[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85253-en":3,"doc-seo-85253-105":29,"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":20,"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},85253,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","On the Existence of Almost Periodic Solutions with Applications to Global Entrainment","This paper establishes two results on existence and stability of almost periodic solutions for dynamical systems, extending periodic-system theory to the almost periodic setting. Local exponential stability of the unforced system and input-to-state stability under sufficiently small inputs yield global entrainment to small almost periodic signals. These findings guarantee global entrainment in Lotka–Volterra systems with a Volterra Lyapunov-stable interaction matrix. The results extend to the uniformly recurrent case.","On the Existence of Almost Periodic Solutions with Applications to Global Entrainment  \nIasson Karafyllis* and Miroslav Krstic**  \n*Dept. of Mathematics, National Technical University of Athens, Zografou Campus, 15780, Athens, Greece,  \nemail: [iasonkar@central.ntua.gr](iasonkar@central.ntua.gr) ; [iasonkaraf@gmail.com](iasonkaraf@gmail.com)  \n**Dept. of Mechanical and Aerospace Eng., University of California, San Diego, La Jolla, CA 92093-0411, U.S.A., [email: ](email: krstic@ucsd.edu)[krstic@ucsd.edu](email: krstic@ucsd.edu)  \nTo Eduardo Sontag on the occasion of his 75th birthday  \nwith gratitude for his deep and inspiring ISS and entrainment results  \nAbstract  \nThis paper provides two results that are useful in the study of the  \nexistence and the stability properties of almost periodic solutions for a  \ngiven dynamical system. The obtained results are generalizations of  \nrecent results for periodic systems and are applied to the global  \nentrainment problem in nonlinear time-invariant control systems. It is  \nshown that local exponential stability for the unforced system and input  \nto-state stability with respect to small inputs can guarantee global  \nentrainment to small almost periodic inputs. In this way, global  \nentrainment is shown in Lotka-Volterra systems with a Volterra  \nLyapunov stable interaction matrix. All results can be extended to the  \nuniformly recurrent case.  \nKeywords: Dynamical systems, almost periodic solutions, entrainment, Lotka-Volterra models.  \nInvited paper for the ISS Special Issue dedicated to Eduardo Sontag  \n1. Introduction  \nThe problem of existence and attractivity of almost periodic solutions of systems described by ordinary differential equations is an old problem that has attracted the attention of many researchers (see for instance [2, 5, 10] and references therein) . It is a difficult mathematical problem which requires special attention because there are counterexamples which show that results in the periodic case are not valid in the almost periodic case.  \nIn this work we are inspired by Eduardo Sontag’s results for Input-to-State Stability and entrainment and we continue our recent efforts in [15] to apply conditions that involve matrix measures of the Jacobian matrix in order to explore existence and attractivity of almost periodic solutions. Under almost periodic forcing, a solution that is defined and bounded on the whole real line need not itself be almost periodic; almost periodic systems possessing bounded entire solutions but no almost periodic solutions exist. Here we provide two novel results (Theorem 1 and Theorem 2) that show that a bounded entire solution can inherit the rhythm of the excitation if a fixed amount of contraction can be accumulated within a finite window, uniformly in time and without thinning  \nas time advances (see properties (I) and (II) in Theorem 1 and Theorem 2, respectively) . The mechanism compares the solution not with a neighboring trajectory but with its own time-shift by an almost period; recurrent contraction collapses that difference to the residual mismatch of the vector field under the shift, which vanishes along the almost periods. The bounded entire solution is thereby almost periodic, with the same frequency module as the forcing. Theorem 1 is a genuine extension of Theorem 2 in [15]: this is shown by Proposition 1.  \nOur main results can be applied directly to a well-known problem: the entrainment problem. Entrainment is an important phenomenon arising in many applications ranging from biological systems to the stability of a power grid; see [3, 19, 23] and the discussion in [20] . Indeed, periodic rhythms are present in nature and regulate the function of all living organisms. From a mathematical viewpoint, the problem of proving that entrainment takes place is a very difficult problem. For nonlinear systems, driving the system by an external periodic input does not guarantee that the solution of the system converges to a period","cbCaivofw2Kmb4DG","https://ap.wps.com/l/cbCaivofw2Kmb4DG","pdf",914250,1,32,"English","en",105,"# Introduction\n## Almost periodic existence and attractivity background\n## Novel contraction-based inheritance of excitation rhythm\n## Applications to global entrainment\n## Verifiable entrainment criterion and consequences","[{\"question\":\"What does the paper prove about almost periodic solutions?\",\"answer\":\"It proves two results characterizing when bounded entire solutions exist and become almost periodic, including their stability properties, for dynamical systems under almost periodic forcing.\"},{\"question\":\"How do the stability assumptions relate to global entrainment?\",\"answer\":\"Local exponential stability of the unforced system together with input-to-state stability for small inputs ensures global entrainment of the system to small almost periodic inputs.\"},{\"question\":\"Where are the results applied, and what is guaranteed there?\",\"answer\":\"The criteria are applied to Lotka–Volterra systems, showing global entrainment when the interaction matrix is Volterra Lyapunov stable; the response converges to a signal with the same frequency module as the input.\"}]",1784202096,81,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"on-the-existence-of-almost-periodic-solutions-with-applications-to-global-entrainment","",{"@graph":35,"@context":85},[36,53,68],{"@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/on-the-existence-of-almost-periodic-solutions-with-applications-to-global-entrainment/85253/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What does the paper prove about almost periodic solutions?","Question",{"text":75,"@type":76},"It proves two results characterizing when bounded entire solutions exist and become almost periodic, including their stability properties, for dynamical systems under almost periodic forcing.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the stability assumptions relate to global entrainment?",{"text":80,"@type":76},"Local exponential stability of the unforced system together with input-to-state stability for small inputs ensures global entrainment of the system to small almost periodic inputs.",{"name":82,"@type":73,"acceptedAnswer":83},"Where are the results applied, and what is guaranteed there?",{"text":84,"@type":76},"The criteria are applied to Lotka–Volterra systems, showing global entrainment when the interaction matrix is Volterra Lyapunov stable; the response converges to a signal with the same frequency module as the input.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]