[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121319-en":3,"doc-seo-121319-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},121319,1099514067438,"River Wang","https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542",8,"Research & Report","Deciphering Complexity: Machine Learning Insights into Chaotic Dynamical Systems","We introduce new machine-learning techniques for analyzing chaotic dynamical systems. The study aims to compute the Lyapunov exponent using only two trajectory data points, avoiding the averaging procedure required by traditional methods. It also explores phase transition graphs to detect “almost integrable” trajectories by observing deviations of conserved quantities from whole numbers, and identifies “integrable regions” within chaotic trajectories. The methods are applied and tested on the “Two objects moving on a rod” and “Henon-Heiles” systems.","arXiv :2408 .02005v1 [nlin .CD] 4 Aug 2024  \nSpringer Nature 2021 LATEX template  \nDeciphering Complexity: Machine Learning Insights into Chaotic Dynamical Systems  \nLazare Osmanov 1*  \n1* School of Physics, Free University of Tbilisi, David  \nAghmashenebeli Alley, Tbilisi, 0159, Georgia.  \nCorresponding author(s) . E-mail(s): [lazare.osmanov1521r@gmail.com](lazare.osmanov1521r@gmail.com) ;  \nAbstract  \nWe introduce new machine-learning techniques for analyzing chaotic dynamical systems. The primary objectives of the study include the development of a new and simple method for calculating the Lyapunov exponent using only two trajectory data points unlike traditional methods that require an averaging procedure, the exploration of phase transition graphs from regular periodic to chaotic dynamics to identify ”almost integrable” trajectories where conserved quantities deviate from whole numbers, and the identification of ”integrable regions” within chaotic trajectories. These methods are applied and tested on two dynamical systems: ”Two objects moving on a rod” and the ”Henon-Heiles” systems.  \nKeywords: Machine learning, Chaos, Lyapunov’s exponent, Coupled  \noscillators  \n1 Introduction  \nIn recent years, artificial intelligence (AI), machine learning, and deep learning have made significant advancements across various scientific disciplines. These techniques have permeated fields such as physics, where they find application in laboratory experiments in high energy and condensed matter physics and astronomical observations [1–5] . This paper focuses on applying machine learning methods to chaos theory within dynamical systems.  \nSpringer Nature 2021 LATEX template  \n2 Deciphering Complexity: Machine Learning Insights into Chaotic Dynamical System  \nFig. 1 Schematic picture from [26] for the ”two bodies swinging on a rod” system. m1 , m2 are masses of the loads that are attached to the ropes ends, l1 , l2 are distances between the loads and touching points of the rope with the rod. The radius of the rod is R and the full length of the rope is L. The rod is a frictionless surface. The arrows indicate oscillation directions of the loads, angle θ grows in the positive, counterclockwise direction, while ϕ in the clockwise direction.  \nNot only in complex systems covering all branches of physics [7–11], but Even seemingly simple systems like the double pendulum, the Duffing oscillator, Chua’s circuit, etc exhibit chaotic behavior under specific conditions [12–20] . Understanding and controlling chaotic dynamics, which are prevalent in everyday phenomena, hold practical significance.  \nTraditionally, physicists have derived equations of motion, conservation laws, and symmetry conditions using a model-driven approach. They then analyze whether the system exhibits chaotic behavior and explore the conditions under which this occurs. This analysis often involves investigating Lyapunov exponents, which are key parameters for assessing chaos and sensitivity to initial conditions. Various computational methods exist for computing Lyapunov exponents, as well as for studying bifurcation diagrams and phase transitions that indicate when a system becomes chaotic. However, these methods typically require significant computational resources. In contrast, this paper takes a different approach by analyzing chaotic systems using data-driven methods. Instead of relying on known equations of motion, we utilize phase space trajectory data as input for our analysis. This data-driven approach offers a novel perspective on understanding chaotic behavior and may provide insights that traditional model-driven approaches overlook.  \nSpringer Nature 2021 LATEX template  \nDeciphering Complexity: Machine Learning Insights into Chaotic Dynamical Systems  \nFig. 2 Trajectories of the test systems are shown in panels (a) and (b) . Panel (a) depicts”Two bodies swinging on a rod,” while panel (b) illustrates the ”Henon-Heiles system.”Panels (c) and (d) display the corre","cbCaipbV7On8efLU","https://ap.wps.com/l/cbCaipbV7On8efLU","pdf",2255198,1,15,"English","en",105,"# Introduction\n## Background and motivation for data-driven chaos analysis\n## Problem 1: Efficient Maximal Lyapunov exponent estimation\n## Problem 2: Phase transitions and almost integrable trajectories\n## Problem 3: Integrable regions inside chaotic dynamics\n# Applications and test systems\n## Two objects moving on a rod\n## Henon-Heiles system","[{\"question\":\"How does the proposed method compute the Lyapunov exponent more efficiently?\",\"answer\":\"It computes the Maximal Lyapunov exponent using just two closely spaced initial conditions derived from trajectory data, removing the extensive averaging required by traditional approaches.\"},{\"question\":\"What is meant by “almost integrable” trajectories in the phase transition analysis?\",\"answer\":\"“Almost integrable” trajectories are identified on phase transition graphs by checking where conserved quantities deviate from whole-number values, separating regular periodic behavior from chaotic dynamics.\"},{\"question\":\"Which dynamical systems are used to apply and test the machine-learning techniques?\",\"answer\":\"The methods are applied and tested on the “Two objects moving on a rod” system and the “Henon-Heiles” system.\"}]","Deciphering Complexity: Machine Learning Insights into Chaotic Dynamical Systems | 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does the proposed method compute the Lyapunov exponent more efficiently?","Question",{"text":75,"@type":76},"It computes the Maximal Lyapunov exponent using just two closely spaced initial conditions derived from trajectory data, removing the extensive averaging required by traditional approaches.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is meant by “almost integrable” trajectories in the phase transition analysis?",{"text":80,"@type":76},"“Almost integrable” trajectories are identified on phase transition graphs by checking where conserved quantities deviate from whole-number values, separating regular periodic behavior from chaotic dynamics.",{"name":82,"@type":73,"acceptedAnswer":83},"Which dynamical systems are used to apply and test the machine-learning techniques?",{"text":84,"@type":76},"The methods are applied and tested on the “Two objects moving on a rod” system and the “Henon-Heiles” 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