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This edited volume gathers selected, peer-reviewed contributions
presented at the fourth International Conference on Differential
& Difference Equations Applications (ICDDEA), which was held in
Lisbon, Portugal, in July 2019. First organized in 2011, the ICDDEA
conferences bring together mathematicians from various countries in
order to promote cooperation in the field, with a particular focus
on applications. The book includes studies on boundary value
problems; Markov models; time scales; non-linear difference
equations; multi-scale modeling; and myriad applications.
This work focuses on the preservation of attractors and saddle
points of ordinary differential equations under discretisation. In
the 1980s, key results for autonomous ordinary differential
equations were obtained - by Beyn for saddle points and by Kloeden
& Lorenz for attractors. One-step numerical schemes with a
constant step size were considered, so the resulting discrete time
dynamical system was also autonomous. One of the aims of this book
is to present new findings on the discretisation of dissipative
nonautonomous dynamical systems that have been obtained in recent
years, and in particular to examine the properties of nonautonomous
omega limit sets and their approximations by numerical schemes -
results that are also of importance for autonomous systems
approximated by a numerical scheme with variable time steps, thus
by a discrete time nonautonomous dynamical system.
This book gathers papers from the International Conference on
Differential & Difference Equations and Applications 2017
(ICDDEA 2017), held in Lisbon, Portugal on June 5-9, 2017. The
editors have compiled the strongest research presented at the
conference, providing readers with valuable insights into new
trends in the field, as well as applications and high-level survey
results. The goal of the ICDDEA was to promote fruitful
collaborations between researchers in the fields of differential
and difference equations. All areas of differential and difference
equations are represented, with a special emphasis on applications.
The authors provide a fast introduction to probabilistic and statistical concepts necessary to understand the basic ideas and methods of stochastic differential equations. The book is based on measure theory which is introduced as smoothly as possible. It is intended for advanced undergraduate students or graduates, not necessarily in mathematics, providing an overview and intuitive background for more advanced studies as well as some practical skills in the use of MAPLE in the context of probability and its applications. As prerequisites the authors assume a familiarity with basic calculus and linear algebra, as well as with elementary ordinary differential equations and, in the final chapter, simple numerical methods for such ODEs. Although statistics is not systematically treated, they introduce statistical concepts such as sampling, estimators, hypothesis testing, confidence intervals, significance levels and p-values and use them in a large number of examples, problems and simulations.
This volume contains the notes from five lecture courses devoted to
nonautonomous differential systems, in which appropriate
topological and dynamical techniques were described and applied to
a variety of problems. The courses took place during the C.I.M.E.
Session "Stability and Bifurcation Problems for Non-Autonomous
Differential Equations," held in Cetraro, Italy, June 19-25 2011.
Anna Capietto and Jean Mawhin lectured on nonlinear boundary value
problems; they applied the Maslov index and degree-theoretic
methods in this context. Rafael Ortega discussed the theory of
twist maps with nonperiodic phase and presented applications. Peter
Kloeden and Sylvia Novo showed how dynamical methods can be used to
study the stability/bifurcation properties of bounded solutions and
of attracting sets for nonautonomous differential and
functional-differential equations. The volume will be of interest
to all researchers working in these and related fields.
There is an extensive literature in the form of papers (but no
books) on lattice dynamical systems. The book focuses on
dissipative lattice dynamical systems and their attractors of
various forms such as autonomous, nonautonomous and random. The
existence of such attractors is established by showing that the
corresponding dynamical system has an appropriate kind of absorbing
set and is asymptotically compact in some way.There is now a very
large literature on lattice dynamical systems, especially on
attractors of all kinds in such systems. We cannot hope to do
justice to all of them here. Instead, we have focused on key areas
of representative types of lattice systems and various types of
attractors. Our selection is biased by our own interests, in
particular to those dealing with biological applications. One of
the important results is the approximation of Heaviside switching
functions in LDS by sigmoidal functions.Nevertheless, we believe
that this book will provide the reader with a solid introduction to
the field, its main results and the methods that are used to obtain
them.
The nature of time in a nonautonomous dynamical system is very
different from that in autonomous systems, which depend only on the
time that has elapsed since starting rather than on the actual time
itself. Consequently, limiting objects may not exist in actual time
as in autonomous systems. New concepts of attractors in
nonautonomous dynamical system are thus required.In addition, the
definition of a dynamical system itself needs to be generalised to
the nonautonomous context. Here two possibilities are considered:
two-parameter semigroups or processes and the skew product flows.
Their attractors are defined in terms of families of sets that are
mapped onto each other under the dynamics rather than a single set
as in autonomous systems. Two types of attraction are now possible:
pullback attraction, which depends on the behaviour from the system
in the distant past, and forward attraction, which depends on the
behaviour of the system in the distant future. These are generally
independent of each other.The component subsets of pullback and
forward attractors exist in actual time. The asymptotic behaviour
in the future limit is characterised by omega-limit sets, in terms
of which form what are called forward attracting sets. They are
generally not invariant in the conventional sense, but are
asymptotically invariant in general and, if the future dynamics is
appropriately uniform, also asymptotically negatively
invariant.Much of this book is based on lectures given by the
authors in Frankfurt and Wuhan. It was written mainly when the
first author held a 'Thousand Expert' Professorship at the Huazhong
University of Science and Technology in Wuhan.
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