Behaviors of chaotic systems are unpredictable. Chaotic systems are deterministic, their evolutions being governed by dynamical equations. Are the two statements contradictory? They are not, because the theory of chaos encompasses two levels of description. On a higher level, unpredictability appears as an emergent property of systems that are predictable on a lower level. In this talk, we exam...

The seminal papers by Pecora and Carrol (PC) [1] and Ott, Grebogi and Yorke (OGY) [2] in 1990 have induced avalanche of research works in the field of chaos control. Chaos synchronization in dynamical systems is one of methods of controling chaos, see, e.g. [1-8] and references therein.The interest to chaos synchronization in part is due to the application of this phenomenen in secure communica...

in this paper, we aresupposed to introduce the definitions of n-fold commutative, andimplicative hyper k-ideals. these definitions are thegeneralizations of the definitions of commutative, andimplicative hyper k-ideals, respectively, which have been definedin [12]. then we obtain some related results. in particular wedetermine the relationships between n-fold implicative hyperk-ideal and n-fold...

Preface During most of the Twentieth Century, physicists have been mainly concerned with linear dynamics. Despite the works of Poincaré, Birkhoff and von Neumann, the paradigm in physics was linear dynamics. Courses in Classical Mechanics systematically ignored intrinsically non-linear phenomena and chaos, restricting Mechanics to Integrable Systems, i.e., dynamical systems with an underlying L...