities of gender, race, class, and sexuality. Using a conceptual bibliographic methodology, the study conceives space as a field of symbolic and institutional disputes, where urban arrangements function as devices of control, discipline, and also resistance. It highlights the subalternization and invisibilization of certain bodies, especially women’s, and the role of violence in the normalization of spatialities. The article critiques Eurocentric and objectivist paradigms of traditional geography, advocating for feminist epistemologies and situated knowledge as theoretical-methodological alternatives. It concludes that thinking Geography through the intersections of body, power, and space is a political and ethical act, essential to understanding inequalities and constructing more just, plural, and transformative geographies.

Given a finite word $ w $, Guibas and Odlyzko (J. Combin. Theory Ser. A, 30, 1981,183-208) showed that the autocorrelation polynomial $ \phi_w(t) $ of $ w $, which records the set of self-overlaps of $ w $, explicitly determines for each $ n $, the number $ |\mathcal{B}_n(w)| $ of words of length $ n $ that avoid $ w $. We consider this and related problems from the viewpoint of symbolic dynamics, focusing on the setting of $ X_{\{w\}} $, the space of all bi-infinite sequences that avoid $ w $. We first summarize and elaborate upon (J. Combin. Theory Ser. A, 30, 1981,183-208) and other work to show that the sequence $ |\mathcal{B}_n(w)| $ is equivalent to several invariants of $ X_{\{w\}} $. We then give a finite-state labeled graphical representation $ L_w $ of $ X_{\{w\}} $ and show that $ w $ can be recovered from the graph isomorphism class of the unlabeled version of $ L_w $. Using $ L_w $, we apply ideas from probability and Perron-Frobenius theory to obtain results comparing features of $ X_{\{w\}} $ for different $ w $. Next, we give partial results on the problem of classifying the spaces $ X_{\{w\}} $ up to conjugacy. Finally, we extend some of our results to spaces of multi-dimensional arrays that avoid a given finite pattern.

Code ecologies: Integrating cultural legitimacy analysis in sustainability transitions