The required text is:
Steven H Strogatz, Nonlinear dynamics and chaos with applications to Physics, Biology, Chemistry and Engineering, ISBN 0-7382-0453-6, Westview Press (Perseus Books Publishing), Cambridge, MA, 2000.
Other recommended reading:
Jim Meiss Differential dynamical systems, SIAM, 2007.
This is an excellent text which includes proofs of many of the important theorems; it also includes many other interesting things.