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We review V.I. Arnold's 1963 celebrated paper \cite{ARV63} {\sl Proof of A.N. Kolmogorov's theorem on the conservation of conditionally periodic motions with a small variation in the Hamiltonian}, and prove that, optimizing Arnold's scheme, one can get "sharp" asymptotic quantitative conditions (as $\varepsilon\to 0$, $\varepsilon$ being the strength of the perturbation). All constants involved are explicitly computed., Comment: To appear in

Chow, Li and Yi in [2] proved that the majority of the unperturbed tori {\it on sub-manifolds} will persist for standard Hamiltonian systems. Motivated by their work, in this paper, we study the persistence and tangent frequencies preservation of lower dimensional invariant tori on smooth sub-manifolds for real analytic, nearly integrable Hamiltonian systems. The surviving tori might be elliptic, hyperbolic, or of mixed type., 25 pages. To appear in Nonlinear Analysis: TMA

We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has to satisfy in order for the perturbed Hamiltonian to be a first order deformation., Comment: minor typo errors fixed, bibliography updated

We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori. ¶ We show that an analytic invariant torus ${\mathcal{T}}_{0}$ with Diophantine frequency $\omega _{0}$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ${\mathcal{T}}_{0}$ satisfies a Rüssmann transversality condition, the torus ${\mathcal{T}}_{0}$ is accumulated by Kolmogorov–Arnold–Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least $d+1$ that is foliated by analytic invariant tori with frequency $\omega _{0}$ . ¶ For frequency vectors $\omega _{0}$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian $H$ satisfies a Kolmogorov nondegeneracy condition at ${\mathcal{T}}_{0}$ , then ${\mathcal{T}}_{0}$ is accumulated by KAM tori of positive total measure. ¶ In four degrees of freedom or more, we construct for any $\omega _{0}\in{\mathbb{R}}^{d}$ , $C^{\infty}$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus ${\mathcal{T}}_{0}$ with frequency $\omega _{0}$ that is not accumulated by a positive measure of invariant tori.

In this paper, we are concerned with the boundedness of all the solutions for a kind of second order differential equations with p-Laplacian and an oscillating term $(\phi_p(x'))'+a\phi_p(x^+)-b\phi_p(x^-)=G_x(x,t)+f(t)$, where$x^+=\max (x,0)$,$x^- =\max(-x,0)$,$\phi_p(s)=|s|^{p-2}s$,$p\geq2$, $a $ and $b$ are positive constants $(a\not=b)$, the perturbation $f(t)\in {\cal C}^{23}(\RR/2\pi_p \ZZ)$, the oscillating term $G\in {\cal C}^{21}(\RR\times\RR/2\pi_p \ZZ)$,where $\pi_p=\frac{2\pi(p-1)^{\frac{1}{p}}}{p\sin\frac{\pi}{p}},$ and $G(x,t)$ satisfies $\label{G} |D_x^iD_t^jG(x,t)|\le C,\quad 0\le i+j\le 21,$ and $\label{hatG} |D_t^j\hat{G}|\le C,\quad 0\le j\le 21$ for some $C>0$, where $\hat{G}$ is some function satisfying $\frac{\pa \hat{G}}{\pa x}=G$.

In this paper, we prove the boundedness of all solutions for the equation $x'' + n^2x + \phi(x) + g''(x) q(t) = 0$, where $n \in \mathbb{N}$, $q(t) = q(t+2\pi)$, $\phi(x)$ and $g(x)$ are bounded.

A Gevrey symplectic normal form of an analytic and more generally Gevrey smooth Hamiltonian near a Lagrangian invariant torus with a Diophantine vector of rotation is obtained. The normal form implies effective stability of the quasi-periodic motion near the torus.

Consider in $L^2(\R^2)$ the operator family $H(\epsilon):=P_0(\hbar,\omega)+\epsilon F_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\om$, $F_0$ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and $\ep\in\C$. Then there exist $\ep^\ast >0$ independent of $\hbar$ and an open set $\Omega\subset\C^2\setminus\R^2$ such that if $|\ep, Comment: 17 pages

Let $R_1, R_2,\ldots,R_m$ be rotations generating ${\mathbb{SO}}_{d+1}$ , $d\geq 2$ , and let $f_1, f_2,\ldots,f_m$ be small smooth perturbations of them. We show that $\{f_\alpha\}$ can be linearized simultaneously if and only if the associated random walk has zero Lyapunov exponents. As a consequence, we obtain stable ergodicity of actions of random rotations in even dimensions

Resonant motions of integrable systems subject to perturbations may continue to exist and to cover surfaces with parametric equations admitting a formal power expansion in the strength of the perturbation. Such series may be, sometimes, summed via suitable sum rules defining $C^\infty$ functions of the perturbation strength: here we find sufficient conditions for the Borel summability of their sums in the case of two-dimensional rotation vectors with Diophantine exponent $\tau=1$ (e. g. with ratio of the two independent frequencies equal to the golden mean)., Comment: 17 pages, 1 figure

Motivated by the Lagrange top coupled to an oscillator, we consider the quasi-periodic Hamiltonian Hopf bifurcation. To this end, we develop the normal linear stability theory of an invariant torus with a generic (i.e., non-semisimple) normal $1:-1$ resonance. This theory guarantees the persistence of the invariant torus in the Diophantine case and makes possible a further quasi-periodic normal form, necessary for investigation of the non-linear dynamics. As a consequence, we find Cantor families of invariant isotropic tori of all dimensions suggested by the integrable approximation., Comment: Published at http://dx.doi.org/10.1214/074921706000000301 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)

We show that near a focus-focus point in a Liouville integrable Hamiltonian system with two degrees of freedom lines of locally constant rotation number in the image of the energy-momentum map are spirals determined by the eigenvalue of the equilibrium. From this representation of the rotation number we derive that the twist condition for the isoenergetic KAM condition vanishes on a curve in the image of the energy-momentum map that is transversal to the line of constant energy. In contrast to this we also show that the frequency map is non-degenerate for every point in a neighborhood of a focus-focus point., 13 pages