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A suspension flow with a Plykin attractor

In this thesis we construct a vector field ~s* on ^CxS1 with a period-one 
and a period-three repelling orbit. This vector field is real analytic everywhere 
except on these two special orbits. We will show that on the vector field ~s* 
there exists a region of uniform hyperbolicity and neighbourhoods of repulsion 
near the two special orbits, and we have a conjecture that we can enlarge the 
region of uniform hyperbolicity and the neighbourhoods of repulsion so that 
they cover the whole space ^C x S1. We prove that if the conjecture is true 
then there exists a suspended Plykin attractor for the vector field ~s*. 
We briefly outline the contents of this thesis. The first chapter reviews 
the basic concepts and tools for the constructions and proofs in the later 
chapters. The second chapter introduces the original Plykin attractor and a 
modern construction of the Plykin attractor on a plane in discrete time. In 
the third chapter we give a construction of the Plykin attractor in continuous 
time, the vector field ~s* on ^CxS1. The fourth chapter discusses and provides 
the proofs of some properties of ~s*. The last chapter gives a summary of the 
construction and contains some discussions on how to make the vector field 
real analytic on the whole of ^C x S1.