In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic. It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic. Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

In this book the theory of hyperbolic sets is developed, both for diffeomorphisms and flows, with an emphasis on shadowing. We show that hyperbolic sets are expansive and have the shadowing property. Then we use shadowing to prove that hyperbolic sets are robust under perturbation, that they have an asymptotic phase property and also that the dynamics near a transversal homoclinic orbit is chaotic. It turns out that chaotic dynamical systems arising in practice are not quite hyperbolic. However, they possess enough hyperbolicity to enable us to use shadowing ideas to give computer-assisted proofs that computed orbits of such systems can be shadowed by true orbits for long periods of time, that they possess periodic orbits of long periods and that it is really true that they are chaotic. Audience: This book is intended primarily for research workers in dynamical systems but could also be used in an advanced graduate course taken by students familiar with calculus in Banach spaces and with the basic existence theory for ordinary differential equations.

The young orphan lass with crimson red hair and emerald green eyes braved the treachery of the Wolfwind Forest when she ran away to seek a better life. Befriended by Quentis, the First Mate aboard a merchant ship called the Lady Janis, and his wife Scarlet, the young girl finally had a home to call her own. Striking out once she had grown up, Charity finds herself in a world where her trickery and fast wits lands her on the opposite side of the law. She flees to the home of Quentis and Scarlet for safety but is forced to leave the land to avoid a hangman's rope. Quentis accompanies her on a voyage to the uncolonized land of Pryham and along the way her passenger vessel is attacked by a notorious pirate. The young girl turns the tables and bests the swarthy scoundrel and receives his ship, which she aptly names the Sea Witch, as her prize. She is awarded a Letter of Marque and thus embarks upon a witty and lighthearted campaign of adventure on the high seas!