MATH 614
Dynamical Systems and Chaos
Lecture 1:
Examples of dynamical systems.
A discrete dynamical system is simply a transformation
f : X X . The set X is regarded the phase space of the
system and the map f is considered the law of evolution over a
per
MATH 614
Dynamical Systems and Chaos
Lecture 11:
Maps of the circle.
Circle S 1 .
S 1 = cfw_(x, y ) R2 : |x|2 + |y |2 = 1
S 1 = cfw_z C : |z| = 1
T1 = R/Z
T1 = R/2Z
: S 1 [0, 2),
angular coordinate
: S 1 R/2Z R
(multi-valued function)
1
: R S 1,
(x) =
MATH 614
Dynamical Systems and Chaos
Lecture 2:
Periodic points.
Hyperbolicity.
Orbit
Let f : X X be a map dening a discrete dynamical
system. We use notation f n for the n-th iteration of f dened
inductively by f 1 = f and f n = f n1 f for n = 2, 3, . .