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Iterations of continuous maps of an interval to itself serve as the simplest examples of models for dynamical systems. These models present an interesting mathematical structure going far beyond the simple equilibrium solutions one might expect. If, in addition, the dynamical system depends on an experimentally controllable parameter, there is a corresponding mathematical structure revealing a great deal about interrelations between the behavior for different parameter values. This work explains some of the early results of this theory to mathematicians and theoretical physicists, with the additional hope of stimulating experimentalists to look for more of these general phenomena of beautiful regularity, which oftentimes seem to appear near the much less understood chaotic systems. Although continuous maps of an interval to itself seem to have been first introduced to model biological systems, they can be found as models in most natural sciences as well as economics. Iterated Maps on the Interval as Dynamical Systems is a classic reference used widely by researchers and graduate students in mathematics and physics, opening up some new perspectives on the study of dynamical systems .
The contents of this volume consist of 15 lectures on mathematics and its applications which include the following topics: dynamics of neural network, phase transition of cellular automata, homoclinic bifurcations, ergodic theories of low dimensional dynamical systems, Anosov endomorphisms and Anosov flows, axiom A systems, complex dynamical systems, multi-dimensional holomorphic dynamical systems and holomorphic vector fields.
"This thesis contains three related articles which are inserted in separate chapters. The chapters are self contained, hence it is possible to read each of them independently. In each of the chapters we discuss iterated function systems, IFSs, generated by finitely many continuous maps f1, . . . , fk on the unit interval I = [0, 1], taken with positive probabilities pi, ∑ki=1 pi =1, randomly at each iterate. We always assume that these maps fix the boundaries of the interval I and study dynamics of the system under some conditions like the sign of the Lyapunov exponent at the boundaries of the interval. The sign of Lyapunov exponent at a boundary determines whether the boundary is, on average, attracting (negative Lyapunov exponent), neutral (zero), or repelling (positive). We discuss occurrence of intriguing dynamical phenomena such as synchronization, intermingled basins and intermittency, in this context."--Samenvatting auteur.
This book provides a comprehensive survey of the Sharkovsky ordering, its different aspects and its role in dynamical systems theory and applications. It addresses the coexistence of cycles for continuous interval maps and one-dimensional spaces, combinatorial dynamics on the interval and multidimensional dynamical systems. Also featured is a short chapter of personal remarks by O.M. Sharkovsky on the history of the Sharkovsky ordering, the discovery of which almost 60 years ago led to the inception of combinatorial dynamics. Now one of cornerstones of dynamics, bifurcation theory and chaos theory, the Sharkovsky ordering is an important tool for the investigation of dynamical processes in nature. Assuming only a basic mathematical background, the book will appeal to students, researchers and anyone who is interested in the subject.