Kolmogorov–Arnold–Moser theorem
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Reducible KAM Tori for the Degasperis–Procesi Equation Open
We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis–Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash–Moser …
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Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite gap tori for the 2D cubic NLS equation Open
We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the twodimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite gap solutions. These are inherited from the inte…
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Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions Open
We consider the evolutionary Hamilton-Jacobi equation depending on the unknown function with the continuous initial condition on a connected closed manifold.Under certain assumptions on $H(x,u,p)$ with respect to $u$ and $p$, we provide an…
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LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS Open
We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indic…
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Rigorous KAM results around arbitrary periodic orbits for Hamiltonian systems Open
We set up a methodology for computer assisted proofs of the existence and the KAM stability of an arbitrary periodic orbit for Hamiltonian systems. We give two examples of application for systems with two and three degrees of freedom. The …
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Normal forms for perturbations of systems possessing a Diophantine invariant torus Open
We give a new proof of Moser's 1967 normal-form theorem for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. The proposed approach, based on an inverse function theorem in anal…
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KAM Theory for Partial Differential Equations Open
In the last years much progress has been achieved in KAM theory concerning bifurcation of quasi-periodic solutions of Hamiltonian or reversible partial differential equations. We provide an overview of the state of the art in this field.
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Localized mode and nonergodicity of a harmonic oscillator chain Open
We present a simple and microscopic physical model that breaks the ergodicity. Our model consists of coupled classical harmonic oscillators, and the motion of the tagged particle obeys the generalized Langevin equation satisfying the secon…
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Force Evolutionary Billiards and Billiard Equivalence of the Euler and Lagrange Cases Open
A class of force evolutionary billiards is discovered that realizes important integrable Hamiltonian systems on all regular isoenergy 3-surfaces simultaneously, i.e., on the phase 4-space. It is proved that the well-known Euler and Lagrang…
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Stable dynamics in forced systems with sufficiently high/low forcing frequency Open
We consider parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability of the dynamics when the frequency of the forcing is relatively high or low. We show that, provided the frequency is suffi…
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Angle-action variables for orbits trapped at a Lindblad resonance Open
The conventional approach to orbit trapping at Lindblad resonances via a pendulum equation fails when the parent of the trapped orbits is too circular. The problem is explained and resolved in the context of the Torus Mapper and a realisti…
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Billiards and integrable systems Open
The survey is devoted to the class of integrable Hamiltonian systems and the class of integrable billiard systems and to the recent results of the authors and their students on the problem of comparison of these classes from the point of v…
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KAM for the Klein Gordon equation on $\mathbb S^d$ Open
Recently the KAM theory has been extended to multidimensional PDEs. Nevertheless all these recent results concern PDEs on the torus, essentially because in that case the corresponding linear PDE is diagonalized in the Fourier basis and the…
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KAM for quasi-linear forced hamiltonian NLS Open
In this paper we prove the existence of quasi-periodic, small-amplitude, solutions for quasi-linear Hamiltonian perturbations of the non-linear Schroedinger equation on the torus in presence of a quasi-periodic forcing. In particular we pr…
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General KAM theorems and their applications to invariant tori with prescribed frequencies Open
In this paper we develop some new KAM-technique to prove two general KAM theorems for nearly integrable hamiltonian systems without assuming any non-degeneracy condition. Many of KAM-type results (including the classical KAM theorem) are s…
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Degenerate lower dimensional invariant tori in reversible system Open
In this paper, we are concerned with the existence of lower dimensional invariant tori in nearly integrable reversible systems. By KAM method, we prove that under some reasonable assumptions, there are many so-called degenerate lower dimen…
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Lyapunov stability for regular equations and applications to the Liebau phenomenon Open
We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the …
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Existence of a Smooth Hamiltonian Circle Action near Parabolic Orbits and Cuspidal Tori Open
We show that every parabolic orbit of a two-degree-of-freedom integrable system admits a $$C^{\infty}$$-smooth Hamiltoniancircle action, which is persistent under small integrable $$C^{\infty}$$ perturbations.We deduce from this result the…
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Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers Open
We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assist…
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Evolutionary force billiards Open
A new class of integrable billiards has been introduced: evolutionary force billiards. They depend on a parameter and change their topology as energy (time) increases. It has been proved that they realize some important integrable systems …
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Non-contractible periodic orbits in Hamiltonian dynamics on tori Open
We show that the presence of one non-degenerate, non-contractible periodic\norbit of a Hamiltonian on the standard symplectic torus implies the existence\nof infinitely many simple non-contractible periodic orbits.\n
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Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map Open
By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system: x n + 1 = 1 y n , y n + 1 = β x n 1 + y n , n = 0 , 1 , 2 , … , where the parameter β >…
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Analysis of Transport and Mixing Phenomenon to Invariant Manifolds Using LCS and KAM Theory Approach in Unsteady Dynamical Systems Open
The Lagrangian approach for the two-dimensional incompressible fluid flows has been studied with the help of dynamical systems techniques: Kolmogorov-Arnold-Moser (KAM) theory, stable, unstable manifold structures, and Lagrangian coherent …
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Small amplitude weak almost periodic solutions for the 1d NLS Open
All the almost periodic solutions for nonintegrable PDEs found in the literature are very regular (at least C-infinity) and, hence, very close to quasiperiodic ones. This fact is deeply exploited in the existing proofs. Proving the existen…
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Complex Dynamics and Statistics of 1-D Hamiltonian Lattices: Long Range Interactions and Supratransmission Open
In this paper, I review a number of results that my co-workers and I have obtained in the field of 1-Dimensional (1D) Hamiltonian lattices. This field has grown in recent years, due to its importance in revealing many phenomena that concer…
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Beltrami fields exhibit knots and chaos almost surely Open
In this paper, we show that, with probability $1$ , a random Beltrami field exhibits chaotic regions that coexist with invariant tori of complicated topologies. The motivation to consider this question, which arises in the study of station…
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Schrödinger spectra and the effective Hamiltonian of weak KAM theory on the flat torus Open
In this paper we investigate the link between the spectrum of some periodic Schrödinger type operators and the effective Hamiltonian of the weak KAM theory. We show that the extension of some local quasimodes is linked to the localization …
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On Co-Orbital Quasi-Periodic Motion in the Three-Body Problem Open
Within the framework of the planar three-body problem we establish the existence of quasi-periodic motions and KAM 4-tori related to the co-orbital motion of two small moons about a large planet where the moons move in nearly circular orbi…
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Differentiable Rigidity for quasiperiodic cocycles in compact Lie groups Open
We study close-to-constants quasiperiodic cocycles in $\\mathbb{T} ^{d} \\times G$, where $d \\in \\mathbb{N} ^{*} $ and $G$ is a compact Lie group, under the assumption that the rotation in the basis satisfies a Diophantine condition. We …
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An infinite dimensional KAM theorem with application to two dimensional completely resonant beam equation Open
In this paper, we consider the two dimensional completely resonant beam equation with cubic nonlinearity on T2. We prove the existence of the quasi-periodic solutions, which lie in a special subspace of L2(T2). After some transformations, …