Different methodologies have been developed for the analysis and study of dynamical systems, including both theoretical models and natural systems. Examples span a wide range of applications, such as astronomy, financial and economic time series, biophysical systems, physiological phenomena, and Earth sciences, including seismicity and climatic processes. The study of these complex systems is commonly based on the analysis of the signals they generate, using mathematical tools to extract relevant information. A broad spectrum of mathematical disciplines converges in this context, including stochastic, probability and statistical theory, entropic and informational measures, fractal and multifractal analysis, natural time analysis, modeling of non-linearity and recurrence methods, generalized entropies, non-extensive systems, machine learning, and high-dimensional and multivariate complexity. Research in this area is largely focused on the characterization of complex systems, providing indicators of determinism or stochasticity, distinguishing between regularity, chaos, and noise, and identifying topological as well as disorder-regularity features. In addition, short- and long-term forecasting, together with the identification of short- and long-range correlations, play a central role in such characterization. To address these objectives, numerous mathematical tools have been developed for the analysis of time series and point processes, each designed to capture specific signal properties. In this work, many of the most important tools used in time series analysis are compiled and reviewed, highlighting their main characteristics and the different types of complex systems to which they have been applied.

This study investigates the dynamic degradation of Evapotranspiration (ET) time series induced by Xylella fastidiosa (X.f.) infection using the Complexity-Entropy Causality Plane (CECP). By evaluating MODIS ET datasets from infected sites (X2015, X2016, and X2017) against a healthy one (Matera), we characterize the "infection" signature as a fundamental transition from structured, deterministic dynamics toward stochastic disorder. Our findings reveal that infected signals undergo a simultaneous increase in permutation entropy (H) and a significant decrease in statistical complexity (C). This "loss of complexity" serves as a robust marker of ecosystem disturbance, overshadowing the underlying deterministic components of the ET signal. We demonstrate that with the embedding dimension d(x) = 4 there is a good diagnostic sensitivity, capturing intricate fourth-order temporal correlations with Mahalanobis distance D-M > 1.3 and classification accuracy AUC = 0.83. These results establish the H-C pair at d(x )= 4 as an optimal diagnostic tool for the detection of subtle anomalies in MODIS ET data, suggesting the CECP as a robust feature space for monitoring environmental signals, providing a reliable framework for the early detection and quantification of site-specific disturbances.

Asymmetry in persistence introduces an additional directed force that alters the dynamics of stochastic processes, potentially affecting the behavior of extremes in their realizations (time series). In this work, we investigate these effect, unexplored so far, using a modified Langevin model that incorporates an asymmetric persistence mechanism. Extremes are defined through run theory and analyzed using informational measures, alongside examining the topological properties of visibility graphs constructed from the point processes of extremes. Our results reveal a systematic influence of asymmetry on the behavior of extremes—both their size and magnitude decrease with increasing asymmetry, while their degree of order, quantified by Shannon entropy, increases.

The statistical method of visibility graph (VG) has been becoming widely employed for analyzing the topological properties of signals in various scientific fields. The VG method is based on transforming time series into graphs or networks, whose nodes are the series values linked between each other by edges drawn on the basis of specific ‘visibility’ criteria. The number of the edges departing from each node is the degree of that particular node. In this paper, the VG is utilized to analyze the topological properties of Moderate Resolution Imaging Spectroradiometer (MODIS) satellite evapotranspiration time series of pixels covering olive orchards in various areas of southern Italy, some of which are affected by Xylella fastidiosa infection. Xylella fastidiosa is a very dangerous phytobacterium that causes desiccation and then death of olive trees. By converting the investigated MODIS time series in networks by using the VG, we focused on evaluating the discrimination capability between infected and uninfected sites by analyzing two informational quantities: the fisher information measure (FIM) and the Shannon entropy of the connection degree distribution. The results of the receiver operating characteristic analysis indicate that the Shannon entropy of the degree distribution demonstrates strong discrimination capability between infected and healthy pixels, while the FIM show a much less discrimination power. These findings suggest that the VG method combined with information theory is highly effective in identifying satellite pixels covering infected vegetated areas and holds significant potential as a valuable tool for infection detection of large-scale areas.

Geophysics represents a dynamic research field that delves into the intricate physical properties and processes that shape the Earth and its surrounding space environment [...]

The complexity–entropy causality plane (CECP), a model-free information-theoretic framework based on permutation entropy H and statistical complexity C, is applied to Sentinel-2 multispectral time series (2017–2024) to discriminate pixels from a Pinus pinea stand infested by Toumeyella parvicornis — an invasive insect first detected in Italy in 2015 that has since caused severe damage to Mediterranean coastal pine forests — from those of a healthy reference forest. The two study sites, Castel Porziano and Follonica, share similar environmental and climatic conditions along the Tyrrhenian coast of central Italy. At embedding dimension dx=3, at which CECP coordinates are robust to the 40% gap rate present in the preprocessed dataset, the NIR band achieves the strongest discrimination (AUC=0.881, ∣d∣≥1.46), followed by the green (AUC=0.824) and blue (AUC=0.738) bands; the red and SWIR bands yield AUC values near the random classifier baseline. A spectrally structured sign inversion of Cohen's d between the blue and the green–NIR bands reflects the distinct physiological mechanisms through which infestation alters the temporal dynamics of reflectance in different spectral regions. At dx=3, H and C carry virtually redundant discriminatory information, so that permutation entropy alone suffices for classification. These results demonstrate that the CECP provides a scalable, assumption-free framework for characterising infestation-driven changes in the temporal dynamics of satellite reflectance in Mediterranean forest ecosystems.

The time fluctuations of forest fires occurring in Basilicata, a region situated in Southern Italy, between 2004 and 2023 were investigated using various analytical approaches. Analysis revealed a clustering of fire occurrences over time, as indicated by a significantly high coefficient of variation. This suggests that the fire sequence does not follow a Poisson distribution and instead exhibits a clustered structure, largely driven by the heightened frequency of events during the summer seasons. The analysis of monthly forest fire occurrences and total burned area indicates a significant correlation between the two. This correlation is reinforced by shared patterns, notably an annual cycle that appears to be influenced by meteorological factors, aligning with the yearly fluctuations in the region’s weather conditions typical of a Mediterranean climate. Furthermore, the relationship between the Standardized Precipitation Evapotranspiration Index (SPEI) and forest fires revealed that the accumulation period of the SPEI corresponds to the cycle length of the fires: longer cycles in fire occurrences align with higher accumulation periods in SPEI data.

The main anthropogenic sources of air pollution in big cities are vehicular traffic and industrial activities. The emissions of primary pollutants are produced directly from the combustion of fossil fuels of vehicles and industry, whilst the secondary pollutants, such as tropospheric ozone (O3), are produced from precursors like Carbon monoxide (CO), among others, and meteorological factors such as radiation. In this study, we analyze the time series of CO and O3 concentrations monitored by the RAMA program between 2019 and 2023 in the southwest of the Mexico City Metropolitan Area, encompassing the COVID-19 lockdown period declared from March to September–October 2020. After removing cyclic patterns and normalizing the data, we applied informational and topological methods to investigate variability changes in the concentration time series, particularly in response to the lockdown. Following the onset of lockdown measures in March 2020—which led to a significant reduction in industrial activity and vehicular traffic—the informational quantities NX and Fisher Information Measure (FIM) for CO revealed significant shifts during the lockdown, while these metrics remained stable for O3. Also, the coefficient of variation of the degree CVk, which was defined for the network constructed for each series by the Visibility Graph, showed marked changes for CO but not for O3. The combined informational and topological analysis highlighted distinct underlying structures: CO exhibited localized, intermittent emission patterns leading to greater structural complexity, while O3 displayed smoother, less organized variability. Also, the temporal variation of the FIM and NX provides a means to monitor the evolving statistical behavior of the CO and O3 time series over time. Finally, the Visibility Graph (VG) method shows a behavioral trend similar to that shown by the informational quantifiers, revealing a significant change during the lockdown for CO, although remaining almost stable for O3.

Super-shear ruptures, characterized by velocities exceeding the shear wave speed, were first predicted theoretically and later observed in laboratory experiments. While a few tectonic earthquakes have been reported as super-shear, most involve strike-slip faults, including the 2023 Mw 7.5 Kahramanmaraş earthquake (Türkiye) and transient phases of the Mw 7.8 event. However, natural ruptures propagate through complex, rough fault systems - deviating from idealized smooth interfaces - resulting in heterogeneous slip, stress drops, and rupture jumps. Additionally, the expected high-frequency spectral signature of super-shear ruptures often conflicts with observations. To reconcile these discrepancies, we propose a generalized interpretation of super-shear events, where observed super-shear velocities arise not only from continuous rupture fronts but also from dynamically triggered multi-focal ruptures along strike. We explore how fault rheology modulates rupture speed and introduce a triggering mechanism driven by P-wave perturbations. Our model also predicts Mach cones detected teleseismically during super-shear earthquakes such as the Kahramanmaraş doublet, while it suggests they should not be observed locally in the case of super-shear cascading rupture envelopes. We show that both the Kahramanmaraş 2023 events initiated cascading instabilities, with dynamic stress transfers propagating rupture across fault patches. High-frequency (>10 Hz) P-wave pulses mark transitions between patches, identified via accelerometric waveform analysis. Our findings support the idea that even minor stress perturbations can trigger near-instantaneous dynamic ruptures, posing implications for early-warning algorithms.

Seismic activity clusters in space and time due to stress accumulation and static and dynamic triggering. Therefore, both moderate and large magnitude events can be preceded by smaller events and also seismic swarms can occur without being succeeded by major shocks – which represents the vast majority of cases. Unveiling if seismic activity can forewarn mainshocks, being somewhat distinguished by swarms, is an issue of crucial importance for the development of short-term seismic hazard. The analysis of thousand clusters of seismicity before mainshocks in Southern California and Italy highlights that the surface over which selected seismic activity spreads is positively correlated with the magnitude of the impending mainshock, as well as the cumulative seismic moment, the number of earthquakes, the variance of magnitude and its entropy, while no significant difference is observed in the duration, seismic rate, and trends of magnitudes and interevent times between foreshocks and swarms. Our interpretation is that crustal volumes and fault interfaces host more and more correlated seismicity as they become unstable, and some properties of seismic clusters may mark their state of stability. For this reason, large mainshocks tend to occur in more extended correlated regions and because of the scaling of maximum magnitudes with the size of unstable faults. Considering this, the recording of more numerous and energetic cluster activity before mainshocks than during swarms is also reasonable. In recent years, our ability to track seismic clusters has improved outstandingly, so that their structural and statistical characterization can be performed almost in real time. Therefore, it may be possible to compare the current features of the active seismic cluster with the cumulative distribution functions of past seismicity. However, we would like to stress that foreshocks should not be considered as precursors in the sense that neither they forewarn mainshocks, nor they are physically different from swarms: the precursor is not in seismic activity itself, but in the development of mechanical instability within crustal volumes.

This study investigates the capability of Sentinel-1 (S1) SAR time series to identify vegetation sites affected by pest infestations. For this purpose, the statistical method of the Fisher–Shannon analysis was employed to discern infected from unifected forest trees. The analysis was performed on a case study (Castel Porziano) located in the urban and peri-urban areas of Rome (Italy), which have been significantly impacted by Toumeyella parvicornis (TP) in recent years. For comparison, the area of Follonica (Italy), which has not yet been affected by this insect, was also analyzed. Two polarizations (VV and VH) and two orbit types (Ascending and Descending) were analyzed. The results, supported by Receiver Operating Characteristic (ROC) analysis, demonstrated that VH polarization in the Descending orbit provided the best performance in identifying TP-infected sites.

In this study, we analyzed the microearthquake seismicity in the Enguri area (Georgia) recorded between 2020 and 2023 using a newly installed seismic network developed within the DAMAST project. The high sensitivity of the network allowed the detection of even very small seismic events, enabling a detailed investigation of the temporal dynamics of local seismicity. Statistical analyses suggest that the seismic activity around the Enguri Dam is influenced by a combination of natural tectonic processes and subtle reservoir-induced stress changes. While the dam does not appear to exert strong seismic forcing, the observed ≈7-month delay between water level variations and seismicity may indicate a triggering effect. Localized stress variations and temporal clustering further support the hypothesis that water level fluctuations modulate seismic activity. Additionally, the mild persistence in interoccurrence times is consistent with a stress accumulation and delayed triggering mechanism associated with reservoir loading.

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This study analyzes the temporal dynamics of instrumental seismicity recorded in the Pertusillo reservoir area (Southern Italy) between 2001 and 2018. The Gutenberg–Richter analysis of the frequency–magnitude distribution reveals that the seismic catalog is complete for events with magnitudes M≥1.1. The time-clustering of the sequence is at both global and local levels with a coefficient of variation Cv and Lv significantly beyond the 95% confidence band. The Allan Factor method, applied to the series of earthquake occurrence times, corroborates the found time-clustering, showing a bi-fractal behavior indicated by the co-existence of two scaling regimes with a cutoff time scale τc≈45 days and two different fractal exponents, α≈0.3 for time scales less than τc and α≈1.2 for larger ones. The application of the correlogram-based periodogram to both the monthly number of events and the monthly mean water level of the Pertusillo reservoir identifies the yearly cycle as the most significant in both variables. The connection between seismicity and the water level is also demonstrated by the value above 0.5 of the Average Edge Overlap (AEO), a topological metric derived from the Visibility Graph method applied to both the monthly variables. Furthermore, the variation in the AEO between the monthly mean water level and the monthly number of events, along with the time delay between them, indicates that the first leads the second by 1 month.

The 2025 Santorini-Amorgos seismic sequence marked a significant episode of volcanic-seismic unrest in the Hellenic Volcanic Arc, offering a unique opportunity to investigate precursory patterns and the dynamic evolution of seismicity in a complex tectonic setting. Here, we analyze the preparatory phase of the crisis using a high-resolution relocated seismic catalog, anomaly detection, and statistical modelling. We identify four distinct stages of seismic activity: 1) An initial volcano-driven phase starting in the summer 2024 with slightly accelerating moment release and focusing towards the Amorgos region; it was followed by 2) a progressive onset of the seismic sequence during January associated with stronger clustering, steady b-value and rocketing magnitude entropy with right-shifting multifractal spectrum. 3) A successive very energetic transitional, five-days-long chaotic phase in early February, with evident breakdown of the Gutenberg-Richter law, apparent decrease of the b-value, multifractality and hypocenters super-diffusion. Finally, 4) a terminal sub-diffusive aftershocks-dominated phase occurring after mid-February. Clustering, fractal and entropy analyses reveal significant premonitory changes marked by progressive migration of seismicity towards the area that hosted the sequence. Our findings suggest that the 2025 sequence was promoted by multiscale crustal weakening processes likely triggered by a magmatic-tectonic interaction and governed by the strong segmentation of the Santorini-Amorgos normal-faulting system preventing the strike of a mainshock with magnitude larger than 6.

Starting from an initial catalogue of 7833 natural and reservoir-induced seismic events collected in Aswan region (south Egypt) from 1982 to 2016, we investigate the fault structure and triggering mechanisms by determining high-resolution hypocenters for 2562 earthquakes. Despite the complex network of intersecting fractures, highprecision earthquake locations reveal numerous discrete fault strands (some not mapped yet), whose kinematics have been determined from the focal mechanisms of Ml >= 2.5 earthquakes. Furthermore, the analysis of the space-time evolution of seismicity indicates an eastward migration of earthquakes and a progressive activation of different faults. This process could reflect fluid migration in the Wadi Kalabsha embayment likely controlled by permeability barriers at depth acting as seals, and the subsequent build-up of the pore pressure as the dominant driving mechanism of the observed seismicity in the region. A strong spatio-temporal earthquake clustering is observed in the region characterized by seismic sequences, repeated earthquakes, and long-lasting swarms. High variability of b-value in correspondence of the occurrence of mainshock-aftershock sequences also suggests the activation of faults and fractures within their respective fault zones under lower differential stress conditions due to fluids. The projection of earthquake hypocenters on an E-W vertical depth section of the study area shows a seismic gap with an approximate length of about 11 km along the Kalabsha fault, which suggests the presence of a locked fault patch that may generate a Mw 5.9 earthquake if this segment were to break in a single rupture episode.

Xylella Fastidiosa has been recently detected for the first time in southern Italy, representing a very dangerous phytobacterium capable of inducing severe diseases in many plants. In particular, the disease induced in olive trees is called olive quick decline syndrome (OQDS), which provokes the rapid desiccation and, ultimately, death of the infected plants. In this paper, we analyse about two thousands pixels of MODIS satellite evapotranspiration time series, covering infected and uninfected olive groves in southern Italy. Our aim is the identification of Xylella Fastidiosa-linked patterns in the statistical features of evapotranspiration data. The adopted methodology is the well-known Fisher–Shannon analysis that allows one to characterize the time dynamics of complex time series by means of two informational quantities, the Fisher information measure (FIM) and the Shannon entropy power (SEP). On average, the evapotranspiration of Xylella Fastidiosa-infected sites is characterized by a larger SEP and lower FIM compared to uninfected sites. The analysis of the receiver operating characteristic curve suggests that SEP and FIM can be considered binary classifiers with good discrimination performance that, moreover, improves if the yearly cycle, very likely linked with the meteo-climatic variability of the investigated areas, is removed from the data. Furthermore, it indicated that FIM exhibits superior effectiveness compared to SEP in discerning healthy and infected pixels.