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Guido Gentile

Publications and source records attributed to Guido Gentile.

At least 55 records · Page 3Linked to original sources
arXiv2007

Periodic solutions for the Schroedinger equation with nonlocal smoothing nonlinearities in higher dimension

We consider the nonlinear Schroedinger equation in higher dimension with Dirichlet boundary conditions and with a non-local smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in solving a "small divisor problem" which we do by using a renormalisation group approach.

math.AP↗
arXiv2006

Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasi-periodic forcing term and in the presence of damping. In the limit of large damping, under some generic non-degeneracy condition on the force, there are quasi-periodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case the proof given simplifies that provided in a previous work of ours. We also show that in any dimension $d$, for the existence of a quasi-periodic solution with the same frequency vector as the forcing term, the standard Diophantine condition can be weakened into the Bryuno condition. In all cases, under a suitable positivity condition, the quasi-periodic solution is proved to describe a local attractor.

math.DS↗
arXiv2006

Bifurcation curves of subharmonic solutions

We revisit a problem considered by Chow and Hale on the existence of subharmonic solutions for perturbed systems. In the analytic setting, under more general (weaker) conditions, we prove their results on the existence of bifurcation curves from the nonexistence to the existence of subharmonic solutions. In particular our results apply also when one has degeneracy to first order -- i.e. when the subharmonic Melnikov function vanishes identically. Moreover we can deal as well with the case in which degeneracy persists to arbitrarily high orders, in the sense that suitable generalisations to higher orders of the subharmonic Melnikov function are also identically zero. In general the bifurcation curves are not analytic, and even when they are smooth they can form cusps at the origin: we say in this case that the curves are degenerate as the corresponding tangent lines coincide. The technique we use is completely different from that of Chow and Hale, and it is essentially based on rigorous perturbation theory.

math.DS↗
arXiv2005

Resummation of perturbation series and reducibility for Bryuno skew-product flows

We consider skew-product systems on T^d x SL(2,R) for Bryuno base flows close to constant coefficients, depending on a parameter, in any dimension d, and we prove reducibility for a large measure set of values of the parameter. The proof is based on a resummation procedure of the formal power series for the conjugation, and uses techniques of renormalisation group in quantum field theory.

math.DS↗
arXiv2005

Globally and locally attractive solutions for quasi-periodically forced systems

We consider a class of differential equations, $\ddot x + γ\dot x + g(x) = f(ωt)$, with $ω\in {\bf R}^{d}$, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of the limit cycle described by the trajectory with the same quasi-periodicity as the forcing. For $g(x)=x^{2p+1}$, $p\in {\bf N}$, we show that, when the dissipation coefficient is large enough, there is only one limit cycle and that it is a global attractor. In the case of other forces, including $g(x)=x^{2p}$ (with $p=1$ describing the varactor equation), we find estimates for the basin of attraction of the limit cycle.

math.DS↗
arXiv2005

Fractional Lindstedt series

The parametric equations of the surfaces on which highly resonant quasi-periodic motions develop (lower-dimensional tori) cannot be analytically continued, in general, in the perturbation parameter, i.e. they are not analytic functions of the perturbation parameter. However rather generally quasi-periodic motions whose frequencies satisfy only one rational relation ("resonances of order 1") admit formal perturbation expansions in terms of a fractional power of the perturbation parameter, depending on the degeneration of the resonance. We find conditions for this to happen, and in such a case we prove that the formal expansion is convergent after suitable resummation.

math-ph↗
arXiv2005

Degenerate lower-dimensional tori under the Bryuno condition

We study the problem of conservation of maximal and lower-dimensional invariant tori for analytic convex quasi-integrable Hamiltonian systems. In the absence of perturbation the lower-dimensional tori are degenerate, in the sense that the normal frequencies vanish, so that the tori are neither elliptic nor hyperbolic. We show that if the perturbation parameter is small enough, for a large measure subset of any resonant submanifold of the action variable space, under some generic non-degeneracy conditions on the perturbation function, there are lower-dimensional tori which are conserved. They are characterised by rotation vectors satisfying some generalised Bryuno conditions involving also the normal frequencies. We also show that, again under some generic assumptions on the perturbation, any torus with fixed rotation vector satisfying the Bryuno condition is conserved for most values of the perturbation parameter in an interval small enough around the origin. According to the sign of the normal frequencies and of the perturbation parameter the torus becomes either hyperbolic or elliptic or of mixed type.

math.DS↗
arXiv2005

Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms

We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that already the case of periodic forcing terms is non-trivial, as the solution is not analytic in a neighbourhood of the origin: it turns out to be Borel-summable. In the case of quasi-periodic forcing terms we need Renormalization Group techniques in order to control the small divisors arising in the perturbation series. We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.

math.DS↗
arXiv2004

Stability for quasi-periodically perturbed Hill's equations

We consider a perturbed Hill's equation of the form $\ddot ϕ+ (p_{0}(t) + εp_{1}(t)) ϕ= 0$, where $p_{0}$ is real analytic and periodic, $p_{1}$ is real analytic and quasi-periodic and $\eps$ is a ``small'' real parameter. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials $p_{0}$ and $p_{1}$ and the proper frequency of the unperturbed ($ε=0$) Hill's equation, but without making non-degeneracy assumptions on the perturbing potential $p_{1}$, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if $ε$ lies in a Cantor set of relatively large measure in $[-ε_0,ε_0]$, where $ε_0$ is small enough. Our method is based on a resummation procedure of a formal Lindstedt series obtained as a solution of a generalized Riccati equation associated to Hill's problem.

math-ph↗
arXiv2004

Conservation of resonant periodic solutions for the one-dimensional nonlinear Schroedinger equation

We consider the one-dimensional nonlinear Schrödinger equation with Dirichlet boundary conditions in the fully resonant case (absence of the zero-mass term). We investigate conservation of small amplitude periodic-solutions for a large set measure of frequencies. In particular we show that there are infinitely many periodic solutions which continue the linear ones involving an arbitrary number of resonant modes, provided the corresponding frequencies are large enough and close enough to each other (wave packets with large wave number).

math.DS↗
arXiv2004

Breakdown of Lindstedt Expansion for Chaotic Maps

In a previous paper of one of us [Europhys. Lett. 59 (2002), 330--336] the validity of Greene's method for determining the critical constant of the standard map (SM) was questioned on the basis of some numerical findings. Here we come back to that analysis and we provide an interpretation of the numerical results by showing that no contradiction is found with respect to Greene's method. We show that the previous results based on the expansion in Lindstedt series do correspond to the transition value but for a different map: the semi-standard map (SSM). Moreover, we study the expansion obtained from the SM and SSM by suppressing the small divisors. The first case turns out to be related to Kepler's equation after a proper transformation of variables. In both cases we give an analytical solution for the radius of convergence, that represents the singularity in the complex plane closest to the origin. Also here, the radius of convergence of the SM's analogue turns out to be lower than the one of the SSM. However, despite the absence of small denominators these two radii are lower than the ones of the true maps for golden mean winding numbers. Finally, the analyticity domain and, in particular, the critical constant for the two maps without small divisors are studied analytically and numerically. The analyticity domain appears to be an perfect circle for the SSM analogue, while it is stretched along the real axis for the SM analogue yielding a critical constant that is larger than its radius of convergence.

cond-mat.other↗
arXiv2004

Periodic solutions for completely resonant nonlinear wave equations

We consider the nonlinear string equation with Dirichlet boundary conditions $u_{xx}-u_{tt}=ϕ(u)$, with $ϕ(u)=Φu^{3} + O(u^{5})$ odd and analytic, $Φ\neq0$, and we construct small amplitude periodic solutions with frequency $ø$ for a large Lebesgue measure set of $ø$ close to 1. This extends previous results where only a zero-measure set of frequencies could be treated (the ones for which no small divisors appear). The proof is based on combining the Lyapunov-Schmidt decomposition, which leads to two separate sets of equations dealing with the resonant and nonresonant Fourier components, respectively the Q and the P equations, with resummation techniques of divergent powers series, allowing us to control the small divisors problem. The main difficulty with respect the nonlinear wave equations $u_{xx}-u_{tt}+ M u = ϕ(u)$, $M\neq0$, is that not only the P equation but also the Q equation is infinite-dimensional

math.DS↗
arXiv2003

Scaling of the Critical Function for the Standard Map: Some Numerical Results

The behavior of the critical function for the breakdown of the homotopically non-trivial invariant (KAM) curves for the standard map, as the rotation number tends to a rational number, is investigated using a version of Greene's residue criterion. The results are compared to the analogous ones for the radius of convergence of the Lindstedt series, in which case rigorous theorems have been proved. The conjectured interpolation of the critical function in terms of the Bryuno function is discussed.

math.DS↗
arXiv1998

Lindstedt series and Hamilton--Jacobi equation for hyperbolic tori in three time scales problems

Interacting systems consisting of two rotators and a pendulum are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of unstable quasi periodic motions in phase space is studied via Lindstedt series. The result is a strong improvement, compared to our previous results, on the domain of validity of bounds that imply existence of invariant tori, large homoclinic angles, long heteroclinic chains and drift--diffusion in phase space.

chao-dyn↗
arXiv1998

Bryuno Function and the Standard Map

For the standard map the homotopically non-trivial invariant curves of rotation number satisfying the Bryuno condition are shown to be analytic in the perturbative parameter, provided the latter is small enough. The radius of convergence of the Lindstedt series -- sometimes called critical function of the standard map -- is studied and the relation with the Bryuno function is derived: the logarithm of the radius of convergence plus twice the Bryuno function is proved to be bounded (from below and from above) uniformily in the rotation number.

chao-dyn↗
arXiv1998

Hamilton-Jacobi equation, heteroclinic chains and Arnol'd diffusion in three time scales systems

Interacting systems consisting of two rotators and a point mass near a hyperbolic fixed point are considered, in a case in which the uncoupled systems have three very different characteristic time scales. The abundance of quasi periodic motions in phase space is studied via the Hamilton-Jacobi equation. The main result, a high density theorem of invariant tori, is derived by the classical canonical transformation method extending previous results. As an application the existence of long heteroclinic chains (and of Arnol'd diffusion) is proved for systems interacting through a trigonometric polynomial in the angle variables.

chao-dyn↗
arXiv1996

Quasi linear flows on tori: regularity of their linearization

Under suitable conditions a flow on a torus $C^{(p)}$--close, with $p$ large enough, to a quasi periodic diophantine rotation is shown to be conjugated to the quasi periodic rotation by a map that is analytic in the perturbation size. This result is parallel to Moser's theorem stating conjugability in class $C^{(p')}$ for some $p'<p$. The extra conditions restrict the class of perturbations that are allowed.

chao-dyn↗