Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom: (AMS-208) (Annals of Mathematics Studies Book 384)

Figure 1.3: Diffusion along a cylinder. 1. (2D NHIC) The Aubry set A 0 H (c) is contained in a 2-dimensional normally hyperbolic invariant cylinder. 2. (1D Graph Theorem) The Aubry set A 0 H (c) is contained in a Lipschitz graph over the circle T. In this case, we say that the pair (H , c) is of the Aubry-Mather type. Under these assumptions, the Aubry set resembles the Aubry-Mather sets for twist maps, and in particular, generically we have the following dichotomy: In the latter case, we can show that the Arnold mechanism applies after an additional perturbation. Since either the Mather or Arnold mechanism applies, we conclude that A(c) is connected to A(c ) for c, c close. Moreover, this argument can be continued if c is also of Aubry-Mather type. Dynamically, the orbit is either diffusing along the heteroclinic orbits of invariant curves or diffusing in a Birkhoff region of instability within the cylinder. See Figure .3. While a cohomology of Aubry-Mather type is robust, namely it can be extended along a continuous curve, in a one-parameter family one may encounter a bifurcation where the Aubry set jumps from one cylinder to another one. At the bifurcation, the Aubry set is contained in both cylinders. We say that the pair (H , c) is of bifurcation Aubry-Mather type if the Aubry set is possibly contained in two cylinders. For technical reasons, we have to involve a different bifurcation type, called asymmetric bifurcation type. This is very similar to the bifurcation Aubry-Mather type, the main difference is, on one side of the bifurcation, the Aubry set is an Aubry-Mather type set contained in an invariant cylinder, while on the other side we have a hyperbolic periodic orbit. This happens when we cross double resonance. Forcing relation and jump mechanism The rigorous formulation of the three diffusion mechanisms will be given using the concept of forcing equivalence defined by Bernard in , which is a general- Chapter Three Normal forms and cohomology classes at single resonances RESONANT COMPONENT AND NON-DEGENERACY CONDITIONS Let (k 1 , Γ k1 ) ∈ K be a resonant segment in the diffusion path. Define the resonant component of H 1 relative to the single resonance k 1 as follows: Definition 3.1. For a given λ > 0, we say that H 1 satisfies the condition SR(k 1 , Γ k1 , λ) if for each p 0 ∈ Γ k1 , at least one of [SR1 λ ] and [SR2 λ ] holds for the function Z k1 (θ s , p).

The first complete proof of Arnold diffusion-one of themost important problems in dynamical systems and mathematicalphysics Arnold diffusion, which concerns the appearance ofchaos in classical mechanics, is one of the most important problemsin the fields of dynamical systems and mathematical physics. Sinceit was discovered by Vladimir Arnold in 1963, it has attracted theefforts of some of the most prominent researchers in mathematics.The question is whether a typical perturbation of a particularsystem will result in chaotic or unstable dynamical phenomena. Inthis groundbreaking book, Vadim Kaloshin and Ke Zhang provide thefirst complete proof of Arnold diffusion, demonstrating that thatthere is topological instability for typical perturbations offive-dimensional integrable systems (two and a half degrees offreedom). This proof realizes a plan John Mather announced in 2003but was unable to complete before his death. Kaloshin and Zhangfollow Mather's strategy but emphasize a more Hamiltonian approach,tying together normal forms theory, hyperbolic theory, Mathertheory, and weak KAM theory. Offering a complete, clean, and modernexplanation of the steps involved in the proof, and a clear accountof background material, this book is designed to be accessible tostudents as well as researchers. The result is a criticalcontribution to mathematical physics and dynamical systems,especially Hamiltonian systems.

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. 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. "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 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." --Publisher's description