Search Results (233)

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. Full article

In complex and unknown processes, global models are fitted over the entire input domain but often tend to perform poorly whenever the response surface exhibits non-stationary behavior and varying smoothness. A common approach is to use local models, which requires partitioning the input domain into subdomains and training multiple models, thereby adding significant complexity. Recognizing this limitation, this study addresses the need for models that represent the input–output relationship consistently over the full domain while still adapting to local variations in the response. It introduces a novel machine learning approach: the Polynomial Chaos Expanded Gaussian Process (PCEGP), leveraging polynomial chaos expansion to calculate input-dependent hyperparameters of the Gaussian process (GP). This provides a mathematically interpretable approach that incorporates non-stationary covariance functions and heteroscedastic noise estimation to generate locally adapted models. The model performance is compared to different algorithms in benchmark tests for regression tasks. The results demonstrate low prediction errors of the PCEGP, highlighting model performance that is often competitive with or better than previous methods. A key advantage of the presented model is its interpretable hyperparameters along with training and prediction runtimes comparable to those of a standard GP. Full article

Lyapunov-exponent-based diagnostics are widely used to quantify deterministic chaos in optically injected semiconductor lasers (OISLs). In most numerical implementations, the optical field is represented either in phasor coordinates $(A,\psi ,N)$ or in Cartesian quadrature coordinates $(X,Y,N)$. Although these representations are mathematically related through a smooth coordinate transformation away from vanishing field amplitude, their numerical realizations can exhibit markedly different robustness in variational calculations, directly impacting the reliability of Lyapunov exponent estimation and chaoticity maps. In this work, we present a systematic assessment of the numerical reliability of Lyapunov-based chaos analysis in master-slave optically injected semiconductor lasers using both phasor and quadrature formulations. The full Lyapunov spectrum was computed via a noise-free variational method that integrates the nonlinear dynamics together with the corresponding Jacobian equations using a fourth-order Runge-Kutta scheme combined with periodic QR orthonormalization. High-resolution Lyapunov maps were constructed in the injection strength-frequency detuning parameter space, and the consistency between both formulations was quantitatively evaluated. While both approaches reproduce the overall structure of chaotic and non-chaotic regions, the phasor formulation may generate spurious positive Lyapunov exponents in regimes where the optical field amplitude approaches low values. These discrepancies originate from singular terms proportional to $1/A$ and $1/A^2$ in the variational Jacobian of the phasor model, which can lead to numerical amplification and artificial chaotic signatures. The quadrature formulation avoids these singularities and provides numerically stable and physically consistent Lyapunov spectra across the explored parameter space. The results establish practical guidelines for robust chaos quantification in optically injected semiconductor lasers and highlight the importance of representation choice in variational Lyapunov analysis of nonlinear photonic systems. Full article

Depth conversion of underwater static electric fields refers to a mathematical approach that indirectly determines the distribution of planar electric fields at larger depths using measured planar electric field data obtained from a shallower region with finite depth and limited area. The complicated environment of high-latitude low-temperature sea areas further increases the difficulty of performing practical large-depth measurements of underwater electric fields. Therefore, depth conversion becomes an important technical strategy for overcoming the constraints of field measurements and for comprehensively understanding the distribution of underwater static electric fields of the target. This study begins with the mathematical formulation of the depth conversion problem, solves the related boundary value problem, and develops the corresponding depth conversion method. Subsequently, based on COMSOL simulation data of the underwater static electric field generated by a scaled-down submersible model, numerical analyses are conducted to investigate the effects of factors such as grid discretization, measurement plane dimensions, conversion depth, and data noise on the conversion accuracy. Finally, the reliability of the conversion method is validated in a laboratory environment by simulating a naturally frozen sea area and employing measured underwater static electric field data from the scaled-down submersible model. The results demonstrate that the developed conversion method can effectively achieve extrapolation of the underwater static electric field of the submersible from shallow regions to deeper water. Even when the noise amplitude is nearly twice that of the effective signal and the conversion depth reaches 8 times the outer diameter of the submersible, the relative root mean square error (RRMSE) of the conversion results can still be maintained below 0.10. These findings provide useful references for the advancement of technologies related to underwater electric field characteristic recognition and electric field stealth performance evaluation in high-latitude low-temperature sea areas. Full article

The cavity-less electro-optic combs (EOCs), recognized for exceptional tunability, stability and high power, are a crucial enabler for the fields such as optical communications, precision measurement and metrology, and microwave photonics. This work systematically investigates the fundamental physical factors that govern the spectral flatness via ultrafast measurements and modeling simulations. The ultrafast analysis results demonstrate that, the finite effective modulation extinction ratio of the electro-optic intensity modulators will result in generation of coherent spectral components with identical frequencies but varying phases and amplitudes in ultrashort temporal scale, finally lead to remarkable spectral interference and further intensity fluctuations across the combs spectrum. Furthermore, the established mathematical relationship between the spectral flatness and the modulation extinction ratio of the intensity modulators exhibits a nonlinear dependence up to the third order. Cascading intensity modulators has been exploited to mitigate the spectral interference and improve the modulation extinction ratio, which has been verified by using home-made high sensitive autocorrelator and frequency-resolved optical gating (FROG), and finely spectral flatness of 0.54 dB among 11 lines has been achieved, which recognized for the first time that modulation extinction ratio related spectral interference phenomenon play a subtle role in EOCs generation. Furthermore, photonic analog-to-digital converters (PADCs) have been investigated and an obvious enhancement in signal-to-noise-and-distortion (SINAD) is achieved, These findings will provide crucial theoretical and experimental support for optimizing EOCs performance, and advance the development and application. Full article

SAR systems are widely used in military, scientific, and commercial missions. The use of these systems makes jamming a focal point for users. To reduce the impact of jamming, this phenomenon needs to be better understood. This paper suggests a method for the categorization of jamming mechanisms in synthetic aperture radar (SAR) systems that builds on the physics of SAR acquisitions and jamming signals. It investigates the effects of different types of jammers on SAR images and systematically categorizes them. The categorization is based on the mathematical representation of the physics of the jamming signal and the SAR image formation process. For all suggested categories, jammers are simulated based on a simple, generic scene to investigate their effects on the resulting SAR image. These effects can vary across a wide range—from adding background noise to unfocused responses similar to point target scattering. Moreover, an intelligent jammer is even able to generate false targets virtually or to reduce the signal intensity of real targets by retransmitting signals fully adapted to the image acquisition scenario. Categorizing jammers is a bottom-up task best based on physics. It will be essential in the future when the sky is crowded with constellations of satellites and interference levels increase significantly and become commonplace. Full article

Complex dynamical systems are characterized by inherent nonlinearity, high dimensionality, spatiotemporal uncertainty, and implicit symmetry, posing fundamental challenges for their mathematical modeling and multi-step evolution prediction. For example, wind power exhibits strong randomness, intermittency, and latent temporal symmetry. To address this, this paper proposes a symmetry-guided deep generative model, the bi-directional recurrent generative adversarial network (BDR-GAN), for the multi-step rolling prediction of such systems. The BDR-GAN formalizes multi-step evolution as a conditional probability distribution learning problem. It systematically integrates three forms of symmetry to enhance modeling validity: bi-directional temporal symmetry captured by a BiLSTM-based generator, structural symmetry within the adversarial learning framework between the generator and a 1D-CNN discriminator, and rolling symmetry enabled by a recursive prediction strategy that supports cyclic state updates. Theoretical analysis demonstrates that this symmetry-embedded adversarial mechanism enables BDR-GAN to effectively approximate the underlying dynamic operators and the conditional distribution of future states, improving the learned model’s generalization. Experimental validation on wind power datasets confirms the framework’s superiority. Compared to benchmark models, BDR-GAN achieves superior prediction accuracy (e.g., RMSE 0.236, MAPE 5.12%), provides reliable uncertainty quantification (PICP 95.5%), and exhibits enhanced robustness against noise and variability. This work provides a generalizable, symmetry-guided modeling framework for the multi-step evolution of complex dynamical systems, offering theoretical and technical support for high-precision prediction in critical applications such as wind power integration and smart grid operation. Full article

The global expansion of continuous GNSS networks has generated large-scale spatiotemporal datasets whose analysis requires robust mathematical and statistical tools. This study introduces a geospatial, multivariate statistical framework for classifying 21,548 GNSS stations from the University of Nevada repository. The methodology integrates harmonic regression, stochastic noise modeling, quality assessment, and slope estimation into a unified feature space suitable for high-dimensional analysis. Using unsupervised learning clustering computed with our custom-developed code, based entirely on free and open-source software, we identify homogeneous station groups that reflect dominant signal properties—periodicity, noise structure, data quality, and long-term velocity—together with their spatial context. The resulting clusters exhibit strong mathematical coherence and reveal continental-scale patterns driven by seasonal forcing, tectonic regime, climatic variability, and monument stability. By grouping stations with similar statistical behavior, the proposed framework improves reference-site selection, enhances deformation-field interpretation, and supports the detection of anomalous or hazard-related behavior. Overall, this approach provides a scalable, data-driven mathematical tool for analyzing complex spatiotemporal signals and contributes to more reliable deformation modeling and environmental risk assessment. Full article

Variational Mode Decomposition (VMD) is a powerful formalism for the time-scale analysis of vibration signals from rotating machinery. However, its performance is often compromised by complex parameter configuration, where subjective manual tuning leads to mode mixing or information loss. In this study, we present a physics-guided framework that generalizes VMD optimization across diverse operating conditions. We utilized a meta-dataset combining three distinct sources (CWRU, HUST, UO) to validate the approach. Through a shaft-normalized segmentation strategy and K-Means++ clustering, we identified six distinct signal archetypes based on spectral morphology. Central to this framework is the Energy-Dispersion Index (EDI), a novel physically interpretable metric designed to differentiate between structured fault transients and stochastic noise. Extensive validation via a full-factorial Design of Experiments (8640 trials) confirmed the statistical superiority of EDI over benchmarks like kurtosis and envelope entropy, yielding an 8.3% improvement in modal fidelity. Furthermore, a rigorous ablation study demonstrated that the proposed archetype-based parameterization is highly efficient. This strategy achieved a $392\times$ speedup over online optimization while maintaining statistically equivalent diagnostic accuracy. Additionally, by generalizing parameters from high-quality archetype representatives, the framework reduced spectral leakage (Orthogonality Index) by 51.4% compared to instance-wise optimization. The resulting framework provides a mathematically rigorous, real-time solution for automated vibration signal decomposition. Full article

Community detection in graphs can be viewed as the estimation of a partition map that remains stable under admissible perturbations of graph topology and node attributes. While modern graph neural networks (GNNs) achieve strong empirical accuracy, they often exhibit severe assignment drift under minor perturbations, leading to illusory community structures. In this work, we propose DISPEL-GNN, a stability-aware graph learning framework that integrates spectral operator regularization, Bayesian uncertainty modeling, and risk-aware dynamic attention for perturbation-bounded community detection. The model explicitly constrains graph operators through uniform spectral norm bounds, high-frequency energy suppression, and commutator alignment while dynamically modulating message passing based on node-level spectral risk and epistemic uncertainty. We further formalize instability via assignment of drift functional and establish perturbation bounds linking drift to operator norms and spectral gaps, complemented by a PAC-Bayesian generalization guarantee. Extensive experiments on real-world benchmarks including Cora, Citeseer, Pubmed, Cora-Full, and DBLP demonstrate that DISPEL-GNN consistently reduces assignment drift by 18–35% under feature noise and edge perturbations while improving clustering quality with up to +3.0 NMI and +0.04 ARI compared to strong baselines such as GAT and Bayesian GNNs. The normalized mutual information (NMI), adjusted Rand index (ARI), and PAC-Bayesian (PAC) constraints serve as evaluative and theoretical instruments in this study. Additional studies on synthetic graphs with controlled spectral gaps confirm that the proposed method maintains stable community assignments in low-gap regimes where classical spectral and GNN-based methods degrade sharply. These results establish DISPEL-GNN as a mathematically grounded and practically effective framework for robust and interpretable community detection. A metric-wise dominance analysis shows that DISPEL-GNN achieves metric-wise dominance across most accuracy and robustness criteria, with minor tradeoffs in modularity on selected datasets. These results indicate that explicitly modeling stability and uncertainty provides a principled pathway toward reliable and interpretable community detection in noisy graph environments. Full article

Efficient retrieval of mathematical and structural similarities in System Dynamics models remains a significant challenge for traditional lexical systems, which often fail to capture the contextual dependencies of simulation processes. This paper presents an architectural approach and implementation of a semantic search module integrated into an existing cloud-based modeling and simulation system. The proposed method employs a strategy for serializing graph structures into textual descriptions, followed by the generation of vector embeddings via local ONNX inference and indexing within a vector database (Qdrant). Experimental validation performed on a diverse corpus of complex dynamic models, compares the proposed approach against traditional information retrieval methods (Full-Text Search, Keyword Search in PostgreSQL, and Apache Lucene with Standard and BM25 scoring). The results demonstrate the distinct advantage of semantic search, achieving high precision (over 90%) within the scope of the evaluated corpus and effectively eliminating information noise. In comparison, keyword search exhibited only 24.8% precision with a significant rate of false positives, while standard full-text analysis failed to identify relevant models for complex conceptual queries (0 results). Despite a recorded increase in latency (~2 s), the study proves that the vector-based approach is a significantly more robust solution for detecting hidden semantic connections in mathematical model databases, providing a foundation for future developments toward multi-vector indexing strategies. Full article

Accurate channel estimation is critical for enabling effective directional beamforming and spectrally efficient transmission in beamspace massive multiple-input multiple-output (MIMO) systems. However, conventional model-driven algorithms are derived from idealized mathematical models and typically suffer severe performance degradation under model mismatches caused by complex and nonideal propagation environments. Although data-driven deep learning (DL) approaches can learn channel characteristics from data, they typically require large-scale training datasets and demonstrate limited generalization capability. To overcome these limitations, we propose a model-data hybrid-driven network (MD-HDN) scheme to address the wideband beamspace channel estimation problem. In the MD-HDN scheme, we unfold the vector approximate message passing (VAMP) algorithm into a trainable network, where a novel shrinkage function is introduced to enhance the estimation accuracy. Extensive numerical results confirm that the proposed MD-HDN scheme can significantly outperform existing schemes under various signal-to-noise ratio (SNR), and achieve substantial improvements in both estimation accuracy and robustness. Full article

We present an advanced statistical framework for estimating the relative intensity of astrophysical event distributions (e.g., Gamma-Ray Bursts, GRBs) on the sky tofacilitate population studies and large-scale structure analysis. In contrast to the traditional approach based on the ratio of Kernel Density Estimation (KDE), which is characterized by numerical instability and bandwidth sensitivity, this work applies a logistic regression embedded in a Bayesian framework to directly model selection effects. It reformulates the problem as a logistic regression task within a Generalized Additive Model (GAM) framework, utilizing isotropic Splines on the Sphere (SOS) to map the conditional probability of redshift measurement. The model complexity and smoothness are objectively optimized using Restricted Maximum Likelihood (REML) and the Akaike Information Criterion (AIC), ensuring a data-driven bias-variance trade-off. We benchmark this approach against an Adaptive Kernel Density Estimator (AKDE) using von Mises–Fisher kernels and Abramson’s square root law. The comparative analysis reveals strong statistical evidence in favor of this Preconditioned (Precon) Estimator, yielding a log-likelihood improvement of $\Delta \mathcal{L}\approx 74.3$ (Bayes factor $>{10}^{30}$) over the adaptive method. We show that this Precon Estimator acts as a spectral bandwidth extender, effectively decoupling the wideband exposure map from the narrowband selection efficiency. This provides a tool for cosmologists to recover high-frequency structural features—such as the sharp cutoffs—that are mathematically irresolvable by direct density estimators due to the bandwidth limitation inherent in sparse samples. The methodology ensures that reconstructions of the cosmic web are stable against Poisson noise and consistent with observational constraints. Full article

The spaceborne full-polarimetric (FP) synthetic aperture radar (SAR) is an advanced sensor for high-resolution Earth observation. However, FP data acquired by such a system are prone to distortions induced by ionospheric Faraday rotation (FR). From the perspective of exploiting these distortions, this enables the estimation of the ionospheric FR angle (FRA), and consequently the total electron content, across most global regions (including the extensive ocean areas) using spaceborne FP SAR measurements. The accuracy of FRA estimation, however, is highly sensitive to noise interference. This study addresses denoising in FRA retrieval based on the Bickel–Bates estimator, with a specific focus on noise reduction methods built upon the adaptive Goldstein filter (AGF) that was originally designed for radar interferometric processing. For the first time, three signal-to-noise ratio (SNR)-based AGFs suitable for FRA estimation are investigated. A key feature of these filters is that their SNRs are all defined using the amplitude of the Bickel–Bates estimator signal rather than the FRA estimates themselves. Accordingly, these AGFs are applied to the estimator signal instead of the estimated FRAs. Two of the three AGFs are developed by adopting the mathematical forms of SNRs and filter parameters consistent with the existing SNR-based AGFs for interferogram. The third AGF is newly proposed by utilizing more general mathematical forms of SNR and filter parameter that differ from the first two. Specifically, its SNR definition aligns with that widely used in image processing, and its filter parameter is derived as a function of the defined SNR plus an additionally introduced adjustable factor. The three SNR-based AGFs tailored for FRA estimation are tested and evaluated against existing AGF variants and classical image denoising methods using three sets of FP SAR Datasets acquired by the L-band ALOS PALSAR sensor, encompassing an ocean-only scene, a plain land–ocean combined scene, and a more complex land–ocean combined scene. Experimental results demonstrate that all three filters can effectively mitigate noise, with the newly proposed AGF achieving the best performance among all denoising methods included in the comparison. Full article

Mathematical models for infectious diseases, particularly autonomous ODE models, are generally known to possess simple dynamics, often converging to stable disease-free or endemic equilibria. This paper investigates the dynamic consequences of a crucial, yet often overlooked, component of pandemic response: the saturation of public health testing. We extend the standard SIR model to include compartments for ‘Confirmed’ (C) and ‘Monitored’ (M) individuals, resulting in a new SICMR model. By fitting the model to U.S. COVID-19 pandemic data (specifically the Omicron wave of late 2021), we demonstrate that capacity constraints in testing destabilize the testing-free endemic equilibrium ($E_1$). This equilibrium becomes an unstable saddle-focus. The instability is driven by a sociological feedback loop, where the rise in confirmed cases drive testing effort, modeled by a nonlinear Holling Type II functional response. We explicitly verify that the eigenvalues for the best-fit model satisfy the Shilnikov condition (${\lambda }_u>{\lambda }_s$), demonstrating the system possesses the necessary ingredients for complex, chaotic-like dynamics. Furthermore, we employ Stochastic Differential Equations (SDEs) to show that intrinsic noise interacts with this instability to generate ’noise-induced bursting,’ replicating the complex wave-like patterns observed in empirical data. Our results suggest that public health interventions, such as testing, are not merely passive controls but active dynamical variables that can fundamentally alter the qualitative stability of an epidemic. Full article