[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83884-en":3,"doc-seo-83884-105":30,"detail-sidebar-cat-0-en-105":84},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83884,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Weighted Phase Volume Method Instability Analysis Integral Criteria and Ellipsoidal Reachable Sets","A method for analysing the stability of dynamical systems is proposed by introducing a weighted phase volume and performing time rescaling through a positive function. The method controls phase-volume contraction via the weighting function and scaling factor while preserving phase-portrait topology. Integral dissipativity conditions yield new notions of integral, asymptotic, and exponential stability. For quadratic weights, ellipsoidal covering and inner ellipsoids provide geometric bounds on reachable sets. Links to classical Lyapunov stability are proved and illustrated with numerical examples.","arXiv :2607 .05033v1 [ ee ss . SY] 6 Jul 2026  \nWEIGHTED PHASE VOLUME METHOD INSTABILITY ANALYSIS: INTEGRAL CRITERIA AND ELLIPSOIDAL REACHABLE SETS  \nIgor B. Furtat  \nJuly 7, 2026  \nInstitute for Problems in Mechanical Engineering, Russian Academy of Sciences,  \nSt. Petersburg, Russia  \n[cainenash@mail.ru](cainenash@mail.ru)  \nAbstract  \nA method for analysing the stability of dynamical systems is proposed, based on the introduction of a weighted phase volume and time rescaling by a positive function. The advantage of the method is the ability to set the contraction properties of the phase volume by choosing the weighting function and the scaling factor, while preserving the topology of the phase portrait. Integral dissipativity conditions are derived, leading to new definitions of integral stability, asymptotic stability, and exponential stability. For quadratic weighting functions, covering and inner ellipsoids are constructed, providing geometric estimates of reachable sets. The connection between the proposed approach and classical Lyapunov stability is established. The efficiency of the method is demonstrated through numerical examples.  \nKeywords: Liouville’s theorem, Reynolds transport theorem, weighted phase volume, dissipativity, ellipsoidal approximation, stability divergence method, Lyapunov stability.  \n1 Introduction  \nThe stability of dynamical systems is still one of the important problems in differential equation theory and mathematical physics. The classical Lyapunov method [1]  \nand its developments in [2–5] provide effective tools for analysing equilibrium stability. However, constructing Lyapunov functions for complex, nonlinear, nonautonomous or discontinuous systems remains a significant challenge.  \nAlternative approaches based on the geometric properties of the vector field go back to the classical Liouville theorem [6–8] and the Reynolds transport theorem [9] . They describe the evolution of phase volume and establish a link between stability and the sign of divergence in vector fields. Pioneering works [10–12] laid the foundations of divergence stability analysis. In [13], index and divergence criteria for the stability of a singular point were obtained. In a series of works by V.P. Zhukov [14–17], the source and sink method is developed, enabling the formulation of necessary and sufficient conditions for the instability and asymptotic stability of nonlinear autonomous systems. However, these results are often limited by the dimension of the phase space, or they require specific assumptions about the structure of the vector field.  \nA significant step is taken by A. Rantzer in [18,19], where the concept of dual Lyapunov functions (density method) is introduced. This stability concept is proposed for almost all initial conditions [18–20] . Linear matrix inequalities are also introduced to verify these conditions.  \nFurther development of [14–20] is presented in [21–26] . These developments include divergence stability conditions and methods for analysing nonautonomous and perturbed systems, as well as the theory of density systems. In contrast to the classical Lyapunov method [1–5], the approaches [14–26] do not provide direct geometric information about reachable sets.  \nSeveral studies have focused on the analysis of dissipativity and stability of nonconservative systems. In [27], the concept of dissipative dynamical systems with quadratic supply rates is introduced. In [28], an energy-based approach to constructing Lyapunov functions for physical systems is proposed. In [29], a dynamical equivalence between a Lyapunov function and a potential function is established. In [30], loss of stability of nonconservative systems in regions of divergence instability is investigated. In [31], a new dissipativity criterion based on the notion of dissipative power was proposed, surpassing the sensitivity of the classical divergence criterion. In [32–36], modern numerical and analytical methods for stability analysis are pres","cbCaisN4kbMoKojR","https://ap.wps.com/l/cbCaisN4kbMoKojR","pdf",442250,5,1,31,"English","en",105,"# Introduction\n## Background and related stability methods\n## Weighted phase volume and time rescaling\n## Main results and differences from prior work","[{\"question\":\"What is the relationship between the new approach and classical Lyapunov stability?\",\"answer\":\"A connection is established between the proposed weighted-phase-volume framework and classical Lyapunov stability, clarifying how the integral and geometric criteria relate to traditional stability notions.\"}]",1784191216,78,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":79,"head_meta":81,"extra_data":83,"updated_unix":28},"weighted-phase-volume-method-instability-analysis-integral-criteria-and-ellipsoidal-reachable-sets","",{"@graph":36,"@context":78},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/weighted-phase-volume-method-instability-analysis-integral-criteria-and-ellipsoidal-reachable-sets/83884/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72],{"name":73,"@type":74,"acceptedAnswer":75},"What is the relationship between the new approach and classical Lyapunov stability?","Question",{"text":76,"@type":77},"A connection is established between the proposed weighted-phase-volume framework and classical Lyapunov stability, clarifying how the integral and geometric criteria relate to traditional stability notions.","Answer","https://schema.org",{"og:url":52,"og:type":80,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":82,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":85},[86,90,94,98,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":87,"show_sort_weight":88,"slug":89},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":20,"slug":130},19,"General","general"]