The ``Fundamental Theorem" given by Arnold in [2] asserts the persistence of full dimensional invariant tori for 2-scale Hamiltonian systems. However, persistence in multi-scale systems is much more complicated and difficult. In this paper, we explore the persistence of lower dimensional invariant tori for multi-scale Hamiltonian systems, which play an important role in dynamics of resonant Hamiltonian systems. Moreover, using the corresponding results we give a quasi-periodic Poincaré Theorem for multi-scale Hamiltonian systems, i.e., at least $2^{m_0}$ families resonant tori survive small perturbations, where the integer $m_0$ is the multiplicity of resonance.
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.
In this paper the problem of persistence of invariant tori under small perturbations of integrable Hamiltonian systems is considered. The existence of one-to-one correspondence between hyperbolic invariant tori and critical points of the function $Ψ$ of two variables defined on semi-cylinder is established. It is proved that if unperturbed Hamiltonian has a saddle point, then under arbitrary perturbations there persists at least one hyperbolic torus.
We compute the critical surface for the existence of invariant tori of a family of Hamiltonian systems with two and three degrees of freedom. We use and compare two methods to compute the critical surfaces: renormalization-group transformations and conjugation in configuration space. We unveil the presence of cusps in the critical surface for the breakup of three-dimensional invariant tori, whereas the critical surface of two-dimensional invariant tori is expected to be smooth.
In this article, we consider the dynamics in a neighborhood of a quasi-periodic torus which is invariant by a Hamiltonian flow, we discuss several notions of stability and we prove several results of instability when the frequency of the invariant torus is resonant.
This paper demonstrates sufficient conditions for the existence of a positive measure set of invariant KAM tori in a singly thermostated, 1 degree-of-freedom hamiltonian vector field. This result is applied to 4 important single thermostats in the literature and it is shown that in each case, if the hamiltonian is real-analytic and well-behaved, then the thermostated system always has a positive measure set of invariant KAM tori for sufficiently weak coupling and high temperature. This extends results of Legoll, Luskin & Moeckel.
In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The proof combines classical Birkhoff normal forms and a new method to obtain generic Nekhoroshev estimates developed by the author and L. Niederman in another paper. We will mainly focus on the neighbourhood of elliptic fixed points, the other cases being completely similar.
For various values of n, d, and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth d-parameter families of invariant n-tori carrying conditionally periodic motions. In the Hamiltonian case, these tori can be isotropic, coisotropic, or atropic (neither isotropic nor coisotropic). The cases of non-compact and compact phase spaces are considered. In particular, for any N no less than 3 and any vector omega in R^N, we present an example of an analytic Hamiltonian system with N degrees of freedom and with an isolated (and even unique) inv…
Consider an integer $n \geq 2$ and real numbers $τ>n-1$ and $l>2(τ+1)$. Using ideas of Moser, Salamon proved that individual Diophantine tori persist for Hamiltonian systems which are of class $C^l$. Under the stronger assumption that the system is a $C^{l+τ}$ perturbation of an analytic integrable system, P{ö}schel proved the persistence of a set of positive measure of Diophantine tori. We improve the last result by showing it is sufficient for the perturbation to be of class $C^{l}$ and the integrable part to be of class $C^{l+2}$.
We obtain a global version of the Hamiltonian KAM theorem for invariant Lagrangean tori by glueing together local KAM conjugacies with help of a partition of unity. In this way we find a global Whitney smooth conjugacy between a nearly-integrable system and an integrable one. This leads to preservation of geometry, which allows us to define all the nontrivial geometric invariants like monodromy or Chern classes of an integrable system also for near integrable systems.
The purpose of this paper is to present a method to compute parameterizations of invariant tori and bundles in non-autonomous quasi-periodic Hamiltonian systems. We generalize flow map parameterization methods to the quasi-periodic setting. To this end, we introduce the notion of fiberwise isotropic tori and sketch definitions and results on fiberwise symplectic deformations and their moment maps. These constructs are vital to work in a suitable setting and lead to the proofs of magic cancellations that guarantee the existence of solutions of cohomological equations. We apply our algorithms in…
In this paper we study families of Lagrangian tori that appear in a neighborhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the "integrable" limit $\varepsilon\to 0$. Dynamics on these tori is oscillatory in the direction of the resonance phases and rotating with respect to the other (non-resonant) phases. We also show that, if multiplicity of the resonance equals one, generically these tori occupy a set of large relative measure in the resonant domains in the sense that the relative measure of the remaining "chaotic" set is of order $\sqrt\varepsilon$. …
The goal of this paper is to present a methodology for the computation of invariant tori in Hamiltonian systems combining flow map methods, parameterization methods, and symplectic geometry. While flow map methods reduce the dimension of the tori to be computed by one (avoiding Poincare maps), parameterization methods reduce the cost of a single step of the derived Newton-like method to be proportional to the cost of a FFT. Symplectic properties lead to some magic cancellations that make the methods work. The multiple shooting version of the methods are applied to the computation of invariant …
In this paper, we prove the existence of full dimensional invariant tori for a non-autonomous, almost-periodically forced nonlinear beam equation with a periodic boundary condition via KAM theory.
In this paper we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems \begin{equation*} H=h(y)+f(x,y,t), \end{equation*} where $y\in D\subseteq\mathbb{R}^n$ with $D$ being a closed bounded domain, $x\in \mathbb{T}^n$, $f(x,y,t)$ is a real analytic almost periodic function in $t$ with the frequency ${ω}=(\cdots,{ω}_λ,\cdots)_{λ\in \mathbb{Z}}\in \mathbb{R}^{\mathbb{Z}}$. As an application, we will prove the existence of almost periodic solutions and the boundedness of all solutions for the second order differential equations with superquadratic potentials …
We prove the existence of real analytic Hamiltonians with topologically unstable quasi-periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic motion: $\quad i)$ Show the existence of topologically unstable tori of arbitrary frequency. Moreover, the Birkhoff Normal Form at the invariant torus can be chosen to be convergent, equal to a planar or non-planar polynomial. $\quad ii)$ Show the optimality of the exponential stability for Diophantine tori. ' $\quad iii)$ Show the existence of real analytic Hami…
In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate. When the Hamiltonian is smooth (respectively Gevrey-smooth, respectively real-analytic), the in-variant tori are smooth (respectively Gevrey-smooth, respectively real-analytic). This answers a question raised in a recent work by Eliasson, Fayad and Krikorian ([EFK]). We also take the opportunity to ask other questions concerning the stability of no…
The KAM (Kolmogorov-Arnold-Moser) theorem guarantees the stability of quasi-periodic invariant tori by perturbation in some Hamiltonian systems. Michel Herman proved a similar result for quasi-periodic motions, with $k$-dimensional involutive manifolds in Hamiltonian systems with $n$ degrees of freedom $n \leq k < 2n $. In this paper, we extend this result to the case of a quasi-periodic motion on symplectic tori $k = 2n$.
In this paper, we establish a KAM-theorem for ordinary differential equations with finitely differentiable vector fields and multiple degeneracies. The theorem can be used to deal with the persistence of quasi-periodic invariant tori in multiple Hopf and zero-multiple Hopf bifurcations, as well as their subordinate bifurcations, of equilibrium points of continuous dynamical systems.
Let H(q,p) = p^2/2 + V(q) be a 1-degree of freedom mechanical Hamiltonian with a C^n periodic potential V where n>4. The Nosé-thermostated system associated to H is shown to have invariant tori near the infinite temperature limit. This is shown to be true for all thermostats similar to Nosé's. These results complement the result of Legoll, Luskin and Moeckel who proved the existence of such tori near the decoupling limit.
We study a new type of normal form at a critical point of an analytic Hamiltonian. Under a Bruno condition on the frequency, we prove a convergence statement to the normal form. Using this result, we prove the Herman invariant tori conjecture namely the existence of a positive measure set of invariant tori near the critical point. This paper is an update of the first 2012 proof of the author. The functional analytic arguments have been simplified using Banach functors, minor points have been clarified. A series of videos is available on the webpage https://www.agtz.mathematik.uni-mainz.de/cate…
We prove a general theorem on the persistence of Whitney infinitely smooth families of invariant tori in the reversible context 2 of KAM theory. This context refers to the situation where dim Fix G < (codim T)/2 where Fix G is the fixed point manifold of the reversing involution G and T is the invariant torus in question. Our result is obtained as a corollary of the theorem by H.W.Broer, M.-C.Ciocci, H.Hanssmann, and A.Vanderbauwhede of 2009 concerning quasi-periodic stability of invariant tori with singular "normal" matrices in reversible systems.
In this paper, we investigate the existence of KAM tori for an infinite dimensional Hamiltonian system with finite number of zero normal frequencies. By constructing a constant quantity we show that, for "most" frequencies in the sense of Lebesgue measure, either if the quantity is zero, there is a KAM tori or if the quantity is not zero, there is no KAM tori in some domain. As application, we show that the nonlinear Schrödinger equation with a zero frequency possesses many quasi-periodic solutions.
Consider a real--analytic nearly--integrable mechanical system with potential $f$, namely, a Hamiltonian system, having a real-analytic Hamiltonian $$ H(y,x)=\frac12 | y |^2 +\e f(x)\ , $$ $y,x$ being $n$--dimensional standard action--angle variables (and $|\cdot|$ the Euclidean norm). Then, for "general" potentials $f$'s and $\e$ small enough, the Liouville measure of the complementary of invariant tori is smaller than $\e|\ln \e|^a$ (for a suitable $a>0$).
We prove that every smooth action of Z^k, k>1, on the (k+1)-dimensional torus homotopic to an action by hyperbolic linear maps preserves an absolutely continuous measure. This is a first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
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 special cases of our theorems under some non-degeneracy condition and some smoothness condition. Moreover, we can obtain some interesting results about KAM tori with prescribed frequencies.
In this paper, we show the existence of non contractible periodic orbits in Hamiltonian systems defined on $T^*\T^n$ separating two Lagrangian tori under certain cone assumption. Our result answers a question of Polterovich in \cite{P} in a sharp way. As an application, we find periodic orbits of almost all the homotopy types on a dense set of energy level in Lorentzian type mechanical Hamiltonian systems defined on $T^*\T^2$. This solves a problem of Arnold in \cite{A}.
We show that the presence of one non-degenerate, non-contractible periodic orbit of a Hamiltonian on the standard symplectic torus implies the existence of infinitely many simple non-contractible periodic orbits.
The purpose of this brief note is twofold. First, we summarize in a very concise form the principal information on Whitney smooth families of quasi-periodic invariant tori in various contexts of KAM theory. Our second goal is to attract (via an informal discussion and a simple example) the experts' attention to the peculiarities of the so-called excitation of elliptic normal modes in the reversible context 2.
A method via the KAM technique is introduced to study the existence of invariant tori and quasiperiodic solutions for impulsive Duffing-type equations with time period 1. Basing on several planar symplectic homeomorphisms and some estimates of impulsive perturbations under each symplectic homeomorphisms, we prove via the Moser's twist theorem the boundedness (Lagrange stability) and the existence of an invariant circle for the equation with area-preserving impulsive terms. And this invariant circle having any rotation number $ω>ω_0$ with some $ω_0>0$, so we obtain also that the solutions start…
We generalize to some PDEs a theorem by Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with $r$ integrals of motion and $n$ degrees of freedom, $r\leq n$. The result we get ensures the persistence of an $r$-parameter family of $r$-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunof-Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of c…
We prove a KAM-type result for the persistence of two-dimensional invariant tori in perturbations of integrable action-angle-angle maps with degeneracy, satisfying the intersection property. Such degenerate action-angle-angle maps arise upon generic perturbation of three-dimensional volume-preserving vector fields, which are invariant under volume-preserving action of $S^1$ when there is no motion in the group action direction for the unperturbed map. This situation is analogous to degeneracy in Hamiltonian systems. The degenerate nature of the map and the unequal number of action and angle va…
This work studies existence and regularity questions for attracting invariant tori in three dimensional dissipative systems of ordinary differential equations. Our main result is a constructive method of computer assisted proof which applies to explicit problems in non-perturbative regimes. We obtain verifiable lower bounds on the regularity of the attractor in terms of the ratio of the expansion rate on the torus with the contraction rate near the torus. We consider separately two important cases of rotational and resonant tori. In the rotational case we obtain $C^k$ lower bounds on the regul…
In the present paper, we consider the following reversible system \begin{equation*} \begin{cases} \dot{x}=ω_0+f(x,y),\\ \dot{y}=g(x,y), \end{cases} \end{equation*} where $x\in\mathbf{T}^{d}$, $y\backsim0\in \mathbf{R}^{d}$, $ω_0$ is Diophantine, $f(x,y)=O(y)$, $g(x,y)=O(y^2)$ and $f$, $g$ are reversible with respect to the involution G: $(x,y)\mapsto(-x,y)$, that is, $f(-x,y)=f(x,y)$, $g(-x,y)=-g(x,y)$. We study the accumulation of an analytic invariant torus $Γ_0$ of the reversible system with Diophantine frequency $ω_0$ by other invariant tori. We will prove that if the Birkhoff normal form …
A method to establish regions of phase space through which pass no invariant tori transverse to a given direction field is applied to the planar circular restricted three-body problem. Implications for the location of stable orbits for planets around a binary star are deduced. It is expected that lessons learnt from this problem will be useful for applications of the method to other contexts such as flux surfaces for magnetic fields, guiding centre motion in magnetic fields, and classical models of chemical reaction dynamics.
This paper demonstrates sufficient conditions for the existence of KAM tori in a singly thermostated, integrable hamiltonian system with $n$ degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2--which include the Nosé thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends Theorem 3.2 of Legoll, Luskin & Moeckel, (Non-ergodicity of Nosé-Hoover dynamics, Nonlinearity, 22 (2009), pp. 1673--1694) to prove that a "typical" singly thermostated, integrable, real-analytic hamiltonian possesses a positive-measure s…
In this paper we prove the persistence of hyperbolic invariant tori in generalized Hamiltonian systems, which may admit a distinct number of action and angle variables. The systems under consideration can be odd dimensional in tangent direction. Our results generalize the well-known results of Graff and Zehnder in standard Hamiltonians. In our case the unperturbed Hamiltonian systems may be degenerate. We also consider the persistence problem of hyperbolic tori on sub-manifolds.
We proved the existence of invariant tori in differentiable Hamiltonian vector fields without action-angle variables. It is a generalization of the result of [Llave, 2005] that deals with analytic vector fields.
We give a proof of the convergence of an algorithm for the construction of lower dimensional elliptic tori in nearly integrable Hamiltonian systems. The existence of such invariant tori is proved by leading the Hamiltonian to a suitable normal form. In particular, we adapt the procedure described in a previous work by Giorgilli and co-workers, where the construction was made so as to be used in the context of the planetary problem. We extend the proof of the convergence to the cases in which the two sets of frequencies, describing the motion along the torus and the transverse oscillations, hav…
We prove that there exist Beltrami fields in Euclidean space, with sharp decay at infinity, which have a prescribed set of invariant tori (possibly knotted or linked) that enclose an arbitrarily large number of hyperbolic periodic orbits. These hyperbolic orbits are cablings over the core curve of each torus. Moreover, the domain bounded by each invariant torus is covered by an almost full measure set of invariant tori. We show that an analogous result holds for high-frequency Beltrami fields on the flat 3-torus.
We show how an object from the combinatorially geometric version of the analytical number theory, namely geometric continued fractions, appears in the classical smooth dynamics, namely in the problem on the topological classification of Anosov diffeomorphisms of tori.
We consider generalized Frenkel-Kontorova models on higher dimensional lattices. We show that the invariant tori which are parameterized by continuous hull functions can be destroyed by small perturbations in the $C^r$ topology with $r<1$.
We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resum…
Invariant tori play a fundamental role in the dynamics of symplectic and volume-preserving maps. Codimension-one tori are particularly important as they form barriers to transport. Such tori foliate the phase space of integrable, volume-preserving maps with one action and $d$ angles. For the area-preserving case, Greene's residue criterion is often used to predict the destruction of tori from the properties of nearby periodic orbits. Even though KAM theory applies to the three-dimensional case, the robustness of tori in such systems is still poorly understood. We study a three-dimensional, rev…
