Corrado Falcolini

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.

Compensations in small divisor problems

A general direct method, alternative to KAM theory, apt to deal with small divisor problems in the real-analytic category, is presented and tested on several small divisor problems including the construction of maximal quasi-periodic solutions for nearly-integrable non-degenerate Hamiltonian or Lagrangian systems and the construction of lower dimensional resonant tori for nearly-integrable Hamiltonian systems. The method is based on an explicit graph theoretical representation of the formal power series solutions, which allows to prove compensations among the monomials forming such representation.

Construction of invariant tori for the spin-orbit problem in the Mercury-Sun system

The stability of spin-orbit resonances, namely commensurabilities between the periods of rotation and revolution of an oblate satellite orbiting around a primary body, is investigated using perturbation theory. We reduce the system to a model described by a one-dimensional, time-dependent Hamiltonian function. By means of KAM theory we rigorously construct bidimensional invariant surfaces, which separate the three dimensional phase space. In particular with a suitable choice of the rotation numbers of the invariant tori we are able to trap the periodic orbit associated with a given resonance in a finite region of the phase space. This technique is applied to the Mercury-Sun system. A connection with the probability of capture in a resonance is also provided.

Singularities of periodic orbits near invariant curves

Abstract We show numerically, for standard-like maps, how the singularities (in the complex parameter) of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number. Furthermore, we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits. The results are compared with analyses performed by Pade approximants, Greene’s method, root criterion and the prediction by renormalization theory.

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems Δu=ɛfx(u, y), whereu ∶y ε ℝM →u(y) ε ℝN,f=f(x, y) is a real-analytic periodic function and ɛ is a small parameter. Kolmogorovs theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusTM+1. In the autonomous case,f=f(x), the above result holds for any ɛ.

A remark on the KAM theorem applied to a four-vortex system

We consider a planar four-vortex system with unit intensities and apply the KAM theorem for two-dimensional tori with fixed frequency. We obtain a rigorous lower bound for the stochasticity threshold of the torus with rotation numberω=(√5—1)/2 and compare our result with numerical experiments.

An Extension of Greene's Criterion for Conformally Symplectic Systems and a Partial Justification

Greenes criterion for twist mappings asserts the existence of smooth invariant circles with preassigned rotation number if and only if the periodic trajectories with frequency approaching that of the quasi-periodic orbit are at the border of linear stability. We formulate an extension of this criterion for conformally symplectic systems in any dimension and prove one direction of the implication, namely, that if there is a smooth invariant attractor, we can predict the eigenvalues of the periodic orbits whose frequencies approximate that of the torus for values of the parameters close to that of the attractor. The proof of this result is very different from the proof in the area-preserving case, since in the conformally symplectic case the existence of periodic orbits requires adjusting parameters. Also, as shown in [R. Calleja, A. Celletti, and R. de la Llave, J. Dynam. Differential Equations, 55 (2013), pp. 821--841], in the conformally symplectic case there are no Birkhoff invariants giving obstructio...

On the convergence of formal series containing small divisors

Poincare—Lindstedt series for the (formal) computation of quasi periodic solutions (in the context of real—analytic, nearly—integrable Hamiltonian dynamical systems) with fixed frequencies have been extensively studied, for over a century, from both the theoretical and applicative point of view. For applications, the Poincare-Lindstedt series provide a simple practical tool to explicitly compute the first few orders of perturbation theory; the problem of convergence has been instead much more controversial (famous is Poincare dubious statement in his Methodes nouvelles de la Mechanique Celeste). The matter was settled indirectly in the sixties thanks to KAM (Kolmogorov, Arnold, Moser) theory. “Indirectly” means that the convergence is obtained as a byproduct of estimates uniform in the smallness parameter rather than directly looking at the formal series and trying to check convergence by studying the rate of growth of coefficients (as in the classical Siegel’s approach to the small divisor problem arising in linearization of germs of complex analytic functions). As it is well known, the main problem with the “direct approach” is that the k th coefficient of the formal power series, if expanded in sums of monomials composed by Fourier coefficients of the Hamiltonian and of “small divisors” (appearing in the denominators of the monomials as linear integer combinations of the basic frequencies), contains, in general, monomials which diverge as k!. Hence, a “direct proof” has necessarily to deal with compensations, i.e., with the problem of grouping together all the “diverging” terms showing that they sum up to much smaller contributions, which can be bounded by a constant to the k th power. Direct proofs (in Hamiltonian setting) were given by H. Eliasson in 1988, by Gallavotti, Gentile and Mastropietro and, independently, by the authors in 1993 (for bibliography and more technical discussions see [1, 2, 3] and references therein).

Periodic orbits approximation of analyticity domains of invariant curves

We show numerically how the singularities of the function which conjugates the standard map to a rotation of rational period behave when the period goes to an irrational number.

Normal Form Invariants Around Spin-Orbit Periodic Orbits

We consider a model of spin-orbit interaction, describing the motion of an oblate satellite rotating about an internal spin-axis and orbiting about a central planet. The resulting second order differential equation depends upon the parameters provided by the equatorial oblateness of the satellite and its orbital eccentricity. Normal form transformations around the main spin-orbit resonances are carried out explicitly. As an outcome, one can compute some invariants; the fact that these quantities are not identically zero is a necessary condition to prove the existence of nearby periodic orbits (Birkhoff fixed point theorem). Moreover, the nonvanishing of the invariants provides also the stability of the spin-orbit resonances, since it guarantees the existence of invariant curves surrounding the periodic orbit.