Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom (Annals of Mathematics Studies)

About the Author\nVadim Kaloshin is the Michael Brin Chair in Mathematics at the University of Maryland, College Park. He was a student of John Mather at Princeton University. Ke Zhang is associate professor of mathematics at the University of Toronto.\nThe first complete proof of Arnold diffusion―one of the most important problems in dynamical systems and mathematical physics\nArnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. In this groundbreaking book, Vadim Kaloshin and Ke Zhang provide the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom).\nThis proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. Kaloshin and Zhang follow Mather's strategy but emphasize a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, this book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.

Science & Math

Zhang, Ke

Kaloshin, Vadim

Mathematics

Princeton University Press

The first complete proof of Arnold diffusion--one of the most important problems in dynamical systems and mathematical physics Arnold diffusion, which concerns the appearance of chaos in classical mechanics, is one of the most important problems in the fields of dynamical systems and mathematical physics. Since it was discovered by Vladimir Arnold in 1963, it has attracted the efforts of some of the most prominent researchers in mathematics. The question is whether a typical perturbation of a particular system will result in chaotic or unstable dynamical phenomena. In this groundbreaking book, Vadim Kaloshin and Ke Zhang provide the first complete proof of Arnold diffusion, demonstrating that that there is topological instability for typical perturbations of five-dimensional integrable systems (two and a half degrees of freedom). This proof realizes a plan John Mather announced in 2003 but was unable to complete before his death. Kaloshin and Zhang follow Mather's strategy but emphasize a more Hamiltonian approach, tying together normal forms theory, hyperbolic theory, Mather theory, and weak KAM theory. Offering a complete, clean, and modern explanation of the steps involved in the proof, and a clear account of background material, this book is designed to be accessible to students as well as researchers. The result is a critical contribution to mathematical physics and dynamical systems, especially Hamiltonian systems.

This book provides an overview of recent advances in the theory of dynamical systems, with a particular emphasis on their connections to other areas of mathematical research, including number theory, geometry, mathematical physics, complex analysis, and celestial mechanics. Compiling&nbsp; the lecture notes from some of the contributions presented at the C.I.M.E. school "Modern Aspects of Dynamical Systems" held in Cetraro in August 2021, the contributions are the following: "Homogeneous dynamics and Diophantine problems" by Manfred Einsiedler, "Effective ergodic theory for translation flow" by Giovanni Forni, "Integrability and rigidity for convex billiards" by Vadim Kaloshin, "Holomorphic dynamics" by Jasmin Raissy and "Exponentially small phenomena and its role in the dynamics" by Tere Martinez-Seara. These notes are suitable for graduate students and young researchers interested in an introduction to some of the modern research areas within the field of dynamical systems.

Modern Aspects of Dynamical Systems: Cetraro, Italy 2021 (C.I.M.E. Foundation Subseries)

Einsiedler, Manfred

Forni, Giovanni

Raissy, Jasmin

Reference

Springer

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