This book presents some exceptional developments in chaotic attractor theory encompassing several new directions of research such as three-dimensional axiom A-diffeomorphisms, Shilnikov attractors, dendrites and finite graphs.

The theory of chaotic attractors has experienced exceptional development over the last fifty years since the revelation of chaos in mathematics (invented by James Yorke) and symbolized by the “butterfly effect”. Relevant new results have been collected in this book, including:

• Some remarks on minimal sets on dendrites and finite graphs and the study of recurrence and nonwandering sets of local dendrite maps.
• Ramified continua as global attractors of C1- smooth self-maps of a cylinder close to skew products
• Chaotic behaviour of countable products of homeomorphism groups and dynamics of three-dimensional axiom A-diffeomorphisms with two-dimensional attractors and repellers.
• The search for invariant sets of the generalized tent map and quasi-hyperbolic regime in a certain family of 2-D piecewise linear map.
• Shilnikov attractors of three-dimensional flows and maps, right fractional calculus to inverse-time chaotic maps and asymptotic stability analysis and diffeomorphisms with infinitely many Smale horseshoes.

The theory of chaotic attractor is also used as a core for evolutionary algorithms and metaheuristic optimizers in this volume.

This book will be of great value to students and researchers in mathematics, physics, engineering, and related disciplines seeking to deepen their understanding of chaotic dynamical systems and their applications.

The chapters in this book were originally published in Journal of Difference Equations and Applications.