Near-integrable Hamiltonian Systems Research Articles

Old perturbative methods for a new problem in Celestial Mechanics: the space debris dynamics

Perturbative methods have been developed and widely used in the XVIII and XIX century to study the behavior of N-body problems in Celestial Mechanics. Such methods apply to nearly-integrable Hamiltonian systems and they have the remarkable property to be constructive. A well-known application of perturbative techniques is represented by the construction of the so-called proper elements, which are quasi-invariants of the dynamics, obtained by removing the perturbing function to higher orders. They have been used to identify families of asteroids; more recently, they have been used in the context of space debris, which is the main core of this work. We describe the dynamics of space debris, considering a model including the Earth’s gravitational attraction, the influence of Sun and Moon, and the Solar radiation pressure. We construct a Lie series normalization procedure and we compute the proper elements associated to the orbital elements. To provide a concrete example, we analyze three different break-up events with nearby initial orbital elements. We use the information coming from proper elements to successfully group the fragments; the clusterization is supported by statistical data analysis and by machine learning methods. These results show that perturbative methods still play an important role in the study of the dynamics of space objects.

Global dynamics visualisation from Lagrangian Descriptors. Applications to discrete and continuous systems

This paper introduces a new global dynamics and chaos indicator based on the method of Lagrangian Descriptor apt for discriminating ordered and deterministic chaotic motions in multidimensional systems. The selected implementation of this method requires only the knowledge of orbits on finite time windows and is free of the computation of the tangent vector dynamics (i.e.,variational equations are not needed). To demonstrate its ability in visualising different dynamical behaviours, in particular for highlighting chaotic regions, several stability maps of classical systems, obtained with different phase space methods, are reproduced. The benchmark examples are rooted in discrete and continuous nearly-integrable dynamical systems, with prominent features played by resonances. These include the Chirikov standard map, higher dimensional symplectic and volume preserving maps, fundamental models of resonances, and a 3 degrees-of-freedom nearly-integrable Hamiltonian system with a dense web of resonances. The indicator thus appears to be relevant for understanding phase space transport mediated by resonances in nearly-integrable system, as ubiquitous in celestial mechanics or astrodynamics.

V. I. Arnold’s “Global” KAM Theorem and Geometric Measure Estimates

This paper continues the discussion started in [ 10 ] concerning Arnold’s legacy on classical KAM theory and (some of) its modern developments. We prove a detailed and explicit “global” Arnold’s KAM theorem, which yields, in particular, the Whitney conjugacy of a non-degenerate, real-analytic, nearly-integrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov set are provided in case the phase space is: (A) a uniform neighbourhood of an arbitrary (bounded) set times the \(d\) -torus and (B) a domain with \(C^{2}\) boundary times the \(d\) -torus. All constants are explicitly given.

Exponentially Small Splitting of Separatrices Associated to 3D Whiskered Tori with Cubic Frequencies

We study the splitting of invariant manifolds of whiskered (hyperbolic) tori with three frequencies in a nearly-integrable Hamiltonian system, whose hyperbolic part is given by a pendulum. We consider a 3-dimensional torus with a fast frequency vector $\omega/\sqrt\varepsilon$, with $\omega=(1,\Omega,\widetilde\Omega)$ where $\Omega$ is a cubic irrational number whose two conjugates are complex, and the components of $\omega$ generate the field $\mathbb Q(\Omega)$. A paradigmatic case is the cubic golden vector, given by the (real) number $\Omega$ satisfying $\Omega^3=1-\Omega$, and $\widetilde\Omega=\Omega^2$. For such 3-dimensional frequency vectors, the standard theory of continued fractions cannot be applied, so we develop a methodology for determining the behavior of the small divisors $\langle k,\omega\rangle$, $k\in{\mathbb Z}^3$. Applying the Poincar\'e-Melnikov method, this allows us to carry out a careful study of the dominant harmonic (which depends on $\varepsilon$) of the Melnikov function, obtaining an asymptotic estimate for the maximal splitting distance, which is exponentially small in $\varepsilon$, and valid for all sufficiently small values of~$\varepsilon$. This estimate behaves like $\exp\{-h_1(\varepsilon)/\varepsilon^{1/6}\}$ and we provide, for the first time in a system with 3 frequencies, an accurate description of the (positive) function $h_1(\varepsilon)$ in the numerator of the exponent, showing that it can be explicitly constructed from the resonance properties of the frequency vector $\omega$, and proving that it is a quasiperiodic function (and not periodic) with respect to $\ln\varepsilon$. In this way, we emphasize the strong dependence of the estimates for the splitting on the arithmetic properties of the frequencies.

On the topology of nearly-integrable Hamiltonians at simple resonances

We show that, in general, averaging at simple resonances a real-analytic, nearly-integrable Hamiltonian, one obtains a one-dimensional system with a cosine-like potential; ‘in general’ means for a generic class of holomorphic perturbations and apart from a finite number of simple resonances with small Fourier modes; ‘cosine-like’ means that the potential depends only on the resonant angle, with respect to which it is a Morse function with one maximum and one minimum. Furthermore, the (full) transformed Hamiltonian is the sum of an effective one-dimensional Hamiltonian (which is, in turn, the sum of the unperturbed Hamiltonian plus the cosine-like potential) and a perturbation, which is uniformly exponentially small. As a corollary, under the above hypotheses, if the unperturbed Hamiltonian is also strictly convex, the effective Hamiltonian at any simple resonance (apart a finite number of low-mode resonances) has the phase portrait of a pendulum. The results presented in this paper are an essential step in the proof (in the ‘mechanical’ case) of a conjecture by Arnold–Kozlov–Neishdadt [Arnold V I, Kozlov V V and Neishtadt A I 2006 Mathematical aspects of classical and celestial mechanics Encyclopaedia of Mathematical Sciences 3rd edn vol 3 (Berlin: Springer), remark 6.8, p 285], claiming that the measure of the ‘non-torus set’ in general nearly-integrable Hamiltonian systems has the same size of the perturbation; compare [Biasco L and Chierchia L 2015 On the measure of Lagrangian invariant tori in nearly-integrable mechanical systems Rendiconti Lincei. Mat. Appl. 26 1–10 and Biasco L and Chierchia L KAM Theory for Secondary Tori (arXiv:1702.06480v1 [math.DS])].

On Resonances in Hamiltonian Systems with Three Degrees of Freedom

We address the dynamics of near-integrable Hamiltonian systems with 3 degrees of freedom in extended vicinities of unperturbed resonant invariant Liouville tori. The main attention is paid to the case where the unperturbed torus satisfies two independent resonance conditions. In this case the average dynamics is 4-dimensional, reduced to a generalised motion under a conservative force on the 2-torus and is generically non-integrable. Methods of differential topology are applied to full description of equilibrium states and phase foliations of the average system. The results are illustrated by a simple model combining the non-degeneracy and non-integrability of the isoenergetically reduced system.

Weak convergence to equilibrium of statistical ensembles in integrable Hamiltonian systems

This article explores the long-time behavior of the bounded orbits associated with an ensemble of initial conditions in a nondegenerate integrable Hamiltonian system. Such systems are inherently nonlinear and subject to highly regular phase space filamentation that can drive the ensemble of orbits toward a stationary state. Describing the statistical ensemble by a probability density on a neighborhood of a family of invariant tori, it is proved that the probability density describing the ensemble at time t converges weakly to an invariant density as t → ∞. More generally, we provide sufficient conditions for convergence to equilibrium of a multiphase system in action-angle form. These ideas are applied to an illustrative exactly soluble example. This work is relevant for understanding the statistical mechanics of integrable and near-integrable Hamiltonian systems.

Transient Invariant and Quasi-Invariant Structures in an Example of an Aperiodically Time Dependent Fluid Flow

Starting from the concept of invariant KAM tori for nearly-integrable Hamiltonian systems with periodic or quasi-periodic nonautonomous perturbation, the paper analyzes the “analogue” of this class of invariant objects when the dependence on time is aperiodic. The investigation is carried out in a model motivated by the problem of a traveling wave in a channel over a smooth, quasi- and asymptotically flat (from which the “transient” feature) bathymetry, representing a case in which the described structures play the role of barriers to fluid transport in phase space. The paper provides computational evidence for the existence of transient structures also for “large” values of the perturbation size, as a complement to the rigorous results already proven by the first author for real-analytic bathymetry functions.

On the chaotic diffusion in multidimensional Hamiltonian systems

We present numerical evidence that diffusion in the herein studied multidimensional near-integrable Hamiltonian systems departs from a normal process, at least for realistic timescales. Therefore, the derivation of a diffusion coefficient from a linear fit on the variance evolution of the unperturbed integrals fails. We review some topics on diffusion in the Arnold Hamiltonian and yield numerical and theoretical arguments to show that in the examples we considered, a standard coefficient would not provide a good estimation of the speed of diffusion. However, numerical experiments concerning diffusion would provide reliable information about the stability of the motion within chaotic regions of the phase space. In this direction, we present an extension of previous results concerning the dynamical structure of the Laplace resonance in Gliese-876 planetary system considering variations of the orbital parameters accordingly to the error introduced by the radial velocity determination. We found that a slight variation of the eccentricity of planet c would destabilize the inner region of the resonance that, though chaotic, shows stable when adopting the best fit values for the parameters.

Quasi-effective stability for nearly integrable Hamiltonian systems

This paper concerns with the stability of the orbits for nearly integrable Hamiltonian systems. Based on Nekehoroshev's original works in [14], we present the definition of quasi-effective stability and prove a theorem on quasi-effective stability under the Russmann's non-degeneracy. Our result gives a relation between KAM theorem and effective stability. A rapidly converging iteration procedure with two parameters is designed.

Lagrangian tori near resonances of near-integrable Hamiltonian systems

We study families of Lagrangian tori that appear in a neighbourhood of a resonance of a near-integrable Hamiltonian system. Such families disappear in the ‘integrable’ limit ε → 0. Dynamics on these tori are 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 a large relative measure in the resonant domains in the sense that the relative measure of the remaining ‘chaotic’ set is of the order . Therefore, for small ε > 0 a random initial condition in a -neighbourhood of a single resonance occurs inside this set (and therefore generates a quasi-periodic motion) with a probability much larger than in the ‘chaotic’ set.We present results of numerical simulations and discuss the form of projection of such tori to the action space.At the end of section 4 we discuss the relationship of our results and a conjecture that tori (in a near-integrable Hamiltonian systems) occupy all the phase space except a set of measure ∼ε.

A λ-lemma for normally hyperbolic invariant manifolds

Let N be a smooth manifold and f: N → N be a C ℓ , ℓ ⩾ 2 diffeomorphism. Let M be a normally hyperbolic invariant manifold, not necessarily compact. We prove an analogue of the λ-lemma in this case. Applications of this result are given in the context of normally hyperbolic invariant annuli or cylinders which are the basic pieces of all geometric mechanisms for diffusion in Hamiltonian systems. Moreover, we construct an explicit class of three-degree-of-freedom near-integrable Hamiltonian systems which satisfy our assumptions.

Continuation of the exponentially small transversality for the splitting of separatrices to a whiskered torus with silver ratio

We study the exponentially small splitting of invariant manifolds of whiskered (hyperbolic) tori with two fast frequencies in nearly-integrable Hamiltonian systems whose hyperbolic part is given by a pendulum. We consider a torus whose frequency ratio is the silver number $\Omega=\sqrt{2}-1$. We show that the Poincar\'e-Melnikov method can be applied to establish the existence of 4 transverse homoclinic orbits to the whiskered torus, and provide asymptotic estimates for the tranversality of the splitting whose dependence on the perturbation parameter $\varepsilon$ satisfies a periodicity property. We also prove the continuation of the transversality of the homoclinic orbits for all the sufficiently small values of $\varepsilon$, generalizing the results previously known for the golden number.

Exponentially Small Lower Bounds for the Splitting of Separatrices to Whiskered Tori with Frequencies of Constant Type

We study the splitting of invariant manifolds of whiskered tori with two frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a two-dimensional torus with a fast frequency vector [Formula: see text], with ω = (1, Ω) where Ω is an irrational number of constant type, i.e. a number whose continued fraction has bounded entries. Applying the Poincaré–Melnikov method, we find exponentially small lower bounds for the maximal splitting distance between the stable and unstable invariant manifolds associated to the invariant torus, and we show that these bounds depend strongly on the arithmetic properties of the frequencies.

Exponentially small asymptotic estimates for the splitting of separatrices to whiskered tori with quadratic and cubic frequencies

We study the splitting of invariant manifolds of whiskered tori with two or three frequencies in nearly-integrable Hamiltonian systems, such that the hyperbolic part is given by a pendulum. We consider a 2-dimensional torus with a frequency vector $\omega=(1,\Omega)$, where $\Omega$ is a quadratic irrational number, or a 3-dimensional torus with a frequency vector $\omega=(1,\Omega,\Omega^2)$, where $\Omega$ is a cubic irrational number. Applying the Poincaré--Melnikov method, we find exponentially small asymptotic estimates for the maximal splitting distance between the stable and unstable manifolds associated to the invariant torus, and we show that such estimates depend strongly on the arithmetic properties of the frequencies. In the quadratic case, we use the continued fractions theory to establish a certain arithmetic property, fulfilled in 24 cases, which allows us to provide asymptotic estimates in a simple way. In the cubic case, we focus our attention to the case in which $\Omega$ is the so-called cubic golden number (the real root of $x^3+x-1=0$), obtaining also asymptotic estimates. We point out the similitudes and differences between the results obtained for both the quadratic and cubic cases.

STABILITY OF NEARLY-INTEGRABLE SYSTEMS WITH DISSIPATION

We study the stability of a vector field associated to a nearly-integrable Hamiltonian dynamical system to which a dissipation is added. Such a system is governed by two parameters, namely the perturbing and dissipative parameters, and it depends on a drift function. Assuming that the frequency of motion satisfies some resonance assumption, we investigate the stability of the dynamics, and precisely the variation of the action variables associated to the conservative model. According to the structure of the vector field, one can find linear and long-term stability times, which are established under smallness conditions of the parameters. We also provide some applications to concrete examples, which exhibit a linear or long-term stability behavior.

New families of symplectic splitting methods for numerical integration in dynamical astronomy

We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coefficients of splitting methods to achieve a prescribed order of accuracy. Splitting methods satisfying such (generalized) order conditions are appropriate in particular for the numerical simulation of the Solar System described in Jacobi coordinates. We show that, when using Poincaré Heliocentric coordinates, the same order of accuracy may be obtained by imposing an additional polynomial equation on the coefficients of the splitting method. We construct several splitting methods appropriate for each of the two sets of coordinates by solving the corresponding systems of polynomial equations and finding the optimal solutions. The experiments reported here indicate that the efficiency of our new schemes is clearly superior to previous integrators when high accuracy is required.

The symmetric parabolic resonance

The parabolic resonance instability emerges in diverse applications ranging from optical systems to simple mechanical ones. It appears persistently in p-parameter families of near-integrable Hamiltonian systems with n degrees of freedom provided n + p ⩾ 3. Here we study the simplest (n = 2, p = 1) symmetric case. The structure and the phase-space volume of the corresponding instability zones are characterized. It is shown that the symmetric case has six distinct non-degenerate normal forms, and two degenerate ones. In the regular cases, the instability zone has the usual extent in the action direction. However, the phase-space volume of this zone is found to be polynomial in the perturbation parameter ε (and not exponentially small as in the elliptic resonance case). Finally, the extent of the instability zone in some of the degenerate cases is explored. Three applications in which the symmetric parabolic resonance arises are presented and analysed.

On the existence of invariant tori in nearly-integrable Hamiltonian systems with finitely differentiable perturbations

We prove the existence of invariant tori in Hamiltonian systems, which are analytic and integrable except a 2n-times continuously differentiable perturbation (n denotes the number of the degrees of freedom), provided that the moduli of continuity of the 2n-th partial derivatives of the perturbation satisfy a condition of finiteness (condition on an integral), which is more general than a Holder condition. So far the existence of invariant tori could be proven only under the condition that the 2n-th partial derivatives of the perturbation are Holder continuous.

KAM stability and celestial mechanics

KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to systems for observable values of the perturbation parameters. We consider the Restricted, Circular, Planar, Three-Body Problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCP3BP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system. The paper is self-contained but does not include the (similar to 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors.