Massimiliano Berti, Philippe Bolle
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the ``splitting determinant'' is exponentially small.
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We improve the global Nekhoroshev stability for analytic quasi-convex nearly integrable Hamiltonian systems. The new stability result is optimal, as it matches the fastest speed of Arnold diffusion.
It is well known that under generic $C^r$ smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the a priori unstable Hamiltonian systems of two and a half degrees of freedom.
We prove the existence of "Arnold diffusion orbits" in cusp-generic nearly integrable a priori stable systems on ${\mathbb A}^3$. The result relies on the cusp-generic existence of chains in nearly integrable a priori stable systems, proved in a previous paper, together with the existence of diffusion orbits along such chains. The latter result was obtained in collaboration with M. Gidea.
The main model studied in this paper is a lattice of nearest neighbors coupled pendula. For certain localized coupling we prove existence of energy transfer and estimate its speed.
We expose some selected topics concerning the instability of the action variables in a priori unstable Hamiltonian systems, and outline a new strategy that may allow to apply these methods to a priori stable systems.
In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$ H(x,y)=h(y)+εP(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3. $$ Under typical perturbation $εP$, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\minα$.
In this paper, we prove that the net of transition chain is $δ$-dense for nearly integrable positive definite Hamiltonian systems with 3 degrees of freedom in the cusp-residual generic sense in $C^r$-topology, $r\ge 6$. The main ingredients of the proof existed in \cite{CZ,C17a,C17b}. As an immediate consequence, Arnold diffusion exists among this class of Hamiltonian systems. The question of \cite{C17c} is answered in Section 9 of the paper.
This paper constructs a certain planar four-body problem which exhibits fast energy growth. The system considered is a quasi-periodic perturbation of the Restricted Planar Circular three-body Problem (RPC3BP). Gelfreich-Turaev's and de la Llave's mechanism is employed to obtain the fast energy growth. The diffusion is created by a heteroclinic cycle formed by two Lyapunov periodic orbits surrounding $L_1$ and $L_2$ Lagrangian points and their heteroclinic intersections. Our model is the first known example in celestial mechanics of the a priori chaotic case of Arnold diffusion.
We assume that a symplectic real-analytic map has an invariant normally hyperbolic cylinder and an associated transverse homoclinic cylinder. It is well known that such cylinder is preserved under small perturbations. We prove that for a generic real- analytic perturbation of this map the boundaries of the cylinder are connected by trajectories of the perturbed map.
In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: $$ H(x,y)=h(y)+\eps P(x,y), \qquad x\in\mathbb{T}^n,\ y\in\mathbb{R}^n,\quad n\geq 3. $$ Under typical perturbation $\eps P$, the system admits "connecting" orbit that passes through any finitely many prescribed small balls in the same energy level $H^{-1}(E)$ provided $E>\min h$.
We prove that for any non-trivial perturbation depending on any two independent harmonics of a pendulum and a rotor there is global instability. The proof is based on the geometrical method and relies on the concrete computation of several scattering maps. A complete description of the different kinds of scattering maps taking place as well as the existence of piecewise smooth global scattering maps is also provided.
Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved that in two dimensions, it is impossible for this angle to tend to zero along trajectories. We prove that such trajectories can exist in higher dimensions. Namely, using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, the existence of trajectories asymptotic…
On the constructivity of the variational approach to Arnold's Diffusion
本文讨论Bessi变分方法处理Arnold扩散的可构造性，在典型模型上获得可在计算机上实现的严格证明，并展示构造的扩散轨迹。
We study the Arnold diffusion in a priori unstable near-integrable systems in a neighbourhood of a resonance of low order. We consider a non-autonomous near-integrable Hamiltonian system with $n+1/2$ degrees of freedom, $n\ge 2$. Let the Hamilton function $H$ of depend on the parameter $\varepsilon$, for $\varepsilon=0$ the system is integrable and has a homoclinic asymptotic manifold $Γ$. Our main result is that for small generic perturbation in an $\varepsilon$-neighborhood of $Γ$ there exist trajectories the projections of which on the space of actions cross the resonance. By ``generic pert…
In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ \[ H_0(p)+\eps H_1(þ,p,t),\quad þ\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(þ_{\e},p_{e})(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}\|p(t)-p(0) \| >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\e$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders…
It is well known that instabilities of nearly integrable Hamiltonian systems occur around resonances. Dynamics near resonances of these systems is well approximated by the associated averaged system, called slow system. Each resonance is defined by a basis (a collection of integer vectors). We introduce a class of resonances whose basis can be divided into two well separated groups and call them dominant. We prove that the associated slow system can be well approximated by a subsystem given by one of the groups, both in the sense of the vector field and weak KAM theory. One of crucial ingredie…
Consider a symplectic map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave in [26], based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that…
We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) + $ε$H1($θ$, p, t), $θ$ $\in$ T n , p $\in$ B n , t $\in$ T = R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with a strictly convex H0. We show that for a "generic" $ε$H1, there exists an orbit ($θ$, p)(t) satisfying p(t) -- p(0) {\textgreater} l(H1) {\textgreater} 0, where l(H1) is independent of $ε$. The diffusion orbit travels along a co-dimension one resonance , and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination geometric a…
We consider the problem of Arnold's diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also develop a new method for measuring the splitting of the separatrices. As an application we justify, for three time scales systems, that the splitting is correctly predicted by the Poincaré-Melnikov function.
Arnold diffusion in the elliptic Hill four-body problem: geometric method and numerical verification
该研究针对椭圆Hill四体问题，基于法向双曲不变流形与散射映射提出两种Arnold扩散论证，结合数值计算与跟踪引理证明了能量扩散轨道的存在。
In this work we illustrate the Arnold diffusion in a concrete example---the \emph{a priori} unstable Hamiltonian system of $2+1/2$ degrees of freedom $H(p,q,I,φ,s) = p^{2}/2+\cos q -1 +I^{2}/2 + h(q,φ,s;\varepsilon)$---proving that for \emph{any} small periodic perturbation of the form $h(q,φ,s;\varepsilon) = \varepsilon\cos q\left( a_{00} + a_{10}\cosφ+ a_{01}\cos s \right)$ ($a_{10}a_{01} \neq 0$) there is global instability for the action. For the proof we apply a geometrical mechanism based in the so-called Scattering map. This work has the following structure: In a first stage, for a more…
In this paper, we study the chaotic motion of a massive particle moving in a perturbed Schwarzschild or Kerr background. We discover three novel orbits that do not exist in the unperturbed cases. First, we find zoom-whirl orbits moving around the photon shell which simultaneously exhibits Arnold diffusion: large oscillations of particle's angular momentum and energy. Next, we show the existence of oscillating orbits between a bounded region and infinity, analogous to Newtonian three-body problem. Thirdly, we find that in perturbed Kerr, there exists chaotic orbits around the event horizon that…
In this article, we prove the existence of Arnold diffusion for an interesting specific system -- discrete nonlinear Schrödinger equation. The proof is for the 5-dimensional case with or without resonance. In higher dimensions, the problem is open. Progresses are made by establishing a complete set of Melnikov-Arnold integrals in higher and infinite dimensions. The openness lies at the concrete computation of these Melnikov-Arnold integrals. New machineries introduced here into the topic of Arnold diffusion are the Darboux transformation and isospectral theory of integrable systems.
We consider a system of infinitely many penduli on an $m$-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasiperiodic tori with …
Arnold diffusion for an a priori unstable Hamiltonian system with 3 $+$ 1/2 degrees of freedom
针对3+1/2自由度先验不稳定哈密顿系统，应用内、外动力学迭代结合的几何机制证明扩散轨道存在，找到保证作用量大漂移的散射映射轨道族Highways，得到与文献一致的显式最优扩散时间，结果由解析方法和数值计算验证。
For a mechanical system consisting of a rotator and a pendulum coupled via a small, time-periodic Hamiltonian perturbation, the Arnold diffusion problem asserts the existence of `diffusing orbits' along which the energy of the rotator grows by an amount independent of the size of the coupling parameter, for all sufficiently small values of the coupling parameter. There is a vast literature on establishing Arnold diffusion for such systems. In this work, we consider the case when an additional, dissipative perturbation is added to the rotator-pendulum system with coupling. Therefore, the system…
We prove the existence of diffusing solutions in the motion of a charged particle in the presence of an ABC magnetic field. The equations of motion are modeled by a 3DOF Hamiltonian system depending on two parameters. For small values of these parameters, we obtain a normally hyperbolic invariant manifold and we apply the so-called geometric methods for a priori unstable systems developed by A. Delshams, R. de la Llave, and T.M. Seara. We characterize explicitly sufficient conditions for the existence of a transition chain of invariant tori having heteroclinic connections, thus obtaining globa…
A degenerate Arnold diffusion mechanism in the Restricted 3 Body Problem
本研究证明限制性平面椭圆三体问题在任意质量比下存在拓扑不稳定性，通过揭示一种涉及指数小现象的退化阿诺德扩散机制，将先前结果从先验不稳定情形推广到先验稳定情形。
In the present paper we prove a strong form of Arnold diffusion. Let $\mathbb{T}^2$ be the two torus and $B^2$ be the unit ball around the origin in $\mathbb{R}^2$. Fix $ρ>0$. Our main result says that for a "generic" time-periodic perturbation of an integrable system of two degrees of freedom \[ H_0(p)+εH_1(θ,p,t),\quad \ θ\in \mathbb{T}^2,\ p\in B^2,\ t\in \mathbb{T}, \] with a strictly convex $H_0$, there exists a $ρ$-dense orbit $(θ_ε,p_ε,t)(t)$ in $\mathbb{T}^2 \times B^2 \times \mathbb{T}$, namely, a $ρ$-neighborhood of the orbit contains $\mathbb{T}^2 \times B^2 \times \mathbb{T}$. Our …
We develop a variational method for constructing positive entropy invariant measures of Lagrangian systems without assuming transversal intersections of stable and unstable manifolds, and without restrictions to the size of non-integrable perturbations. We apply it to a family of two and a half degrees of freedom a-priori unstable Lagrangians, and show that if we assume that there is no topological obstruction to diffusion (precisely formulated in terms of topological non-degeneracy of minima of the Peierl's barrier function), then there exists a vast family of "horsheshoes", such as "shadowin…
We consider a class of autonomous Hamiltonian systems subject to small, time-periodic perturbations. When the perturbation parameter is set to zero, the energy of the system is preserved. This is no longer the case when the perturbation parameter is non-zero. We describe a topological method to establish orbits which diffuse in energy for every suitably small perturbation parameter $\varepsilon>0$. The method yields quantitative estimates: (i) The existence of orbits along which the energy drifts by an amount independent of $\varepsilon$. The time required by such orbits to drift is $O(1/\vare…
We present a diffusion mechanism for time-dependent perturbations of autonomous Hamiltonian systems introduced in [25]. This mechanism is based on shadowing of pseudo-orbits generated by two dynamics: an `outer dynamics', given by homoclinic trajectories to a normally hyperbolic invariant manifold, and an `inner dynamics', given by the restriction to that manifold. On the inner dynamics the only assumption is that it preserves area. Unlike other approaches, [25] does not rely on the KAM theory and/or Aubry-Mather theory to establish the existence of diffusion. Moreover, it does not require to …
Arnold Diffusion in the Full Three-Body Problem
本文证明了全三体问题存在 Arnold 扩散这一强不稳定性，通过将其表述为 Kepler 问题和限制性三体问题的摄动，利用正确对齐窗口的拓扑方法和计算机辅助证明，并以 Neptune-Triton-asteroid 系统为例获得了显式估计。
Arnold's diffusion in quasi integrable hamiltonian systems occurs in exponentially large time. We study an initially hyperbolic system which admits diffusion in polynomial time.
In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold. Using a {\it separatrix map}, introduced in a low dimensional case by Zaslavskii-Filonenko and studied in a multidimensional case by Treschev and Piftankin, for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [GKZ], we prove that the system restric…
This is a short survey on Nekhoroshev theory, KAM theory, and Arnold's diffusion.
On the Arnold diffusion mechanism in Medium Earth Orbit
本文以 Galileo 中地轨道为对象，利用含地球扁率与月球摄动的 Hamiltonian 模型，证明 Arnold 扩散可通过同宿轨道序列引发长期偏心率增长。
We study the problem of Arnold's diffusion in an example of isochronous system by using a geometrical method known as Windows Method. Despite the simple features of this example, we show that the absence of an anisochrony term leads to several substantial difficulties in the application of the method, requiring some additional devices as non-equally spaced transition chains and variable windows. In this way we are able to obtain a set of fast orbits whose drifting time matches, up to a constant, the time obtained via variational methods.
We consider the problem of Arnold's diffusion for nearly integrable isochronous Hamiltonian systems. We prove a shadowing theorem which improves the known estimates for the diffusion time. We also justify for three time scales systems that the splitting of the separatrices is correctly predicted by the Poincare'-Melnikov function.
Poincaré's work more than one century ago, or Laskar's numerical simulations from the 1990's on, have irrevocably impaired the long-held belief that the Solar System should be stable. But mathematical mechanisms explaining this instability have remained mysterious. In 1968, Arnold conjectured the existence of "Arnold diffusion" in celestial mechanics. We prove Arnold's conjecture in the planetary spatial $4$-body problem as well as in the corresponding hierarchical problem (where the bodies are increasingly separated), and show that this diffusion leads, on a long time interval, to some large-…
We study the dynamics of the restricted planar three-body problem near mean motion resonances, i.e. a resonance involving the Keplerian periods of the two lighter bodies revolving around the most massive one. This problem is often used to model Sun--Jupiter--asteroid systems. For the primaries (Sun and Jupiter), we pick a realistic mass ratio $μ=10^{-3}$ and a small eccentricity $e_0>0$. The main result is a construction of a variety of non local diffusing orbits which show a drastic change of the osculating (instant) eccentricity of the asteroid, while the osculating semi major axis is kept a…
We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(μ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ μ) \log (1/ μ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.
Consider a sufficiently smooth nearly integrable Hamiltonian system of two and a half degrees of freedom in action-angle coordinates \[ H_ε(φ,I,t)=H_0(I)+εH_1(φ,I,t), φ\in T^2,\ I\in U\subset R^2,\ t\in T=R/Z. \] Kolmogorov-Arnold-Moser Theorem asserts that a set of nearly full measure in phase space consists of three dimensional invariant tori carrying quasiperiodic dynamics. In this paper we prove that for a class of nearly integrable Hamiltonian systems there is an orbit which contains these KAM tori in its $ω$-limit set. This implies that the closure of the orbit has almost full measure in…
We consider a nearly integrable, non-isochronous, a-priori unstable Hamiltonian system with a (trigonometric polynomial) $O(μ)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $T_d = O((1/ μ) \log (1/ μ))$ by a variational method which does not require the existence of ``transition chains of tori'' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d$ is optimal as a consequence of a general stability result proved via classical perturbation theory.
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