arXiv subjects

Guido Gentile

Publications and source records attributed to Guido Gentile.

At least 37 records · Page 2Linked to original sources

arXiv2014

Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Since Moser's seminal work it is well known that the invariant curves of smooth nearly integrable twist maps of the cylinder with Diophantine rotation number are preserved under perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno rotation numbers. First, we will show that the series expansion for the invariant curves in powers of the perturbation parameter can be formally defined, then we shall prove that the series converges absolutely in a neighbourhood of the origin. This will be achieved using multiscale analysis and renormalisation group techniques to express the coefficients of the series as sums of values which are represented graphically as tree diagrams and then exploit cancellations between terms contributing to the same perturbation order. As a byproduct we shall see that, when perturbing linear maps, the series expansion for an analytic invariant curve converges for all perturbations if and only if the corresponding rotation number satisfies the Bryuno condition.

math.DS↗

arXiv2014

The effects of time-dependent dissipation on the basins of attraction for the pendulum with oscillating support

We consider a pendulum with vertically oscillating support and time-dependent damping coefficient which varies until reaching a finite final value. The sizes of the corresponding basins of attraction are found to depend strongly on the full evolution of the dissipation. In order to predict the behaviour of the system, it is essential to understand how the sizes of the basins of attraction for constant dissipation depend on the damping coefficient. For values of the parameters in the perturbation regime, we characterise analytically the conditions under which the attractors exist and study numerically how the sizes of their basins of attraction depend on the damping coefficient. Away from the perturbation regime, a numerical study of the attractors and the corresponding basins of attraction for different constant values of the damping coefficient produces a much more involved scenario: changing the magnitude of the dissipation causes some attractors to disappear either leaving no trace or producing new attractors by bifurcation, such as period doubling and saddle-node bifurcation. For an initially non-constant damping coefficient, both increasing and decreasing to some finite final value, we numerically observe that, when the damping coefficient varies slowly from a finite initial value to a different final value, without changing the set of attractors, the slower the variation the closer the sizes of the basins of attraction are to those they have for constant damping coefficient fixed at the initial value. If during the variation of the damping coefficient attractors appear or disappear, remarkable additional phenomena may occur. For instance, a fixed point asymptotically may attract the entire phase space, up to a zero measure set, even though no attractor with such a property exists for any value of the damping coefficient between the extreme values.

math.DS↗

arXiv2014

Resonant tori of arbitrary codimension for quasi-periodically forced systems

We consider a system of rotators subject to a small quasi-periodic forcing. We require the forcing to be analytic and satisfy a time-reversibility property and we assume its frequency vector to be Bryuno. Then we prove that, without imposing any non-degeneracy condition on the forcing, there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases.

math.DS↗

arXiv2012

Domains of analyticity for response solutions in strongly dissipative forced systems

We study the ordinary differential equation $\varepsilon\ddot x + \dot x + \varepsilon g(x) = \e f(ωt)$, where $g$ and $f$ are real-analytic functions, with $f$ quasi-periodic in $t$ with frequency vector $ω$. If $c_{0} \in \mathbb{R}$ is such that $g(c_0)$ equals the average of $f$ and $g'(c_0)\neq0$, under very mild assumptions on $ω$ there exists a quasi-periodic solution close to $c_0$. We show that such a solution depends analytically on $\varepsilon$ in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin.

math.DS↗

arXiv2012

Convergent series for quasi-periodically forced strongly dissipative systems

We study the ordinary differential equation ${\varepsilon}\ddot x+\dot x + {\varepsilon} g(x) = {\varepsilon} f(ωt)$, with $f$ and $g$ analytic and $f$ quasi-periodic in $t$ with frequency vector $ω\in R^{d}$. We show that if there exists $c_0\in R$ such that $g(c_0)$ equals the average of $f$ and the first non-zero derivative of $g$ at $c_0$ is of odd order $n$, then, for ${\varepsilon}$ small enough and under very mild Diophantine conditions on $ω$, there exists a quasi-periodic solution close to $c_0$, with the same frequency vector as $f$. In particular if $f$ is a trigonometric polynomial the Diophantine condition on $ω$ can be completely removed. This extends results previously available in the literature for $n=1$. We also point out that, if $n=1$ and the first derivative of $g$ at $c_0$ is positive, then the quasi-periodic solution is locally unique and attractive.

math.DS↗

arXiv2012

Lower-dimensional invariant tori for perturbations of a class of non-convex Hamiltonian functions

We consider a class of quasi-integrable Hamiltonian systems obtained by adding to a non-convex Hamiltonian function of an integrable system a perturbation depending only on the angle variables. We focus on a resonant maximal torus of the unperturbed system, foliated into a family of lower-dimensional tori of codimension 1, invariant under a quasi-periodic flow with rotation vector satisfying some mild Diophantine condition. We show that at least one lower-dimensional torus with that rotation vector always exists also for the perturbed system. The proof is based on multiscale analysis and resummation procedures of divergent series. A crucial role is played by suitable symmetries and cancellations, ultimately due to the Hamiltonian structure of the system.

math.DS↗

arXiv2012

Attractiveness of periodic orbits in parametrically forced systemswith time-increasing friction

We consider dissipative one-dimensional systems subject to a periodic force and study numerically how a time-varying friction affects the dynamics. As a model system, particularly suited for numerical analysis, we investigate the driven cubic oscillator in the presence of friction. We find that, if the damping coefficient increases in time up to a final constant value, then the basins of attraction of the leading resonances are larger than they would have been if the coefficient had been fixed at that value since the beginning. From a quantitative point of view, the scenario depends both on the final value and the growth rate of the damping coefficient. The relevance of the results for the spin-orbit model are discussed in some detail.

math.DS↗

arXiv2012

Resonant motions in the presence of degeneracies for quasi-periodically perturbed systems

We consider one-dimensional systems in the presence of a quasi-periodic perturbation, in the analytical setting, and study the problem of existence of quasi-periodic solutions which are resonant with the frequency vector of the perturbation. We assume that the unperturbed system is locally integrable and anisochronous, and that the frequency vector of the perturbation satisfies the Bryuno condition. Existence of resonant solutions is related to the zeroes of a suitable function, called the Melnikov function - by analogy with the periodic case. We show that, if the Melnikov function has a zero of odd order and under some further condition on the sign of the perturbation parameter, then there exists at least one resonant solution which continues an unperturbed solution. If the Melnikov function is identically zero then one can push perturbation theory up to the order where a counterpart of Melnikov function appears and does not vanish identically: if such a function has a zero of odd order and a suitable positiveness condition is met, again the same persistence result is obtained. If the system is Hamiltonian, then the procedure can be indefinitely iterated and no positiveness condition must be required: as a byproduct, the result follows that at least one resonant quasi-periodic solution always exists with no assumption on the perturbation. Such a solution can be interpreted as a (parabolic) lower-dimensional torus.

math.DS↗

arXiv2011

Resonances within Chaos

A chaotic system under periodic forcing can develop a periodically visited strange attractor. We discuss simple models in which the phenomenon, quite easy to see in numerical simulations, can be completely studied analytically.

nlin.CD↗

arXiv2010

Response solutions for arbitrary quasi-periodic perturbations with Bryuno frequency vector

We study the problem of existence of response solutions for a real-analytic one-dimensional system, consisting of a rotator subjected to a small quasi-periodic forcing. We prove that at least one response solution always exists, without any assumption on the forcing besides smallness and analyticity. This strengthens the results available in the literature, where generic non-degeneracy conditions are assumed. The proof is based on a diagrammatic formalism and relies on renormalisation group techniques, which exploit the formal analogy with problems of quantum field theory; a crucial role is played by remarkable identities between classes of diagrams.

math.DS↗

arXiv2009

KAM theory in configuration space and cancellations in the Lindstedt series

The KAM theorem for analytic quasi-integrable anisochronous Hamiltonian systems yields that the perturbation expansion (Lindstedt series) for quasi-periodic solutions with Diophantine frequency vector converges. If one studies the Lindstedt series, one finds that convergence is ultimately related to the presence of cancellations between contributions of the same perturbation order. In turn, this is due to symmetries in the problem. Such symmetries are easily visualised in action-angle coordinates, where KAM theorem is usually formulated, by exploiting the analogy between Lindstedt series and perturbation expansions in quantum field theory and, in particular, the possibility of expressing the solutions in terms of tree graphs, which are the analogue of Feynman diagrams. If the unperturbed system is isochronous, Moser's modifying terms theorem ensures that an analytic quasi-periodic solution with the same Diophantine frequency vector as the unperturbed Hamiltonian exists for the system obtained by adding a suitable constant (counterterm) to the vector field. Also in this case, one can follow the alternative approach of studying the perturbation expansion for both the solution and the counterterm, and again convergence of the two series is obtained as a consequence of deep cancellations between contributions of the same order. We revisit Moser's theorem, by studying the perturbation expansion one obtains by working in Cartesian coordinates. We investigate the symmetries giving rise to the cancellations which makes possible the convergence of the series. We find that the cancellation mechanism works in a completely different way in Cartesian coordinates. The interpretation of the underlying symmetries in terms of tree graphs is much more subtle than in the case of action-angle coordinates.

math.DS↗

arXiv2009

Construction of quasi-periodic response solutions in forced strongly dissipative systems

We consider a class of ordinary differential equations describing one-dimensional quasiperiodically forced systems in the presence of large damping. We give a fully constructive proof of the existence of response solutions, that is quasi-periodic solutions which have the same frequency vector as the forcing. This requires dealing with a degenerate implicit function equation: we prove that the latter has a unique solution, which can be explicitly determined. As a by-product we obtain an explicit estimate of the minimal size of the damping coefficient.

math.DS↗

arXiv2009

Arnold tongues for a resonant injection-locked frequency divider: analytical and numerical results

In this paper we consider a resonant injection-locked frequency divider which is of interest in electronics, and we investigate the frequency locking phenomenon when varying the amplitude and frequency of the injected signal. We study both analytically and numerically the structure of the Arnold tongues in the frequency-amplitude plane. In particular, we provide exact analytical formulae for the widths of the tongues, which correspond to the plateaux of the devil's staircase picture. The results account for numerical and experimental findings presented in the literature for special driving terms and, additionally, extend the analysis to a more general setting.

math.DS↗

arXiv2009

Quasi-periodic motions in dynamical systems. Review of a renormalisation group approach

Power series expansions naturally arise whenever solutions of ordinary differential equations are studied in the regime of perturbation theory. In the case of quasi-periodic solutions the issue of convergence of the series is plagued of the so-called small divisor problem. In this paper we review a method recently introduced to deal with such a problem, based on renormalisation group ideas and multiscale techniques. Applications to both quasi-integrable Hamiltonian systems (KAM theory) and non-Hamiltonian dissipative systems are discussed. The method is also suited to situations in which the perturbation series diverges and a resummation procedure can be envisaged, leading to a solution which is not analytic in the perturbation parameter: we consider explicitly examples of solutions which are only infinitely differentiable in the perturbation parameter, or even defined on a Cantor set.

math.DS↗

arXiv2009

Quasi-periodic motions in strongly dissipative forced systems

We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.

math.DS↗

arXiv2008

Frequency locking in the injection-locked frequency divider equation

We consider a model for the injection-locked frequency divider, and study analytically the locking onto rational multiples of the driving frequency. We provide explicit formulae for the width of the plateaux appearing in the devil's staircase structure of the lockings, and in particular show that the largest plateaux correspond to even integer values for the ratio of the frequency of the driving signal to the frequency of the output signal. Our results prove the experimental and numerical results available in the literature.

math.DS↗

arXiv2008

Melnikov theory to all orders and Puiseux series for subharmonic solutions

We study the problem of subharmonic bifurcations for analytic systems in the plane with perturbations depending periodically on time, in the case in which we only assume that the subharmonic Melnikov function has at least one zero. If the order of zero is odd, then there is always at least one subharmonic solution, whereas if the order is even in general other conditions have to be assumed to guarantee the existence of subharmonic solutions. Even when such solutions exist, in general they are not analytic in the perturbation parameter. We show that they are analytic in a fractional power of the perturbation parameter. To obtain a fully constructive algorithm which allows us not only to prove existence but also to obtain bounds on the radius of analyticity and to approximate the solutions within any fixed accuracy, we need further assumptions. The method we use to construct the solution -- when this is possible -- is based on a combination of the Newton-Puiseux algorithm and the tree formalism. This leads to a graphical representation of the solution in terms of diagrams. Finally, if the subharmonic Melnikov function is identically zero, we show that it is possible to introduce higher order generalisations, for which the same kind of analysis can be carried out.

math.DS↗

arXiv2008

Periodic solutions for a class of nonlinear partial differential equations in higher dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schroedinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases where the bifurcation equation is infinite-dimensional, such as the nonlinear Schroedinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.

math.AP↗