[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82312-en":3,"doc-seo-82312-105":29,"detail-sidebar-cat-0-en-105":90},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},82312,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Global Continuation as a Complement to Traditional Continuation and Bifurcation Analysis","Multistable dynamical systems model ecosystems, power grids, climate elements, neurons, and other complex processes where perturbations can induce tipping between coexisting states. Traditional bifurcation-diagram approaches rely on local continuation, which tracks individual attractors using linearized response to infinitesimal perturbations and may become inaccurate for finite disturbances. Global continuation is introduced to simultaneously find and continue system attractors under finite perturbations by exploiting information across the full state space and emphasizing practitioner-relevant observables. An open-source implementation in DynamicalSystems.jl supports accessibility, reliability, and practical troubleshooting.","Global continuation as a complement to traditional continuation and  \nbifurcation analysis  \nGeorge Datseris 1, * , Andreas Morr2, 3 , Muhammed Fadera4 , and J¨urgen Kurths3, 5  \n1 Department of Mathematics and Statistics, University of Exeter  \n2 Department of Mathematics, School of Computation, Information and Technology, Technical University of Munich, Boltzmannstraße 3, Garching, Germany  \n3 Potsdam Institute for Climate Impact Research, 14412, Potsdam, Germany  \n4 Global Systems Institute, University of Exeter  \n5 Department of Physics, Humboldt University, 10099, Berlin, Germany  \n* Correspondence: [g.datseris@exeter.ac.uk](g.datseris@exeter.ac.uk)  \nJuly 13, 2026  \narXiv :2607 .09332v1 [nlin .CD] 10 Jul 2026  \nAbstract  \nMultistable dynamical systems are ever-prevalent, used to model for example ecosystems, power grids, climate elements, neurons, and more. When perturbed, such systems may “tip” from one state of operation to another, often with abrupt, irreversible, and high-impact consequences in each context. Traditionally, these systems are analysed via bifurcation diagrams, the result of a process we refer to as local continuation, as it only captures the linear (local) system response to infinitesimal perturbations. Local continuation requires substantial expertise, constant interventions, and may yield inaccurate assessment of the system’s response to large perturbations that is crucial for tipping analysis. To address some inherent challenges of local continuation and to provide fundamentally new information during a continuation, this paper introduces global continuation as a complement suitable for the study of multistability, critical transitions and real-world-oriented applications. Global continuation finds and continues in parallel (practically) all system attractors and their response to finite perturbations by synthesising information from the whole state space, while placing a focus on the qualities or observables of a dynamical system that the practitioner cares about in context. Global continuation does not require deep expertise and is effortless to use and troubleshoot, making it attractive to applied scientists from different disciplines. We highlight several unique advantages that allow global continuation to complement the status quo and exemplify them through a plethora of representative examples. Global continuation is also implemented as open source software in DynamicalSystems.jl, enhancing its accessibility.  \n1 Introduction  \nMultistable dynamical systems are used in science and technology to model a wide range of natural systems, ranging from the Earth’s climate [1] to cell biology [2] . Whenever natural systems admit alternative coexisting stable states or exhibit critical transitions (also called regime  \nshifts or tipping points [3]), multistable dynamical systems provide a framework for understanding and analysing the respective system behaviour. This modelling approach has enjoyed success not only in climate science (popularized in [4] but already discussed in [5, 6 , 7]) and ecology [8, 9], but also in power grids [10], fluid flows [11], atmospheric flows [12], lasers [13, 14], and many more [15] . In general, critical transitions are more than bifurcations; they are fundamental changes in system behaviour that maybe triggered by bifurcations or by perturbations in system parameters or state, either close or even far away from a bifurcation point (known as noise [3], rate [16], or shock tipping [17]) .  \nThe most commonly employed approach to study dynamical models and their critical transitions is local continuation, hitherto called just “continuation”. Continuation is a numerical technique for tracking a system’s steady state(s) and other properties as parameter(s) are varied. Local continuation, specifically, achieves this by tracking an individual fixed point or limit cycle over primarily a single parameter and recording the local (linearized) stability of the state [18] . The key","cbCaipVgad3LZf17","https://ap.wps.com/l/cbCaipVgad3LZf17","pdf",3884856,1,23,"English","en",105,"# Introduction\n## Multistable dynamical systems and critical transitions\n## Local continuation: limitations\n## Global continuation: core idea and accessibility\n# Method philosophy and formulation","[{\"question\":\"What problem does global continuation address compared with traditional local continuation?\",\"answer\":\"Local continuation tracks individual fixed points or limit cycles using linearized stability and can miss dynamics away from the attractor, making it less reliable for finite perturbations. Global continuation focuses on finding and continuing all attractors while capturing behavior beyond the linear regime.\"},{\"question\":\"How does global continuation handle tipping and critical transitions?\",\"answer\":\"It is designed to study multistability and stability beyond linearity, which is essential for analyzing critical transitions. By synthesizing information across the full state space, it tracks attractors’ responses to finite perturbations.\"},{\"question\":\"What makes the global continuation approach practical for applied scientists?\",\"answer\":\"The method automatically finds and tracks (practically) all attractors while ignoring unstable states, and it requires only basic familiarity with attractors rather than deep expertise. The paper also notes an open-source implementation in DynamicalSystems.jl to improve usability and troubleshooting.\"}]",1784179540,58,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"global-continuation-as-a-complement-to-traditional-continuation-and-bifurcation-analysis","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/global-continuation-as-a-complement-to-traditional-continuation-and-bifurcation-analysis/82312/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does global continuation address compared with traditional local continuation?","Question",{"text":74,"@type":75},"Local continuation tracks individual fixed points or limit cycles using linearized stability and can miss dynamics away from the attractor, making it less reliable for finite perturbations. Global continuation focuses on finding and continuing all attractors while capturing behavior beyond the linear regime.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does global continuation handle tipping and critical transitions?",{"text":79,"@type":75},"It is designed to study multistability and stability beyond linearity, which is essential for analyzing critical transitions. By synthesizing information across the full state space, it tracks attractors’ responses to finite perturbations.",{"name":81,"@type":72,"acceptedAnswer":82},"What makes the global continuation approach practical for applied scientists?",{"text":83,"@type":75},"The method automatically finds and tracks (practically) all attractors while ignoring unstable states, and it requires only basic familiarity with attractors rather than deep expertise. The paper also notes an open-source implementation in DynamicalSystems.jl to improve usability and troubleshooting.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]