The global dynamics of the quadratic map are well understood for 
, namely, almost all orbits beginning on the unit interval are asymptotic to a
period one fixed point. We will next show that the orbit structure is also
well understood for 
. This is known as the hyperbolic
regime. This parameter regime is ``fully developed'' in the sense that all
of the possible periodic orbits exist and they are all
unstable.
No chaotic attractor exists in this parameter regime, but
rather a chaotic repeller . Almost all initial
conditions eventually leave, or are repelled from, the unit interval. However, a
small set remains. This remaining invariant set is an example of a fractal.
The analysis found in this book is based substantially on sections 1.5 to 1.8 of Devaney's An Introduction to Chaotic Dynamical Systems [10]. This section is more advanced mathematically than previous sections. The reader should consult Devaney's book for a complete treatment. Section 2.12 contains a more pragmatic description of the symbolic dynamics of the quadratic map and can be read independently of the current section.