Iterated Maps On The Interval As Dynamical Systems

Download or read book Iterated Maps on the Interval as Dynamical Systems written by Pierre Collet and published by . This book was released on 1986 with total page 248 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Iterated Maps on the Interval as Dynamical Systems written by Pierre Collet and published by Springer Science & Business Media. This book was released on 2009-08-25 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 .

Download or read book Ordinary Differential Equations and Dynamical Systems written by Gerald Teschl and published by American Mathematical Society. This book was released on 2024-01-12 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm–Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincaré–Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman–Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale–Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.

Download or read book One-Dimensional Dynamics written by Welington de Melo and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: One-dimensional dynamics has developed in the last decades into a subject in its own right. Yet, many recent results are inaccessible and have never been brought together. For this reason, we have tried to give a unified ac count of the subject and complete proofs of many results. To show what results one might expect, the first chapter deals with the theory of circle diffeomorphisms. The remainder of the book is an attempt to develop the analogous theory in the non-invertible case, despite the intrinsic additional difficulties. In this way, we have tried to show that there is a unified theory in one-dimensional dynamics. By reading one or more of the chapters, the reader can quickly reach the frontier of research. Let us quickly summarize the book. The first chapter deals with circle diffeomorphisms and contains a complete proof of the theorem on the smooth linearizability of circle diffeomorphisms due to M. Herman, J.-C. Yoccoz and others. Chapter II treats the kneading theory of Milnor and Thurstonj also included are an exposition on Hofbauer's tower construction and a result on fuB multimodal families (this last result solves a question posed by J. Milnor).

Download or read book 30 Years Of The Landau Institute - Selected Papers written by Isaak M Khalatnikov and published by World Scientific. This book was released on 1996-08-15 with total page 795 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Landau Institute for Theoretical Physics was created in 1965 by a group of LD Landau's pupils. Very soon, it was widely recognized as one of the world's leading centers in theoretical physics. According to Science Magazine, the Institute in the eighties had the highest citation index among all the scientific organizations in the former Soviet Union. This collection of the best papers of the Institute reflects the development of the many directions in the exact sciences during the last 30 years. The reader can find the original formulations of well-known notions in condensed matter theory, quantum field theory, mathematical physics and astrophysics, which were introduced by members of the Landau Institute.The following are some of the achievements described in this book: monopoles (A Polyakov), instantons (A Belavin et al.), weak crystallization (S Brazovskii), spin superfluidity (I Fomin), finite band potentials (S Novikov) and paraconductivity (A Larkin, L Aslamasov).

Download or read book Sharkovsky Ordering written by Alexander M. Blokh and published by Springer Nature. This book was released on 2022-09-05 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Iteration Theory - Proceedings Of The European Conference written by W Forg-rob and published by World Scientific. This book was released on 1996-07-03 with total page 298 pages. Available in PDF, EPUB and Kindle. Book excerpt: Iteration theory has its roots in the operation of substituting functions into itself. This has led to questions like that of the behaviour of functions by repeating this substitution and when the number of iterations tends to infinity. The terms 'orbit' and 'chaos' appropriately describe this behaviour. Dynamical systems and the theory of functional equations play important roles in this field.