In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various natural and socioeconomic systems. They are also used in engineering. Many natural and almost all socioeconomic systems possess memory which, in many cases, is power-law-like memory. Generalized fractional maps, in which memory is not exactly the power-law memory but the asymptotically power-law-like memory, are used to model and investigate general properties of these systems. In this paper we extend the definition of the notion of generalized fractional maps of arbitrary positive orders that previously was defined only for maps which, in the case of integer orders, converge to area/volume-preserving maps. Fractional generalizations of H'enon and Lozi maps belong to the newly defined class of generalized fractional maps. We derive the equations which define periodic points in generalized fractional maps. We consider applications of our results to the fractional and fractional difference H'enon and Lozi maps.
{"title":"Asymptotic cycles in fractional generalizations of multidimensional maps","authors":"Mark Edelman","doi":"arxiv-2408.00134","DOIUrl":"https://doi.org/arxiv-2408.00134","url":null,"abstract":"In regular dynamics, discrete maps are model presentations of discrete\u0000dynamical systems, and they may approximate continuous dynamical systems. Maps\u0000are used to investigate general properties of dynamical systems and to model\u0000various natural and socioeconomic systems. They are also used in engineering.\u0000Many natural and almost all socioeconomic systems possess memory which, in many\u0000cases, is power-law-like memory. Generalized fractional maps, in which memory\u0000is not exactly the power-law memory but the asymptotically power-law-like\u0000memory, are used to model and investigate general properties of these systems.\u0000In this paper we extend the definition of the notion of generalized fractional\u0000maps of arbitrary positive orders that previously was defined only for maps\u0000which, in the case of integer orders, converge to area/volume-preserving maps.\u0000Fractional generalizations of H'enon and Lozi maps belong to the newly defined\u0000class of generalized fractional maps. We derive the equations which define\u0000periodic points in generalized fractional maps. We consider applications of our\u0000results to the fractional and fractional difference H'enon and Lozi maps.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"68 E-2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141883986","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By means of numerical analysis conducted with the aid of the computer, the collective synchronization of coupled phase oscillators in the Kuramoto model in the connected regime of random networks of various sizes is studied. The oscillators synchronize and achieve phase coherence, and this process is not significantly affected by the level of connectivity of the network. If the probability that two oscillators are coupled is around the network connectivity threshold synchronization still occurs, although in a more attenuated way. If the size of the network is sufficiently large the oscillators have a phase transition.
{"title":"Kuramoto oscillators in random networks","authors":"Agostino Funel","doi":"arxiv-2407.21513","DOIUrl":"https://doi.org/arxiv-2407.21513","url":null,"abstract":"By means of numerical analysis conducted with the aid of the computer, the\u0000collective synchronization of coupled phase oscillators in the Kuramoto model\u0000in the connected regime of random networks of various sizes is studied. The\u0000oscillators synchronize and achieve phase coherence, and this process is not\u0000significantly affected by the level of connectivity of the network. If the\u0000probability that two oscillators are coupled is around the network connectivity\u0000threshold synchronization still occurs, although in a more attenuated way. If\u0000the size of the network is sufficiently large the oscillators have a phase\u0000transition.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christof Schötz, Alistair White, Maximilian Gelbrecht, Niklas Boers
Predicting chaotic dynamical systems is critical in many scientific fields such as weather prediction, but challenging due to the characterizing sensitive dependence on initial conditions. Traditional modeling approaches require extensive domain knowledge, often leading to a shift towards data-driven methods using machine learning. However, existing research provides inconclusive results on which machine learning methods are best suited for predicting chaotic systems. In this paper, we compare different lightweight and heavyweight machine learning architectures using extensive existing databases, as well as a newly introduced one that allows for uncertainty quantification in the benchmark results. We perform hyperparameter tuning based on computational cost and introduce a novel error metric, the cumulative maximum error, which combines several desirable properties of traditional metrics, tailored for chaotic systems. Our results show that well-tuned simple methods, as well as untuned baseline methods, often outperform state-of-the-art deep learning models, but their performance can vary significantly with different experimental setups. These findings underscore the importance of matching prediction methods to data characteristics and available computational resources.
{"title":"Machine Learning for predicting chaotic systems","authors":"Christof Schötz, Alistair White, Maximilian Gelbrecht, Niklas Boers","doi":"arxiv-2407.20158","DOIUrl":"https://doi.org/arxiv-2407.20158","url":null,"abstract":"Predicting chaotic dynamical systems is critical in many scientific fields\u0000such as weather prediction, but challenging due to the characterizing sensitive\u0000dependence on initial conditions. Traditional modeling approaches require\u0000extensive domain knowledge, often leading to a shift towards data-driven\u0000methods using machine learning. However, existing research provides\u0000inconclusive results on which machine learning methods are best suited for\u0000predicting chaotic systems. In this paper, we compare different lightweight and\u0000heavyweight machine learning architectures using extensive existing databases,\u0000as well as a newly introduced one that allows for uncertainty quantification in\u0000the benchmark results. We perform hyperparameter tuning based on computational\u0000cost and introduce a novel error metric, the cumulative maximum error, which\u0000combines several desirable properties of traditional metrics, tailored for\u0000chaotic systems. Our results show that well-tuned simple methods, as well as\u0000untuned baseline methods, often outperform state-of-the-art deep learning\u0000models, but their performance can vary significantly with different\u0000experimental setups. These findings underscore the importance of matching\u0000prediction methods to data characteristics and available computational\u0000resources.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"414 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869551","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We implement a Szilard engine that uses a 2-bit logical unit consisting of coupled quantum flux parametrons -- Josephson-junction superconducting circuits originally designed for quantum computing. Detailed simulations show that it is highly thermodynamically efficient while functioning as a Maxwell demon. The physically-calibrated design is targeted to direct experimental exploration. However, variations in Josephson junction fabrication introduce asymmetries that result in energy inefficiency and low operational fidelity. We provide a design solution that mitigates these practical challenges. The resulting platform is ideally suited to probe the thermodynamic foundations of information processing devices far from equilibrium.
{"title":"Nonequilibrium Thermodynamics of a Superconducting Szilard Engine","authors":"Kuen Wai Tang, Kyle J. Ray, James P. Crutchfield","doi":"arxiv-2407.20418","DOIUrl":"https://doi.org/arxiv-2407.20418","url":null,"abstract":"We implement a Szilard engine that uses a 2-bit logical unit consisting of\u0000coupled quantum flux parametrons -- Josephson-junction superconducting circuits\u0000originally designed for quantum computing. Detailed simulations show that it is\u0000highly thermodynamically efficient while functioning as a Maxwell demon. The\u0000physically-calibrated design is targeted to direct experimental exploration.\u0000However, variations in Josephson junction fabrication introduce asymmetries\u0000that result in energy inefficiency and low operational fidelity. We provide a\u0000design solution that mitigates these practical challenges. The resulting\u0000platform is ideally suited to probe the thermodynamic foundations of\u0000information processing devices far from equilibrium.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The interdependence and high dimensionality of multivariate signals present significant challenges for denoising, as conventional univariate methods often struggle to capture the complex interactions between variables. A successful approach must consider not only the multivariate dependencies of the desired signal but also the multivariate dependencies of the interfering noise. In our previous research, we introduced a method using machine learning to extract the maximum portion of ``predictable information" from univariate signal. We extend this approach to multivariate signals, with the key idea being to properly incorporate the interdependencies of the noise back into the interdependent reconstruction of the signal. The method works successfully for various multivariate signals, including chaotic signals and highly oscillating sinusoidal signals which are corrupted by spatially correlated intensive noise. It consistently outperforms other existing multivariate denoising methods across a wide range of scenarios.
{"title":"Unsupervised Reservoir Computing for Multivariate Denoising of Severely Contaminated Signals","authors":"Jaesung Choi, Pilwon Kim","doi":"arxiv-2407.18759","DOIUrl":"https://doi.org/arxiv-2407.18759","url":null,"abstract":"The interdependence and high dimensionality of multivariate signals present\u0000significant challenges for denoising, as conventional univariate methods often\u0000struggle to capture the complex interactions between variables. A successful\u0000approach must consider not only the multivariate dependencies of the desired\u0000signal but also the multivariate dependencies of the interfering noise. In our\u0000previous research, we introduced a method using machine learning to extract the\u0000maximum portion of ``predictable information\" from univariate signal. We extend\u0000this approach to multivariate signals, with the key idea being to properly\u0000incorporate the interdependencies of the noise back into the interdependent\u0000reconstruction of the signal. The method works successfully for various\u0000multivariate signals, including chaotic signals and highly oscillating\u0000sinusoidal signals which are corrupted by spatially correlated intensive noise.\u0000It consistently outperforms other existing multivariate denoising methods\u0000across a wide range of scenarios.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141869550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This study focuses on extending the concept of weak signal enhancement from dynamical systems based on vibrational resonance of nonlinear systems, to non-smooth systems. A Van der Pol- Duffing oscillator with a one-sided barrier, subjected to harmonic excitations, has been considered an archetypical low-order model, whose response is weak. It is shown that the system response can be significantly enhanced by applying an additional harmonic excitation but with much higher frequencies. The reasons for the underlying physics are investigated analytically using multiple-scale analysis and the Blekham perturbation approach (direct partition motion). The analytical predictions are qualitatively validated using numerical simulations. This approach yields valuable insights into the intricate interplay between fast and slow excitations in non-smooth systems.
{"title":"Vibrational resonance in vibro-impact oscillator through fast harmonic excitation","authors":"Somnath Roy, Sayan Gupta","doi":"arxiv-2407.17849","DOIUrl":"https://doi.org/arxiv-2407.17849","url":null,"abstract":"This study focuses on extending the concept of weak signal enhancement from\u0000dynamical systems based on vibrational resonance of nonlinear systems, to\u0000non-smooth systems. A Van der Pol- Duffing oscillator with a one-sided barrier,\u0000subjected to harmonic excitations, has been considered an archetypical\u0000low-order model, whose response is weak. It is shown that the system response\u0000can be significantly enhanced by applying an additional harmonic excitation but\u0000with much higher frequencies. The reasons for the underlying physics are\u0000investigated analytically using multiple-scale analysis and the Blekham\u0000perturbation approach (direct partition motion). The analytical predictions are\u0000qualitatively validated using numerical simulations. This approach yields\u0000valuable insights into the intricate interplay between fast and slow\u0000excitations in non-smooth systems.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mechanisms of resistivity can be divided into two basic classes: one is dissipative (like scattering on phonons) and another is quasi-elastic (like scattering on static impurities). They are often treated by the empirical Matthiessen rule, which says that total resistivity is just the sum of these two contributions, which are computed separately. This is quite misleading for two reasons. First, the two mechanisms are generally correlated. Second, computing the elastic resistivity alone masks the fundamental fact that the linear-response approximation has a vanishing validity interval at vanishing dissipation. Limits of zero electric field and zero dissipation do not commute for the simple reason that one needs to absorb the Joule heat quadratic in the applied field. Here, we present a simple model that illustrates these two points. The model also illuminates the role of variational principles for non-equilibrium steady states.
{"title":"Interplay between two mechanisms of resistivity","authors":"Anton Kapustin, Gregory Falkovich","doi":"arxiv-2407.16284","DOIUrl":"https://doi.org/arxiv-2407.16284","url":null,"abstract":"Mechanisms of resistivity can be divided into two basic classes: one is\u0000dissipative (like scattering on phonons) and another is quasi-elastic (like\u0000scattering on static impurities). They are often treated by the empirical\u0000Matthiessen rule, which says that total resistivity is just the sum of these\u0000two contributions, which are computed separately. This is quite misleading for\u0000two reasons. First, the two mechanisms are generally correlated. Second,\u0000computing the elastic resistivity alone masks the fundamental fact that the\u0000linear-response approximation has a vanishing validity interval at vanishing\u0000dissipation. Limits of zero electric field and zero dissipation do not commute\u0000for the simple reason that one needs to absorb the Joule heat quadratic in the\u0000applied field. Here, we present a simple model that illustrates these two\u0000points. The model also illuminates the role of variational principles for\u0000non-equilibrium steady states.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"310 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Synchronization is a widespread phenomenon observed across natural and artificial networked systems. It often manifests itself by clusters of units exhibiting coincident dynamics. These clusters are a direct consequence of the organization of the Laplacian matrix eigenvalues into spectral localized blocks. We show how the concept of spectral blocks can be leveraged to design straightforward yet powerful controllers able to fully manipulate cluster synchronization of a generic network, thus shaping at will its parallel functioning. Specifically, we demonstrate how to induce the formation of spectral blocks in networks where such structures would not exist, and how to achieve precise mastering over the synchronizability of individual clusters by dictating the sequence in which each of them enters or exits the synchronization stability region as the coupling strength varies. Our results underscore the pivotal role of cluster synchronization control in shaping the parallel operation of networked systems, thereby enhancing their efficiency and adaptability across diverse applications.
{"title":"Taming Cluster Synchronization","authors":"Cinzia Tomaselli, Lucia Valentina Gambuzza, Gui-Quan Sun, Stefano Boccaletti, Mattia Frasca","doi":"arxiv-2407.10638","DOIUrl":"https://doi.org/arxiv-2407.10638","url":null,"abstract":"Synchronization is a widespread phenomenon observed across natural and\u0000artificial networked systems. It often manifests itself by clusters of units\u0000exhibiting coincident dynamics. These clusters are a direct consequence of the\u0000organization of the Laplacian matrix eigenvalues into spectral localized\u0000blocks. We show how the concept of spectral blocks can be leveraged to design\u0000straightforward yet powerful controllers able to fully manipulate cluster\u0000synchronization of a generic network, thus shaping at will its parallel\u0000functioning. Specifically, we demonstrate how to induce the formation of\u0000spectral blocks in networks where such structures would not exist, and how to\u0000achieve precise mastering over the synchronizability of individual clusters by\u0000dictating the sequence in which each of them enters or exits the\u0000synchronization stability region as the coupling strength varies. Our results\u0000underscore the pivotal role of cluster synchronization control in shaping the\u0000parallel operation of networked systems, thereby enhancing their efficiency and\u0000adaptability across diverse applications.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"321 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Fast scrambling is a distinctive feature of quantum gravity, which by means of holography is closely tied to the behaviour of large$-c$ conformal field theories. We study this phenomenon in the context of semiclassical Liouville theory, providing both insights into the mechanism of scrambling in CFTs and into the structure of Liouville theory, finding that it exhibits a maximal Lyapunov exponent despite not featuring the identity in its spectrum. However, as we show, the states contributing to the relevant correlation function can be thought of as dressed scramblons. At a technical level we we first use the path integral picture in order to derive the Euclidean four-point function in an explicit compact form. Next, we demonstrate its equivalence to a conformal block expansion, revealing an explicit but non-local map between path integral saddles and conformal blocks. By analytically continuing both expressions to Lorentzian times, we obtain two equivalent formulations of the OTOC, which we use to study the onset of chaos in Liouville theory. We take advantage of the compact form in order to extract a Lyapunov exponent and a scrambling time. From the conformal block expansion formulation of the OTOC we learn that scrambling shifts the dominance of conformal blocks from heavy primaries at early times to the lightest primary at late times. Finally, we discuss our results in the context of holography.
{"title":"Quantum Chaos in Liouville CFT","authors":"Julian Sonner, Benjamin Strittmatter","doi":"arxiv-2407.11124","DOIUrl":"https://doi.org/arxiv-2407.11124","url":null,"abstract":"Fast scrambling is a distinctive feature of quantum gravity, which by means\u0000of holography is closely tied to the behaviour of large$-c$ conformal field\u0000theories. We study this phenomenon in the context of semiclassical Liouville\u0000theory, providing both insights into the mechanism of scrambling in CFTs and\u0000into the structure of Liouville theory, finding that it exhibits a maximal\u0000Lyapunov exponent despite not featuring the identity in its spectrum. However,\u0000as we show, the states contributing to the relevant correlation function can be\u0000thought of as dressed scramblons. At a technical level we we first use the path\u0000integral picture in order to derive the Euclidean four-point function in an\u0000explicit compact form. Next, we demonstrate its equivalence to a conformal\u0000block expansion, revealing an explicit but non-local map between path integral\u0000saddles and conformal blocks. By analytically continuing both expressions to\u0000Lorentzian times, we obtain two equivalent formulations of the OTOC, which we\u0000use to study the onset of chaos in Liouville theory. We take advantage of the\u0000compact form in order to extract a Lyapunov exponent and a scrambling time.\u0000From the conformal block expansion formulation of the OTOC we learn that\u0000scrambling shifts the dominance of conformal blocks from heavy primaries at\u0000early times to the lightest primary at late times. Finally, we discuss our\u0000results in the context of holography.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"77 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722320","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tony Albers, Lukas Hille, David Müller-Bender, Günter Radons
We demonstrate that standard delay systems with a linear instantaneous and a delayed nonlinear term show weak chaos, asymptotically subdiffusive behavior, and weak ergodicity breaking if the nonlinearity is chosen from a specific class of functions. In the limit of large constant delay times, anomalous behavior may not be observable due to exponentially large crossover times. A periodic modulation of the delay causes a strong reduction of the effective dimension of the chaotic phases, leads to hitherto unknown types of solutions, and the occurrence of anomalous diffusion already at short times. The observed anomalous behavior is caused by non-hyperbolic fixed points in function space.
{"title":"Weak Chaos, Anomalous Diffusion, and Weak Ergodicity Breaking in Systems with Delay","authors":"Tony Albers, Lukas Hille, David Müller-Bender, Günter Radons","doi":"arxiv-2407.09449","DOIUrl":"https://doi.org/arxiv-2407.09449","url":null,"abstract":"We demonstrate that standard delay systems with a linear instantaneous and a\u0000delayed nonlinear term show weak chaos, asymptotically subdiffusive behavior,\u0000and weak ergodicity breaking if the nonlinearity is chosen from a specific\u0000class of functions. In the limit of large constant delay times, anomalous\u0000behavior may not be observable due to exponentially large crossover times. A\u0000periodic modulation of the delay causes a strong reduction of the effective\u0000dimension of the chaotic phases, leads to hitherto unknown types of solutions,\u0000and the occurrence of anomalous diffusion already at short times. The observed\u0000anomalous behavior is caused by non-hyperbolic fixed points in function space.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720005","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：