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.