Dynamical system
A dynamic system can be described by a pair
where
is a nonempty set called also state space and
is a function called law of motion from
into
. Thus, if
is the state of the system at time
, then
Is the state of the system at time
. As examples we can take
to be the set
of real numbers, and defined as:
Where
and
are real numbers. The evolution of the dynamical system
is described by the difference equation:
Once the initial state
is defined, one ca write
,
and for every positive integer
:
In the study of the dynamical systems one can define a fixed and periodic points which capture the intuitive idea of stationary or an equilibrium of a dynamical system. A point
is a fixed point if
. A point
is a periodic point of a period
if
and
for
.
The attractive
points are a special case of mathematical attractor concept.
