Alessandra Celletti

The measure of chaos by the numerical analysis of the fundamental frequencies. Application to the standard mapping

Abstract The method of analysis of the chaotic behaviour of a dynamical system by the numerical analysis of the fundamental frequencies developed for the study of the stability of the solar system (J. Laskar, Icarus 88, 1990) is presented here with application to the standard mapping. This method is well suited for weakly chaotic motion with any number of degrees of freedom and is based on the analysis of the variations with time of the fundamental frequencies of an hamiltonian system. It allows to give an analytical representation of the solution when it is regular, to detect if an orbit is chaotic over a smaller time span than with the Lyapunov exponents and gives also an estimate of the size of the chaotic zones in the frequency domain. The frequency analysis also provides a numerical criterion for the destruction of invariant curves. Its application to the standard mapping shows that the golden curve does not survive for a = 0.9718 which is very close and compatible with Greenes value ac = 0.971635.

Analysis of resonances in the spin-orbit problem in Celestial Mechanics: the synchronous resonance (part I)

We study the stability of spin-orbit resonances in Celestial Mechanics, namely the exact commensurabilities between the periods of rotation and revolution of satellites or planets. We introduce a mathematical model describing an approximation of the physical situation and we select a set of satellites for which such simplified model provides a good approximation.Applying the Kolmogorov-Arnold-Moser theory we are able to construct invariant surfaces trapping the synchronous resonance from above and below. The existence of such surfaces, established for the natural values of the physical and orbital parameters, allows to prove the stability of the 1∶1 resonance. Furthermore we try to construct KAM tori with frequencies as close as possible to one so to trap the synchronous resonance in a finer region.

On the stability of the lagrangian points in the spatial restricted problem of three bodies

The problem of stability of the Lagrangian equilibrium point of the circular restricted problem of three bodies is investigated in the light of Nekhoroshev-like theory. Looking for stability over a time interval of the order of the estimated age of the universe, we find a physically relevant stability region. An application of the method to the Sun-Jupiter and the Earth-Moon systems is made. Moreover, we try to compare the size of our stability region with that of the region where the Trojan asteroids are actually found; the result in such case is negative, thus leaving open the problem of the stability of these asteroids.

Construction of Analytic KAM Surfaces and Effective Stability Bounds

A class of analytic (possibly) time-dependent Hamiltonian systems withd degrees of freedom and the “corresponding” class of area-preserving, twist diffeomorphisms of the plane are considered. Implementing a recent scheme due to Moser, Salamon and Zehnder, we provide a method that allows us to construct “explicitly” KAM surfaces and, hence, to give lower bounds on their breakdown thresholds. We, then, apply this method to the HamiltonianH≡y2/2+ε(cosx+cos(x−t)) and to the map (y,x)→(y+ε sinx,x+y+ε sinx) obtaining, with the aid of computer-assisted estimations, explicit approximations (within an error of ∼10−5) of the golden-mean KAM surfaces for complex values of ε with |ε| less or equal than, respectively, 0.015 and 0.65. (The experimental numerical values at which such surfaces are expected to disappear are about, respectively, 0.027 and 0.97.) A possible connection between break-down thresholds and singularities in the complex ε-plane is pointed out.

Stability and Chaos in Celestial Mechanics

This book presents classical celestial mechanics and its interplay with dynamical systems in a way suitable for advance level undergraduate students as well as postgraduate students and researchers. First paradigmatic models are used to introduce the reader to the concepts of order, chaos, invariant curves, cantori. Next the main numerical methods to investigate a dynamical system are presented. Then the author reviews the classical two-body problem and proceeds to explore the three-body model in order to investigate orbital resonances and Lagrange solutions. In rotational dynamics the author details the derivation of the rigid body motion, and continues by discussing related topics, from spin-orbit resonances to dumbbell satellite dynamics. Perturbation theory is then explored in full detail including practical examples of its application to finding periodic orbits, computation of the libration in longitude of the Moon. The main ideas of KAM theory are provided including a presentation of long-term stability and converse KAM results. Celletti then explains the implementation of computer-assisted techniques, which allow the user to obtain rigorous results in good agreement with the astronomical expectations. Finally the study of collisions in the solar system is approached.

A constructive theory of Lagrangian tori and computer-assisted applications

Perturbative techniques are among the most powerful tools in the theory of conservative dynamical systems. Besides giving finite time predictions (something well known to the astronomers of the eighteenth century), perturbation methods may be used to establish the existence of regular motions. H. Poincare used thoroughly such methods in his investigation in Celestial Mechanics [Po], obtaining, e.g., his celebrated results on periodic orbits for Hamiltonian systems. A more recent success of perturbation ideas is the so called “KAM (Kolmogorov [Ko]-Arnold [A1]-Moser [Mo1]) theory”, which ensures, under suitable smoothness assumptions, the survival under a small perturbation of “most” of the invariant maximal tori which foliate the phase-space of “integrable” conservative systems (see [B] for review and exhaustive references and [ChG] for recent developments).

Natural Boundaries for Area-Preserving Twist Maps

We consider KAM invariant curves for generalizations of the standard map of the form (x′, y′)=(x+y′, y+ɛf(x)), wheref(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Padé approximant method applied to the function which conjugates the dynamics to a rotation θ↦θ+ω. In the complexɛ plane, natural boundaries of different shapes are found. In the complexθ plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 asɛ tends to the critical value.

Improved estimates on the existence of invariant tori for Hamiltonian systems

The existence of invariant tori in nearly integrable Hamiltonian systems is investigated. We focus our attention on a particular one-dimensional, time-dependent model, known as the forced pendulum . We present a KAM algorithm which allows us to derive explicit estimates on the perturbing parameter ensuring the existence of invariant tori. Moreover, we introduce some technical novelties in the proof of the KAM theorem which allow us to provide results in good agreement with the experimental breakdown threshold. In particular, we have been able to prove the existence of the golden torus with frequency ½((5)1/2 -1) for values of the perturbing parameter equal to 92% of the numerical threshold, thus significantly improving the previous calculations.

Non-integrability of the problem of motion around an oblate planet

We provide a result of non-analytic integrability of the so-called J2-problem. Precisely by using the Lerman theorem we are able to prove the existence of a region of the phase space, where the dynamical system exhibits chaotic motions.

Rigorous estimates for a computer‐assisted KAM theory

Nonautonomous Hamiltonian systems of one degree of freedom close to integrable ones are considered. Let e be a positive parameter measuring the strength of the perturbation and denote by e c the critical value at which a given KAM (Kolmogorov–Arnold–Moser) torus breaks down. A computer‐assisted method that allows one to give rigorous lower bounds for e c is presented. This method has been applied in Celletti–Falcolini–Porzio (to be published in Ann. Inst. H. Poincare) to the Escande and Doveil pendulum yielding a bound which is within a factor 40.2 of the value indicated by numerical experiments.