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:20260422T225843Z UID:deg1webinar/1 DESCRIPTION:Title: Reaction-diffusion equations on graphs: stationary states and Lyapunov functions\nby Antonín Slavík (Charles University\, Prague) as part of DEG1 webinar\n\n\nAbstract\nWe focus on reaction-diffusion systems on d iscrete spatial domains represented by finite graphs (networks). In some s ituations\, such systems are more natural than their continuous-space coun terparts\, and their qualitative behavior might be different. For example\ , unlike the continuous-space model\, the discrete-space Lotka-Volterra co mpetition model has stable spatially heterogeneous stationary states. For a fairly general class of reaction-diffusion systems\, the existence of sp atially heterogeneous stationary states is guaranteed by the implicit func tion theorem\, provided that the diffusion is sufficiently weak. In some a pplications\, the only relevant stationary states are those with nonnegati ve components. We present a criterion for determining which states obtaine d from the implicit function theorem are nonnegative. Finally\, we conside r the problem of constructing Lyapunov functions for reaction-diffusion eq uations on graphs. The results will be illustrated on examples from mathem atical biology.\n LOCATION:https://researchseminars.org/talk/deg1webinar/1/ END:VEVENT BEGIN:VEVENT SUMMARY:Cinzia Soresina (University of Graz) DTSTART:20220310T133000Z DTEND:20220310T143000Z DTSTAMP:20260422T225843Z UID:deg1webinar/2 DESCRIPTION:Title: Multistability and time-periodic spatial patterns in the cross-diffusi on SKT model\nby Cinzia Soresina (University of Graz) as part of DEG1 webinar\n\n\nAbstract\nThe Shigesada-Kawasaki-Teramoto model (SKT) was pro posed to account for stable inhomogeneous steady states exhibiting spatial segregation\, which describes a situation of coexistence of two competing species. Even though the reaction part does not present the activator-inh ibitor structure\, the cross-diffusion terms are the key ingredient for th e appearance of spatial patterns. We provide a deeper understanding of the conditions required on both the cross-diffusion and the reaction coeffici ents for non-homogeneous steady states to exist\, by combining a detailed linearised and weakly non-linear analysis with advanced numerical bifurcat ion methods via the continuation software pde2path. We study the role of t he additional cross-diffusion term in pattern formation\, focusing on mult istability regions and on the presence of time-periodic spatial patterns a ppearing via Hopf bifurcation points.\n LOCATION:https://researchseminars.org/talk/deg1webinar/2/ END:VEVENT BEGIN:VEVENT SUMMARY:Lei Zhao (University of Augsburg) DTSTART:20220324T133000Z DTEND:20220324T143000Z DTSTAMP:20260422T225843Z UID:deg1webinar/3 DESCRIPTION:Title: Conformal Transformations and Integrable Mechanical Billiards\nby Lei Zhao (University of Augsburg) as part of DEG1 webinar\n\n\nAbstract\nT he models we shall discuss are motions of a particle in the\nplane moving under the influence of a conservative force field which in\naddition refle ct elastically against certain smooth reflection "wall".\nThe dynamics of such a system depends on the force field and the shape\nof the reflection wall. While one could believe that the dynamics should\ngenerally be compl icated\, some of these systems are actually integrable\nand thus carry dyn amics with order. In this talk we shall explain how\nconformal corresponde nce of natural mechanical sytems extends to\ncorrespondence between integr able mechanical billiards. This provides a\nlink between some apparently d ifferent integrable mechanical billiards\,\nand also allows us to identify certain new integrable mechanical\nbilliards defined with the Kepler and the two-center problems.\n\nThe talk is based on joint work with Airi Take uchi from Karlsruhe\nInstitute of Technology.\n LOCATION:https://researchseminars.org/talk/deg1webinar/3/ END:VEVENT BEGIN:VEVENT SUMMARY:Alessandro Fonda (University of Trieste) DTSTART:20220407T123000Z DTEND:20220407T133000Z DTSTAMP:20260422T225843Z UID:deg1webinar/4 DESCRIPTION:Title: The Poincaré-Birkhoff theorem: coupling twist with lower and upper so lutions\nby Alessandro Fonda (University of Trieste) as part of DEG1 w ebinar\n\n\nAbstract\nIn 1983\, Conley and Zehnder proved a remarkable the orem on the periodic problem associated with a general Hamiltonian system\ , giving a partial answer to a conjecture by V.I. Arnold. In the same pape r they also mentioned a possible relation of their result with the Poincar é-Birkhoff Theorem\, which was first conjectured by Poincaré in 1912\, s hortly before his death\, and then proved by Birkhoff some years later. Th e pioneering paper by Conley and Zehnder has then been extended in differe nt directions by several authors.\n\nMore recently\, in 2017\, a deeper re lation between these results and the Poincaré-Birkhoff Theorem has been e stablished by A.J. Urena jointly with myself. Our theorem has found severa l 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 extension of this fertile theory to Hamiltonian systems which\, besides the periodi city-twist conditions always required in the Poincaré-Birkhoff Theorem\, also present a pair of well-ordered lower and upper solutions.\n LOCATION:https://researchseminars.org/talk/deg1webinar/4/ END:VEVENT BEGIN:VEVENT SUMMARY:Oleg Makarenkov (UT Dallas) DTSTART:20220428T143000Z DTEND:20220428T153000Z DTSTAMP:20260422T225843Z 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 passive b iped \nwalking down a slope given by the equations\nof switched coupled pe ndula. Following\nthe fundamental work by Garcia et al. \n[J. Biomech. En g. 120 (1998)]\, we\nview the slope of the ground as a small parameter $\\ gamma\\geq 0$. When $\\gamma=0$\, the system can be solved in closed form\ nand the existence of a family of cycles (i.e. potential\nwalking cycles) can be computed in closed form. \nAs observed in the paper by Garcia et al .\, \nthe family of cycles disappears when $\\gamma$ increases and only is olated\nasymptotically stable cycles (walking cycles) persist.\nThe talk p resents a proof of this statement using a \nsuitable perturbation theorem for maps. I will also\nnote that the above-mentioned occurrence of limit c ycles \nobserved by Garcia et al. is a so-called border-collision\nbifurca tion in the modern language of nonsmooth dynamical\nsystems.\n LOCATION:https://researchseminars.org/talk/deg1webinar/5/ END:VEVENT BEGIN:VEVENT SUMMARY:Alfonso Ruiz-Herrera (University of Oviedo) DTSTART:20220512T123000Z DTEND:20220512T133000Z DTSTAMP:20260422T225843Z UID:deg1webinar/6 DESCRIPTION:Title: Topology of Attractors and Periodic Points\nby Alfonso Ruiz-Herrer a (University of Oviedo) as part of DEG1 webinar\n\n\nAbstract\nThe dynami cs of a dissipative and area contracting planar homeomorphism is described in terms of the attractor. This is a subset of the plane defined as the m aximal compact invariant set. We prove that the coexistence of two fixed p oints and an $N$-cycle produces some topological complexity: the attractor cannot be arcwise connected. The proofs are based on the theory of prime ends. We discuss several applications in periodic systems of differential equations. This is a joint work with Rafael Ortega.\n LOCATION:https://researchseminars.org/talk/deg1webinar/6/ END:VEVENT BEGIN:VEVENT SUMMARY:Eduardo Liz (University of Vigo) DTSTART:20220526T123000Z DTEND:20220526T133000Z DTSTAMP:20260422T225843Z 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 per sonal well-being of an individual can be evaluated numerically. The relate d hedonic utility (happiness) profile would give at each instant $t$ the d egree $u(t)$ of happiness. The moment-based approach to the evaluation of happiness introduced by the Nobel laureate Daniel Kahneman establishes th at the experienced utility of an episode can be derived from real-time me asures of the pleasure and pain that the subject experienced during that e pisode. Since these evaluations consist of two types of utility concepts: instant utility and remembered utility\, a dynamical model of happiness ba sed on this approach must be defined by a delay differential equation. Fur thermore\, the application of the peak-end rule leads to a class of delay- differential equations called differential equations with maxima. We propo se a dynamical model for happiness based on differential equations with ma xima and provide rigorous mathematical results which support some experime ntal observations such as the U-shape of happiness over the life cycle and the unpredictability of happiness.\nThe talk is based on joint work with Elena Trofimchuk and Sergei Trofimchuk.\n LOCATION:https://researchseminars.org/talk/deg1webinar/7/ END:VEVENT END:VCALENDAR