Arithmetic dynamics :
Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics,
dynamical systems and
number theory. Classically, discrete dynamics refers to the study of the
iteration of self-maps of the
complex plane or
real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, -adic, and/or algebraic points under repeated application of a
polynomial or
rational function.
Chaos theory :
Chaos theory describes the behavior of certain
dynamical systems – that is, systems whose state evolves with time – that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the
butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations in the initial conditions, the behavior of chaotic systems appears
random. This happens even though these systems are
deterministic, meaning that their future dynamics are fully defined by their initial conditions, with no random elements involved. This behavior is known as deterministic chaos, or simply
chaos.
Complex systems :
Complex systems is a scientific field that studies the common properties of
systems considered
complex in
nature,
society, and
science. It is also called
complex systems theory,
complexity science,
study of complex systems and/or
sciences of complexity. The key problems of such systems are difficulties with their formal
modeling and
simulation. From such perspective, in different research contexts complex systems are defined on the base of their different attributes. :The study of complex systems is bringing new vitality to many areas of science where a more typical
reductionist strategy has fallen short.
Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including
neurosciences,
social sciences,
meteorology,
chemistry,
physics,
computer science,
psychology,
artificial life,
evolutionary computation,
economics, earthquake prediction,
molecular biology and inquiries into the nature of living
cells themselves.
Control theory :
Control theory is an interdisciplinary branch of
engineering and
mathematics, in part it deals with influencing the behavior of
dynamical systems.
Ergodic theory :
Ergodic theory is a branch of
mathematics that studies
dynamical systems with an
invariant measure and related problems. Its initial development was motivated by problems of
statistical physics.
Functional analysis :
Functional analysis is the branch of
mathematics, and specifically of
analysis, concerned with the study of
vector spaces and
operators acting upon them. It has its historical roots in the study of
functional spaces, in particular transformations of
functions, such as the
Fourier transform, as well as in the study of
differential and
integral equations. This usage of the word
functional goes back to the
calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to mathematician and physicist
Vito Volterra and its founding is largely attributed to mathematician
Stefan Banach.
Graph dynamical systems :The concept of
graph dynamical systems (GDS) can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of graph dynamical systems is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result.
Projected dynamical systems :
Projected dynamical systems is a
mathematical theory investigating the behaviour of
dynamical systems where solutions are restricted to a constraint set. The discipline shares connections to and applications with both the static world of
optimization and
equilibrium problems and the dynamical world of
ordinary differential equations. A projected dynamical system is given by the
flow to the projected differential equation.
Symbolic dynamics :
Symbolic dynamics is the practice of modelling a topological or smooth
dynamical system by a discrete space consisting of infinite
sequences of abstract symbols, each of which corresponds to a state of the system, with the dynamics (evolution) given by the
shift operator.
System dynamics :
System dynamics is an approach to understanding the behaviour of systems over time. It deals with internal feedback loops and time delays that affect the behaviour and state of the entire system. What makes using system dynamics different from other approaches to studying systems is the language used to describe
feedback loops with
stocks and flows. These elements help describe how even seemingly simple systems display baffling
nonlinearity.
Topological dynamics :
Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of
general topology. == Applications ==