We investigate chaos in the dynamics of outgoing massless particles near the horizon of static spherically symmetric (SSS) black holes in two well-motivated models of $f(R)$ gravity. In both these models, we probe chaos in the particle trajectories (under suitable harmonic confinement) in the vicinity of the black hole horizons, for a set of initial conditions. The particle trajectories, associated Poincar$acute{e}$ sections, and Lyapunov exponents clearly illustrate the role played by the black hole horizon in the growth of chaos within a specific energy range. We demonstrate how this energy range is controlled by the parameters of the modified gravity theory under consideration. The growth of chaos in such a classical setting is known to respect a surface gravity bound arising from universal aspects of particle dynamics close to the black hole horizon [K. Hashimoto and N. Tanahashi, Phys. Rev. D 95, 024007 (2017)], analogous to the quantum MSS bound [J. Maldacena, S.H. Shenker and D. Stanford, JHEP 08 (2016) 106]. Interestingly, both models studied in our work respect the bound, in contrast to some of the other models of $f(R)$ gravity in the existing literature. The work serves as a motivation to use chaos as an additional tool to probe Einstein gravity in the strong gravity regime in the vicinity of black hole horizons.
我们在两个动机良好的 $f(R)$ 引力模型中研究了静态球对称（SSS）黑洞视界附近出射无质量粒子动力学中的混沌。在这两个模型中，我们在一组初始条件下探测了黑洞视界附近粒子轨迹（在适当的谐波约束下）的混沌。粒子轨迹、相关的Poincar$acute{e}$截面和Lyapunov指数清楚地表明了黑洞视界在特定能量范围内的混沌增长中所起的作用。我们证明了这一能量范围是如何被所考虑的修正引力理论的参数所控制的。众所周知，在这样的经典环境中，混沌的增长要尊重由靠近黑洞视界的粒子动力学的普遍方面所产生的表面引力约束[K. Hashimoto and N. Tanahashi, Phys.Rev. D 95, 024007 (2017)]，类似于量子 MSS 约束[J. Maldacena,S.H. Shenker and D. Stanford, JHEP 08 (2016) 106]。有趣的是，我们工作中研究的两个模型都遵守了这个约束，这与现有文献中的其他一些 $f(R)$ 引力模型形成了鲜明对比。这项工作促使我们把混沌作为一种额外的工具，来探测黑洞视界附近强引力体系中的爱因斯坦引力。
{"title":"Near-horizon chaos beyond Einstein gravity","authors":"Surajit Das, Surojit Dalui, Rickmoy Samanta","doi":"arxiv-2405.09945","DOIUrl":"https://doi.org/arxiv-2405.09945","url":null,"abstract":"We investigate chaos in the dynamics of outgoing massless particles near the\u0000horizon of static spherically symmetric (SSS) black holes in two well-motivated\u0000models of $f(R)$ gravity. In both these models, we probe chaos in the particle\u0000trajectories (under suitable harmonic confinement) in the vicinity of the black\u0000hole horizons, for a set of initial conditions. The particle trajectories,\u0000associated Poincar$acute{e}$ sections, and Lyapunov exponents clearly\u0000illustrate the role played by the black hole horizon in the growth of chaos\u0000within a specific energy range. We demonstrate how this energy range is\u0000controlled by the parameters of the modified gravity theory under\u0000consideration. The growth of chaos in such a classical setting is known to\u0000respect a surface gravity bound arising from universal aspects of particle\u0000dynamics close to the black hole horizon [K. Hashimoto and N. Tanahashi, Phys.\u0000Rev. D 95, 024007 (2017)], analogous to the quantum MSS bound [J. Maldacena,\u0000S.H. Shenker and D. Stanford, JHEP 08 (2016) 106]. Interestingly, both models\u0000studied in our work respect the bound, in contrast to some of the other models\u0000of $f(R)$ gravity in the existing literature. The work serves as a motivation\u0000to use chaos as an additional tool to probe Einstein gravity in the strong\u0000gravity regime in the vicinity of black hole horizons.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062344","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}
Vismaya V S, Alok Hareendran, Bharath V Nair, Sishu Shankar Muni, Martin Lellep
This paper explores the prediction of subsequent steps in H'enon Map using various machine learning techniques. The H'enon map, well known for its chaotic behaviour, finds applications in various fields including cryptography, image encryption, and pattern recognition. Machine learning methods, particularly deep learning, are increasingly essential for understanding and predicting chaotic phenomena. This study evaluates the performance of different machine learning models including Random Forest, Recurrent Neural Network (RNN), Long Short-Term Memory (LSTM) networks, Support Vector Machines (SVM), and Feed Forward Neural Networks (FNN) in predicting the evolution of the H'enon map. Results indicate that LSTM network demonstrate superior predictive accuracy, particularly in extreme event prediction. Furthermore, a comparison between LSTM and FNN models reveals the LSTM's advantage, especially for longer prediction horizons and larger datasets. This research underscores the significance of machine learning in elucidating chaotic dynamics and highlights the importance of model selection and dataset size in forecasting subsequent steps in chaotic systems.
{"title":"Comparative Analysis of Predicting Subsequent Steps in Hénon Map","authors":"Vismaya V S, Alok Hareendran, Bharath V Nair, Sishu Shankar Muni, Martin Lellep","doi":"arxiv-2405.10190","DOIUrl":"https://doi.org/arxiv-2405.10190","url":null,"abstract":"This paper explores the prediction of subsequent steps in H'enon Map using\u0000various machine learning techniques. The H'enon map, well known for its\u0000chaotic behaviour, finds applications in various fields including cryptography,\u0000image encryption, and pattern recognition. Machine learning methods,\u0000particularly deep learning, are increasingly essential for understanding and\u0000predicting chaotic phenomena. This study evaluates the performance of different\u0000machine learning models including Random Forest, Recurrent Neural Network\u0000(RNN), Long Short-Term Memory (LSTM) networks, Support Vector Machines (SVM),\u0000and Feed Forward Neural Networks (FNN) in predicting the evolution of the\u0000H'enon map. Results indicate that LSTM network demonstrate superior predictive\u0000accuracy, particularly in extreme event prediction. Furthermore, a comparison\u0000between LSTM and FNN models reveals the LSTM's advantage, especially for longer\u0000prediction horizons and larger datasets. This research underscores the\u0000significance of machine learning in elucidating chaotic dynamics and highlights\u0000the importance of model selection and dataset size in forecasting subsequent\u0000steps in chaotic systems.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062373","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}
In this paper, the Parameter Switching (PS) algorithm is used to approximate numerically attractors of a Hopfield Neural Network (HNN) system. The PS algorithm is a convergent scheme designed for approximating attractors of an autonomous nonlinear system, depending linearly on a real parameter. Aided by the PS algorithm, it is shown that every attractor of the HNN system can be expressed as a convex combination of other attractors. The HNN system can easily be written in the form of a linear parameter dependence system, to which the PS algorithm can be applied. This work suggests the possibility to use the PS algorithm as a control-like or anticontrol-like method for chaos.
{"title":"Approximation and decomposition of attractors of a Hopfield neural network system","authors":"Marius-F. Danca, Guanrong Chen","doi":"arxiv-2405.07567","DOIUrl":"https://doi.org/arxiv-2405.07567","url":null,"abstract":"In this paper, the Parameter Switching (PS) algorithm is used to approximate\u0000numerically attractors of a Hopfield Neural Network (HNN) system. The PS\u0000algorithm is a convergent scheme designed for approximating attractors of an\u0000autonomous nonlinear system, depending linearly on a real parameter. Aided by\u0000the PS algorithm, it is shown that every attractor of the HNN system can be\u0000expressed as a convex combination of other attractors. The HNN system can\u0000easily be written in the form of a linear parameter dependence system, to which\u0000the PS algorithm can be applied. This work suggests the possibility to use the\u0000PS algorithm as a control-like or anticontrol-like method for chaos.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"2015 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937746","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}
Jordan Orchard, Federico Frascoli, Lamberto Rondoni, Carlos Mejía-Monasterio
Polygonal billiards exhibit a rich and complex dynamical behavior. In recent years polygonal billiards have attracted great attention due to their application in the understanding of anomalous transport, but also at the fundamental level, due to its connections with diverse fields in mathematics. We explore this complexity and its consequences on the properties of particle transport in infinitely long channels made of the repetitions of an elementary open polygonal cell. Borrowing ideas from the Zemlyakov-Katok construction, we construct an interval exchange transformation classified by the singular directions of the discontinuities of the billiard flow over the translation surface associated to the elementary cell. From this, we derive an exact expression of a scattering map of the cell connecting the outgoing flow of trajectories with the unconstrained incoming flow. The scattering map is defined over a partition of the coordinate space, characterized by different families of trajectories. Furthermore, we obtain an analytical expression for the average speed of propagation of ballistic modes, describing with high accuracy the speed of propagation of ballistic fronts appearing in the tails of the distribution of the particle displacement. The symbolic hierarchy of the trajectories forming these ballistic fronts is also discussed.
{"title":"Particle transport in open polygonal billiards: a scattering map","authors":"Jordan Orchard, Federico Frascoli, Lamberto Rondoni, Carlos Mejía-Monasterio","doi":"arxiv-2405.07179","DOIUrl":"https://doi.org/arxiv-2405.07179","url":null,"abstract":"Polygonal billiards exhibit a rich and complex dynamical behavior. In recent\u0000years polygonal billiards have attracted great attention due to their\u0000application in the understanding of anomalous transport, but also at the\u0000fundamental level, due to its connections with diverse fields in mathematics.\u0000We explore this complexity and its consequences on the properties of particle\u0000transport in infinitely long channels made of the repetitions of an elementary\u0000open polygonal cell. Borrowing ideas from the Zemlyakov-Katok construction, we\u0000construct an interval exchange transformation classified by the singular\u0000directions of the discontinuities of the billiard flow over the translation\u0000surface associated to the elementary cell. From this, we derive an exact\u0000expression of a scattering map of the cell connecting the outgoing flow of\u0000trajectories with the unconstrained incoming flow. The scattering map is\u0000defined over a partition of the coordinate space, characterized by different\u0000families of trajectories. Furthermore, we obtain an analytical expression for\u0000the average speed of propagation of ballistic modes, describing with high\u0000accuracy the speed of propagation of ballistic fronts appearing in the tails of\u0000the distribution of the particle displacement. The symbolic hierarchy of the\u0000trajectories forming these ballistic fronts is also discussed.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937731","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}
Anjana S. Nair, Indranil Ghosh, Hammed O. Fatoyinbo, Sishu S. Muni
We put forward the dynamical study of a novel higher-order small network of Chialvo neurons arranged in a ring-star topology, with the neurons interacting via linear diffusive couplings. This model is perceived to imitate the nonlinear dynamical properties exhibited by a realistic nervous system where the neurons transfer information through higher-order multi-body interactions. We first analyze our model using the tools from nonlinear dynamics literature: fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the coexistence of chaotic attractors, and also an intriguing route to chaos starting from a fixed point, to period-doubling, to cyclic quasiperiodic closed invariant curves, to ultimately chaos. We numerically observe the existence of codimension-1 bifurcation patterns: saddle-node, period-doubling, and Neimark Sacker. We also qualitatively study the typical phase portraits of the system and numerically quantify chaos and complexity using the 0-1 test and sample entropy measure respectively. Finally, we study the collective behavior of the neurons in terms of two synchronization measures: the cross-correlation coefficient, and the Kuramoto order parameter.
{"title":"On the higher-order smallest ring star network of Chialvo neurons under diffusive couplings","authors":"Anjana S. Nair, Indranil Ghosh, Hammed O. Fatoyinbo, Sishu S. Muni","doi":"arxiv-2405.06000","DOIUrl":"https://doi.org/arxiv-2405.06000","url":null,"abstract":"We put forward the dynamical study of a novel higher-order small network of\u0000Chialvo neurons arranged in a ring-star topology, with the neurons interacting\u0000via linear diffusive couplings. This model is perceived to imitate the\u0000nonlinear dynamical properties exhibited by a realistic nervous system where\u0000the neurons transfer information through higher-order multi-body interactions.\u0000We first analyze our model using the tools from nonlinear dynamics literature:\u0000fixed point analysis, Jacobian matrix, and bifurcation patterns. We observe the\u0000coexistence of chaotic attractors, and also an intriguing route to chaos\u0000starting from a fixed point, to period-doubling, to cyclic quasiperiodic closed\u0000invariant curves, to ultimately chaos. We numerically observe the existence of\u0000codimension-1 bifurcation patterns: saddle-node, period-doubling, and Neimark\u0000Sacker. We also qualitatively study the typical phase portraits of the system\u0000and numerically quantify chaos and complexity using the 0-1 test and sample\u0000entropy measure respectively. Finally, we study the collective behavior of the\u0000neurons in terms of two synchronization measures: the cross-correlation\u0000coefficient, and the Kuramoto order parameter.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937613","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 propose a novel nonlinear bidirectionally coupled heterogeneous chain network whose dynamics evolve in discrete time. The backbone of the model is a pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This model is assumed to proximate the intricate dynamical properties of neurons in the widely complex nervous system. The model is first realized via various nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian matrix, and bifurcation diagrams. We observe the coexistence of chaotic and period-4 attractors. Various codimension-1 and -2 patterns for example saddle-node, period-doubling, Neimark-Sacker, double Neimark-Sacker, flip- and fold-Neimark Sacker, and 1:1 and 1:2 resonance are also explored. Furthermore, the study employs two synchronization measures to quantify how the oscillators in the network behave in tandem with each other over a long number of iterations. Finally, a time series analysis of the model is performed to investigate its complexity in terms of sample entropy.
{"title":"Dynamical properties of a small heterogeneous chain network of neurons in discrete time","authors":"Indranil Ghosh, Anjana S. Nair, Hammed Olawale Fatoyinbo, Sishu Shankar Muni","doi":"arxiv-2405.05675","DOIUrl":"https://doi.org/arxiv-2405.05675","url":null,"abstract":"We propose a novel nonlinear bidirectionally coupled heterogeneous chain\u0000network whose dynamics evolve in discrete time. The backbone of the model is a\u0000pair of popular map-based neuron models, the Chialvo and the Rulkov maps. This\u0000model is assumed to proximate the intricate dynamical properties of neurons in\u0000the widely complex nervous system. The model is first realized via various\u0000nonlinear analysis techniques: fixed point analysis, phase portraits, Jacobian\u0000matrix, and bifurcation diagrams. We observe the coexistence of chaotic and\u0000period-4 attractors. Various codimension-1 and -2 patterns for example\u0000saddle-node, period-doubling, Neimark-Sacker, double Neimark-Sacker, flip- and\u0000fold-Neimark Sacker, and 1:1 and 1:2 resonance are also explored. Furthermore,\u0000the study employs two synchronization measures to quantify how the oscillators\u0000in the network behave in tandem with each other over a long number of\u0000iterations. Finally, a time series analysis of the model is performed to\u0000investigate its complexity in terms of sample entropy.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937610","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}
R. Shashangan, S. Sudharsan, A. Venkatesan, M. Senthilvelan
Formulating mitigation strategies is one of the main aspect in the dynamical study of extreme events. Apart from the effective control, easy implementation of the devised tool should also be given importance. In this work, we analyze the mitigation of extreme events in a coupled FitzHugh-Nagumo (FHN) neuron model utilizing an easily implementable constant bias analogous to a constant DC stimulant. We report the route through which the extreme events gets mitigated in $Two$, $Three$ and $N-$coupled FHN systems. In all the three cases, extreme events in the observable $bar{x}$ gets suppressed. We confirm our results with the probability distribution function of peaks, $d_{max}$ plot and probability plots. Here $d_{max}$ is a measure of number of standard deviations that crosses the average amplitude corresponding to $bar{x}_{max}$. Interestingly, we found that constant bias suppresses the extreme events without changing the collective frequency of the system.
制定缓解策略是极端事件动态研究的主要内容之一。除了有效的控制之外，所设计工具的易于实施也应受到重视。在这项工作中，我们分析了在耦合 FitzHugh-Nagumo 神经元（FHN）模型中利用类似于恒定 DC 兴奋剂的易实现恒定偏置来缓解极端事件的问题。我们报告了在两元、三元和 N 元耦合 FHN 系统中极端事件得到缓解的途径。在所有三种情况下，观测值 $bar{x}$ 中的极端事件都会被抑制。我们用峰值概率分布函数、$d_{max}$图和概率图证实了我们的结果。这里的$d_{max}$是衡量与$bar{x}_{max}$对应的平均振幅相交的标准偏差的数量。有趣的是，我们发现恒定偏差会抑制极端事件，而不会改变系统的集体频率。
{"title":"Mitigation of extreme events in an excitable system","authors":"R. Shashangan, S. Sudharsan, A. Venkatesan, M. Senthilvelan","doi":"arxiv-2405.05994","DOIUrl":"https://doi.org/arxiv-2405.05994","url":null,"abstract":"Formulating mitigation strategies is one of the main aspect in the dynamical\u0000study of extreme events. Apart from the effective control, easy implementation\u0000of the devised tool should also be given importance. In this work, we analyze\u0000the mitigation of extreme events in a coupled FitzHugh-Nagumo (FHN) neuron\u0000model utilizing an easily implementable constant bias analogous to a constant\u0000DC stimulant. We report the route through which the extreme events gets\u0000mitigated in $Two$, $Three$ and $N-$coupled FHN systems. In all the three\u0000cases, extreme events in the observable $bar{x}$ gets suppressed. We confirm\u0000our results with the probability distribution function of peaks, $d_{max}$ plot\u0000and probability plots. Here $d_{max}$ is a measure of number of standard\u0000deviations that crosses the average amplitude corresponding to $bar{x}_{max}$.\u0000Interestingly, we found that constant bias suppresses the extreme events\u0000without changing the collective frequency of the system.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"2015 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937670","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}
Small heavy particles cannot get attracted into a region of closed streamlines in a non-accelerating frame (Sapsis & Haller 2010). In a rotating system, however, particles can get trapped (Angilella 2010) near vortices. We perform numerical simulations examining trapping of inertial particles in a prototypical rotating flow: an identical pair of rotating Lamb-Oseen vortices, without gravity. Our parameter space includes the particle Stokes number $St$, measuring the particle's inertia, and a density parameter $R$, measuring the particle-to-fluid relative density. We focus on inertial particles that are finitely denser than the fluid. Particles can get indefinitely trapped near the vortices and display extreme clustering into smaller dimensional objects: attracting fixed-points, limit cycles and chaotic attractors. As $St$ increases for a given $R$, we may have an incomplete or complete period-doubling route to chaos, as well as an unusual period-halving route back to a fixed-point attractor. The fraction of trapped particles can vary non-monotonically with $St$. We may even have windows in $St$ for which no particle trapping occurs. At $St$ larger than a critical value, beyond no trapping occurs, significant fractions of particles can spend long but finite times in the vortex vicinity. The inclusion of the Basset-Boussinesq history (BBH) force is imperative in our study due to particle's finite density. BBH force significantly increases the basin of attraction as well as the range of $St$ where trapping can occur. Extreme clustering can be physically significant in planetesimal formation by dust aggregation in protoplanetary disks, phytoplankton aggregation in oceans, etc.
{"title":"Trapping and extreme clustering of finitely-dense inertial particles near a rotating vortex pair","authors":"Saumav Kapoor, Divya Jaganathan, Rama Govindarajan","doi":"arxiv-2405.04949","DOIUrl":"https://doi.org/arxiv-2405.04949","url":null,"abstract":"Small heavy particles cannot get attracted into a region of closed\u0000streamlines in a non-accelerating frame (Sapsis & Haller 2010). In a rotating\u0000system, however, particles can get trapped (Angilella 2010) near vortices. We\u0000perform numerical simulations examining trapping of inertial particles in a\u0000prototypical rotating flow: an identical pair of rotating Lamb-Oseen vortices,\u0000without gravity. Our parameter space includes the particle Stokes number $St$,\u0000measuring the particle's inertia, and a density parameter $R$, measuring the\u0000particle-to-fluid relative density. We focus on inertial particles that are\u0000finitely denser than the fluid. Particles can get indefinitely trapped near the\u0000vortices and display extreme clustering into smaller dimensional objects:\u0000attracting fixed-points, limit cycles and chaotic attractors. As $St$ increases\u0000for a given $R$, we may have an incomplete or complete period-doubling route to\u0000chaos, as well as an unusual period-halving route back to a fixed-point\u0000attractor. The fraction of trapped particles can vary non-monotonically with\u0000$St$. We may even have windows in $St$ for which no particle trapping occurs.\u0000At $St$ larger than a critical value, beyond no trapping occurs, significant\u0000fractions of particles can spend long but finite times in the vortex vicinity.\u0000The inclusion of the Basset-Boussinesq history (BBH) force is imperative in our\u0000study due to particle's finite density. BBH force significantly increases the\u0000basin of attraction as well as the range of $St$ where trapping can occur.\u0000Extreme clustering can be physically significant in planetesimal formation by\u0000dust aggregation in protoplanetary disks, phytoplankton aggregation in oceans,\u0000etc.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"137 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937733","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}
It is well known that bursting activity plays an important role in the processes of transmission of neural signals. In terms of population dynamics, macroscopic bursting can be described using a mean-field approach. Mean field theory provides a useful tool for analysis of collective behavior of a large populations of interacting units, allowing to reduce the description of corresponding dynamics to just a few equations. Recently a new phenomenological model was proposed that describes bursting population activity of a big group of excitatory neurons, taking into account short-term synaptic plasticity and the astrocytic modulation of the synaptic dynamics [1]. The purpose of the present study is to investigate various bifurcation scenarios of the appearance of bursting activity in the phenomenological model. We show that the birth of bursting population pattern can be connected both with the cascade of period doubling bifurcations and further development of chaos according to the Shilnikov scenario, which leads to the appearance of a homoclinic attractor containing a homoclinic loop of a saddle-focus equilibrium with the two-dimensional unstable invariant manifold. We also show that the homoclinic spiral attractors observed in the system under study generate several types of bursting activity with different properties.
{"title":"Spiral Attractors in a Reduced Mean-Field Model of Neuron-Glial Interaction","authors":"Sergey Olenin, Sergey Stasenko, Tatiana Levanova","doi":"arxiv-2405.04291","DOIUrl":"https://doi.org/arxiv-2405.04291","url":null,"abstract":"It is well known that bursting activity plays an important role in the\u0000processes of transmission of neural signals. In terms of population dynamics,\u0000macroscopic bursting can be described using a mean-field approach. Mean field\u0000theory provides a useful tool for analysis of collective behavior of a large\u0000populations of interacting units, allowing to reduce the description of\u0000corresponding dynamics to just a few equations. Recently a new phenomenological\u0000model was proposed that describes bursting population activity of a big group\u0000of excitatory neurons, taking into account short-term synaptic plasticity and\u0000the astrocytic modulation of the synaptic dynamics [1]. The purpose of the\u0000present study is to investigate various bifurcation scenarios of the appearance\u0000of bursting activity in the phenomenological model. We show that the birth of\u0000bursting population pattern can be connected both with the cascade of period\u0000doubling bifurcations and further development of chaos according to the\u0000Shilnikov scenario, which leads to the appearance of a homoclinic attractor\u0000containing a homoclinic loop of a saddle-focus equilibrium with the\u0000two-dimensional unstable invariant manifold. We also show that the homoclinic\u0000spiral attractors observed in the system under study generate several types of\u0000bursting activity with different properties.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937673","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}
Neural networks have become a widely adopted tool for tackling a variety of problems in machine learning and artificial intelligence. In this contribution we use the mathematical framework of local stability analysis to gain a deeper understanding of the learning dynamics of feed forward neural networks. Therefore, we derive equations for the tangent operator of the learning dynamics of three-layer networks learning regression tasks. The results are valid for an arbitrary numbers of nodes and arbitrary choices of activation functions. Applying the results to a network learning a regression task, we investigate numerically, how stability indicators relate to the final training-loss. Although the specific results vary with different choices of initial conditions and activation functions, we demonstrate that it is possible to predict the final training loss, by monitoring finite-time Lyapunov exponents or covariant Lyapunov vectors during the training process.
{"title":"On the weight dynamics of learning networks","authors":"Nahal Sharafi, Christoph Martin, Sarah Hallerberg","doi":"arxiv-2405.00743","DOIUrl":"https://doi.org/arxiv-2405.00743","url":null,"abstract":"Neural networks have become a widely adopted tool for tackling a variety of\u0000problems in machine learning and artificial intelligence. In this contribution\u0000we use the mathematical framework of local stability analysis to gain a deeper\u0000understanding of the learning dynamics of feed forward neural networks.\u0000Therefore, we derive equations for the tangent operator of the learning\u0000dynamics of three-layer networks learning regression tasks. The results are\u0000valid for an arbitrary numbers of nodes and arbitrary choices of activation\u0000functions. Applying the results to a network learning a regression task, we\u0000investigate numerically, how stability indicators relate to the final\u0000training-loss. Although the specific results vary with different choices of\u0000initial conditions and activation functions, we demonstrate that it is possible\u0000to predict the final training loss, by monitoring finite-time Lyapunov\u0000exponents or covariant Lyapunov vectors during the training process.","PeriodicalId":501167,"journal":{"name":"arXiv - PHYS - Chaotic Dynamics","volume":"60 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831650","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}
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