BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Antonín Slavík (Charles University\, Prague) DTSTART:20220225T150000Z DTEND:20220225T160000Z DTSTAMP:20260404T111245Z UID:deg1webinar/1 DESCRIPTION:Title: Reaction-diffusion equations on graphs: stationary states and L yapunov functions\nby Antonín Slavík (Charles University\, Prague) a s part of DEG1 webinar\n\n\nAbstract\nWe focus on reaction-diffusion syste ms on discrete spatial domains represented by finite graphs (networks). In some situations\, such systems are more natural than their continuous-spa ce counterparts\, and their qualitative behavior might be different. For e xample\, unlike the continuous-space model\, the discrete-space Lotka-Volt erra competition model has stable spatially heterogeneous stationary state s. For a fairly general class of reaction-diffusion systems\, the existenc e of spatially heterogeneous stationary states is guaranteed by the implic it function theorem\, provided that the diffusion is sufficiently weak. In some applications\, the only relevant stationary states are those with no nnegative components. We present a criterion for determining which states obtained from the implicit function theorem are nonnegative. Finally\, we consider the problem of constructing Lyapunov functions for reaction-diffu sion equations on graphs. The results will be illustrated on examples from mathematical biology.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Cinzia Soresina (University of Graz) DTSTART:20220310T133000Z DTEND:20220310T143000Z DTSTAMP:20260404T111245Z UID:deg1webinar/2 DESCRIPTION:Title: Multistability and time-periodic spatial patterns in the cross- diffusion SKT model\nby Cinzia Soresina (University of Graz) as part o f DEG1 webinar\n\n\nAbstract\nThe Shigesada-Kawasaki-Teramoto model (SKT) was proposed to account for stable inhomogeneous steady states exhibiting spatial segregation\, which describes a situation of coexistence of two co mpeting species. Even though the reaction part does not present the activa tor-inhibitor structure\, the cross-diffusion terms are the key ingredient for the appearance of spatial patterns. We provide a deeper understanding of the conditions required on both the cross-diffusion and the reaction c oefficients for non-homogeneous steady states to exist\, by combining a de tailed linearised and weakly non-linear analysis with advanced numerical b ifurcation methods via the continuation software pde2path. We study the ro le of the additional cross-diffusion term in pattern formation\, focusing on multistability regions and on the presence of time-periodic spatial pat terns appearing via Hopf bifurcation points.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Lei Zhao (University of Augsburg) DTSTART:20220324T133000Z DTEND:20220324T143000Z DTSTAMP:20260404T111245Z UID:deg1webinar/3 DESCRIPTION:Title: Conformal Transformations and Integrable Mechanical Billiards\nby Lei Zhao (University of Augsburg) as part of DEG1 webinar\n\n\nAbst ract\nThe models we shall discuss are motions of a particle in the\nplane moving under the influence of a conservative force field which in\nadditio n reflect elastically against certain smooth reflection "wall".\nThe dynam ics of such a system depends on the force field and the shape\nof the refl ection wall. While one could believe that the dynamics should\ngenerally b e complicated\, some of these systems are actually integrable\nand thus ca rry dynamics with order. In this talk we shall explain how\nconformal corr espondence of natural mechanical sytems extends to\ncorrespondence between integrable mechanical billiards. This provides a\nlink between some appar ently different integrable mechanical billiards\,\nand also allows us to i dentify certain new integrable mechanical\nbilliards defined with the Kepl er and the two-center problems.\n\nThe talk is based on joint work with Ai ri Takeuchi from Karlsruhe\nInstitute of Technology.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandro Fonda (University of Trieste) DTSTART:20220407T123000Z DTEND:20220407T133000Z DTSTAMP:20260404T111245Z UID:deg1webinar/4 DESCRIPTION:Title: The Poincaré-Birkhoff theorem: coupling twist with lower and u pper solutions\nby Alessandro Fonda (University of Trieste) as part of DEG1 webinar\n\n\nAbstract\nIn 1983\, Conley and Zehnder proved a remarka ble theorem on the periodic problem associated with a general Hamiltonian system\, giving a partial answer to a conjecture by V.I. Arnold. In the sa me paper they also mentioned a possible relation of their result with the Poincaré-Birkhoff Theorem\, which was first conjectured by Poincaré in 1 912\, shortly before his death\, and then proved by Birkhoff some years la ter. The pioneering paper by Conley and Zehnder has then been extended in different directions by several authors.\n\nMore recently\, in 2017\, a de eper relation between these results and the Poincaré-Birkhoff Theorem has been established by A.J. Urena jointly with myself. Our theorem has found several applications and has been further extended in two papers written jointly with P. Gidoni. It is the aim of this talk to propose a further ex tension of this fertile theory to Hamiltonian systems which\, besides the periodicity-twist conditions always required in the Poincaré-Birkhoff The orem\, also present a pair of well-ordered lower and upper solutions.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Oleg Makarenkov (UT Dallas) DTSTART:20220428T143000Z DTEND:20220428T153000Z DTSTAMP:20260404T111245Z UID:deg1webinar/5 DESCRIPTION:Title: The occurrence of stable limit cycles in the model of a planar passive biped walking down a slope\nby Oleg Makarenkov (UT Dallas) as part of DEG1 webinar\n\n\nAbstract\nWe consider the simplest model of a pa ssive biped \nwalking down a slope given by the equations\nof switched cou pled pendula. Following\nthe fundamental work by Garcia et al. \n[J. Biom ech. Eng. 120 (1998)]\, we\nview the slope of the ground as a small parame ter $\\gamma\\geq 0$. When $\\gamma=0$\, the system can be solved in close d form\nand the existence of a family of cycles (i.e. potential\nwalking c ycles) can be computed in closed form. \nAs observed in the paper by Garci a et al.\, \nthe family of cycles disappears when $\\gamma$ increases and only isolated\nasymptotically stable cycles (walking cycles) persist.\nThe talk presents a proof of this statement using a \nsuitable perturbation t heorem for maps. I will also\nnote that the above-mentioned occurrence of limit cycles \nobserved by Garcia et al. is a so-called border-collision\n bifurcation in the modern language of nonsmooth dynamical\nsystems.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alfonso Ruiz-Herrera (University of Oviedo) DTSTART:20220512T123000Z DTEND:20220512T133000Z DTSTAMP:20260404T111245Z UID:deg1webinar/6 DESCRIPTION:Title: Topology of Attractors and Periodic Points\nby Alfonso Ruiz -Herrera (University of Oviedo) as part of DEG1 webinar\n\n\nAbstract\nThe dynamics of a dissipative and area contracting planar homeomorphism is de scribed in terms of the attractor. This is a subset of the plane defined a s the maximal compact invariant set. We prove that the coexistence of two fixed points and an $N$-cycle produces some topological complexity: the at tractor cannot be arcwise connected. The proofs are based on the theory of prime ends. We discuss several applications in periodic systems of differ ential equations. This is a joint work with Rafael Ortega.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Eduardo Liz (University of Vigo) DTSTART:20220526T123000Z DTEND:20220526T133000Z DTSTAMP:20260404T111245Z UID:deg1webinar/7 DESCRIPTION:Title: A dynamical model of happiness\nby Eduardo Liz (University of Vigo) as part of DEG1 webinar\n\n\nAbstract\nIt is now recognized that the personal well-being of an individual can be evaluated numerically. The related hedonic utility (happiness) profile would give at each instant $t $ the degree $u(t)$ of happiness. The moment-based approach to the evalua tion of happiness introduced by the Nobel laureate Daniel Kahneman establi shes that the experienced utility of an episode can be derived from real- time measures of the pleasure and pain that the subject experienced during that episode. Since these evaluations consist of two types of utility con cepts: instant utility and remembered utility\, a dynamical model of happi ness based on this approach must be defined by a delay differential equati on. Furthermore\, the application of the peak-end rule leads to a class of delay-differential equations called differential equations with maxima. W e propose a dynamical model for happiness based on differential equations with maxima and provide rigorous mathematical results which support some e xperimental observations such as the U-shape of happiness over the life cy cle and the unpredictability of happiness.\nThe talk is based on joint wor k with Elena Trofimchuk and Sergei Trofimchuk.\n LOCATION:https://stable.researchseminars.org/talk/deg1webinar/7/ END:VEVENT END:VCALENDAR