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This book is intended to make recent results on the derivation of
higher order numerical schemes for random ordinary differential
equations (RODEs) available to a broader readership, and to
familiarize readers with RODEs themselves as well as the closely
associated theory of random dynamical systems. In addition, it
demonstrates how RODEs are being used in the biological sciences,
where non-Gaussian and bounded noise are often more realistic than
the Gaussian white noise in stochastic differential equations
(SODEs). RODEs are used in many important applications and play a
fundamental role in the theory of random dynamical systems. They
can be analyzed pathwise with deterministic calculus, but require
further treatment beyond that of classical ODE theory due to the
lack of smoothness in their time variable. Although classical
numerical schemes for ODEs can be used pathwise for RODEs, they
rarely attain their traditional order since the solutions of RODEs
do not have sufficient smoothness to have Taylor expansions in the
usual sense. However, Taylor-like expansions can be derived for
RODEs using an iterated application of the appropriate chain rule
in integral form, and represent the starting point for the
systematic derivation of consistent higher order numerical schemes
for RODEs. The book is directed at a wide range of readers in
applied and computational mathematics and related areas as well as
readers who are interested in the applications of mathematical
models involving random effects, in particular in the biological
sciences.The level of this book is suitable for graduate students
in applied mathematics and related areas, computational sciences
and systems biology. A basic knowledge of ordinary differential
equations and numerical analysis is required.
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 is intended to make recent results on the derivation of
higher order numerical schemes for random ordinary differential
equations (RODEs) available to a broader readership, and to
familiarize readers with RODEs themselves as well as the closely
associated theory of random dynamical systems. In addition, it
demonstrates how RODEs are being used in the biological sciences,
where non-Gaussian and bounded noise are often more realistic than
the Gaussian white noise in stochastic differential equations
(SODEs). RODEs are used in many important applications and play a
fundamental role in the theory of random dynamical systems. They
can be analyzed pathwise with deterministic calculus, but require
further treatment beyond that of classical ODE theory due to the
lack of smoothness in their time variable. Although classical
numerical schemes for ODEs can be used pathwise for RODEs, they
rarely attain their traditional order since the solutions of RODEs
do not have sufficient smoothness to have Taylor expansions in the
usual sense. However, Taylor-like expansions can be derived for
RODEs using an iterated application of the appropriate chain rule
in integral form, and represent the starting point for the
systematic derivation of consistent higher order numerical schemes
for RODEs. The book is directed at a wide range of readers in
applied and computational mathematics and related areas as well as
readers who are interested in the applications of mathematical
models involving random effects, in particular in the biological
sciences.The level of this book is suitable for graduate students
in applied mathematics and related areas, computational sciences
and systems biology. A basic knowledge of ordinary differential
equations and numerical analysis is required.
This book offers an introduction to the theory of non-autonomous
and stochastic dynamical systems, with a focus on the importance of
the theory in the Applied Sciences. It starts by discussing the
basic concepts from the theory of autonomous dynamical systems,
which are easier to understand and can be used as the motivation
for the non-autonomous and stochastic situations. The book
subsequently establishes a framework for non-autonomous dynamical
systems, and in particular describes the various approaches
currently available for analysing the long-term behaviour of
non-autonomous problems. Here, the major focus is on the novel
theory of pullback attractors, which is still under development. In
turn, the third part represents the main body of the book,
introducing the theory of random dynamical systems and random
attractors and revealing how it may be a suitable candidate for
handling realistic models with stochasticity. A discussion of
future research directions serves to round out the coverage.
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.
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