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SUMMARY:Antonín Slavík (Charles University\, Prague)
DTSTART;VALUE=DATE-TIME:20220225T150000Z
DTEND;VALUE=DATE-TIME:20220225T160000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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SUMMARY:Cinzia Soresina (University of Graz)
DTSTART;VALUE=DATE-TIME:20220310T133000Z
DTEND;VALUE=DATE-TIME:20220310T143000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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SUMMARY:Lei Zhao (University of Augsburg)
DTSTART;VALUE=DATE-TIME:20220324T133000Z
DTEND;VALUE=DATE-TIME:20220324T143000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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SUMMARY:Alessandro Fonda (University of Trieste)
DTSTART;VALUE=DATE-TIME:20220407T123000Z
DTEND;VALUE=DATE-TIME:20220407T133000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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BEGIN:VEVENT
SUMMARY:Oleg Makarenkov (UT Dallas)
DTSTART;VALUE=DATE-TIME:20220428T143000Z
DTEND;VALUE=DATE-TIME:20220428T153000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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SUMMARY:Alfonso Ruiz-Herrera (University of Oviedo)
DTSTART;VALUE=DATE-TIME:20220512T123000Z
DTEND;VALUE=DATE-TIME:20220512T133000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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SUMMARY:Eduardo Liz (University of Vigo)
DTSTART;VALUE=DATE-TIME:20220526T123000Z
DTEND;VALUE=DATE-TIME:20220526T133000Z
DTSTAMP;VALUE=DATE-TIME:20220816T032131Z
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/
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