Michael Rudnev
1. TX 78712, Austin, United States of America
2. Senate House, Tyndall Avenue, BS8 1TW, Bristol, United Kingdom
2. Senate House, Tyndall Avenue, BS8 1TW, Bristol, United Kingdom
1. Department of Mathematics/C1200 UT Austin
2. Department of Mathematics, University of Bristol
2. Department of Mathematics, University of Bristol
Publications:
Rudnev M., Ten V. V.
A model for separatrix splitting near multiple resonances
2006, vol. 11, no. 1, pp. 83102
Abstract
We propose and study a model for local dynamics of a perturbed convex realanalytic Liouvilleintegrable Hamiltonian system near a resonance of multiplicity $1+m$, $m \geqslant 0$. Physically, the model represents a toroidal pendulum, coupled with a Liouvilleintegrable system of $n$ nonlinear rotators via a small analytic potential. The global bifurcation problem is setup for the $n+1$ dimensional isotropic manifold, corresponding to a specific homoclinic orbit of the toroidal pendulum. The splitting of this manifold can be described by a scalar function on the $n$torus. A sharp estimate for its Fourier coefficients is proven. It generalizes to a multiple resonance normal form of a convex analytic Liouville nearintegrable Hamiltonian system. The bound then is exponentially small.

Rudnev M., Ten V. V.
Sharp upper bounds for splitting of separatrices near a simple resonance
2004, vol. 9, no. 3, pp. 299336
Abstract
General theory for the splitting of separatrices near simple resonances of nearLiouvilleintegrable Hamiltonian systems is developed in the convex realanalytic setting. A generic estimate
$$\mathfrak{S}_k\,\leqslant\,O(\sqrt{\varepsilon}) \, \times \, \exp\left[ \,{\left k\cdot\left(c_1{\omega\over\sqrt{\varepsilon}}+c_2\right)\right}  k\sigma\right],\;\,k\in\mathbb{Z}^n\setminus\{0\}$$ is proved for the
Fourier coefficients of the splitting distance measure $\mathfrak{S}(\phi),\,\phi\in\mathbb{T}^n,$ describing the intersections of Lagrangian manifolds, asymptotic to invariant $n$tori, $\varepsilon$ being the perturbation parameter.
The constants $\omega\in\mathbb{R}^n,$ $c_1,\sigma>0,\,c_2\in\mathbb{R}^n$ are characteristic of the given problem (the Hamiltonian and the resonance), cannot be improved and can be calculated explicitly, given an example. The theory allows for optimal parameter dependencies in the smallness condition for $\varepsilon$. 
Rudnev M., Wiggins S.
On a Homoclinic Splitting Problem
2000, vol. 5, no. 2, pp. 227242
Abstract
We study perturbations of Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$ in the realanalytic case, such that in the absence of the perturbation they contain a partially hyperbolic (whiskered) $n$torus with the Kronecker flow on it with a Diophantine frequency, connected to itself by a homoclinic exact Lagrangian submanifold (separatrix), formed by the coinciding unstable and stable manifolds (whiskers) of the torus. Typically, a perturbation causes the separatrix to split. We study this phenomenon as an application of the version of the KAM theorem, proved in [13]. The theorem yields the representations of global perturbed separatrices as exact Lagrangian submanifolds in the phase space. This approach naturally leads to a geometrically meaningful definition of the splitting distance, as the gradient of a scalar function on a subset of the configuration space, which satisfies a first order linear homogeneous PDE. Once this fact has been established, we adopt a simple analytic argument, developed in [15] in order to put the corresponding vector field into a normal form, convenient for further analysis of the splitting distance. As a consequence, we argue that in the systems, which are Normal forms near simple resonances for the perturbations of integrable systems in the actionangle variables, the splitting is exponentially small.

Rudnev M., Wiggins S.
On a Partially Hyperbolic KAM Theorem
1999, vol. 4, no. 4, pp. 3958
Abstract
We prove structural stability under small perturbations of a family of real analytic Hamiltonian systems of $n+1$ degrees of freedom $(n \geqslant 2)$, comprising an invariant partially hyperbolic $n$torus with the Kronecker flow on it with a diophantine frequency, and an unstable (stable) exact Lagrangian submanifold (whisker), containing this torus. This is the preservation of the exact Lagrangian properties of the whisker that we focus upon. Hence, we develop a Normal form, which is valid globally in the neighborhood of the perturbed whisker and enables its representation as an exact Lagrangian submanifold in the original coordinates, whose generating function solves the Hamilton–Jacobi equation.
