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## Christos Efthymiopoulos (Research Center for Astronomy and Applied Mathematics Academy of Athens)

### Arnold diffusion estimates through optimal normal form computations in Hamiltonian systems

We will review a series of recent results arising by performing high-order computer-algebraic computations of normal forms in Hamiltonian nonlinear systems expressed in action-angle variables. Such computations are currently possible to perform up to a so-called `optimal' normalization order. This is defined as the order $r_opt$ at which the normal form remainder reaches its minimum possible size with respect to all possible orders $r$. Using this approach: i) we will show a specific (non-schematic) example of the phase-space structure in a domain of crossing of two resonances, allowing to visualize Arnold diffusion in a particular model, in a set of variables proposed originally by Benettin and Gallavotti (1986). ii) We will discuss how estimates on the size of the optimal remainder are translated into quantitative estimates on the speed of Arnold diffusion. In this, we explore a theory due to Chirikov (1979). This enables us to show that the diffusion coefficient $D$ scales as a power-law of the size of the optimal normal form remainder $R_{opt}$, i.e., $D~R_{opt}^p$, with $2 \leq p \leq 3$.