Small Divisor Problem Research Articles

Hopf-like bifurcation in a wave equation at a removable singularity

It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and accumulates to zero. The upshot is that the singularity of the linearized operator at criticality which stems from the well known small divisor problem for the wave operator, is entirely removed without the need to exclude parameters via Diophantine conditions, nor the use of accelerated convergence schemes. Only the contraction mapping principle is used.

Small divisors in discrete local holomorphic dynamics

We give an introduction to the study of local dynamics of iterated holomorphic mappings near a fixed point via local conjugations in one and several complex variables. Starting with the systematic construction of formal conjugations to Poincaré–Dulac normal form in general and formal linearisations in particular, we discuss conditions for convergence of the normalising series in terms of the linear part. The convergence is closely related to the size of denominators that show up in the normalising series and depend only on the linear part of the mapping. Hence, we speak of a small-divisor problem. The central result on the linearisation problem is the Brjuno condition, that ensures holomorphic linearisability. The first part of these notes is a survey of the local dynamics in one variable from the viewpoint of local normalisations. In this case, the picture is fairly complete. In the second part we introduce the additional obstacles to both formal and holomorphic normalisations that emerge from interactions of multiple eigenvalues, such as resonances, that preclude the Brjuno condition in particular. For these cases, we proceed with several generalisations of the Brjuno condition, that allow us to find convergent conjugations to at least partial normalisations. The last part reviews a recent application of such a partial normalisation to illuminate the local dynamics near certain one-resonant fixed points.

Spatially quasi-periodic water waves of finite depth

We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and travelling wave settings. We adopt a quasi-periodic conformal mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions are represented by periodic functions on a higher-dimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasi-periodic bottom boundary and observe the formation of quasi-periodic patterns on the free surface. Two types of quasi-periodic travelling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic travelling waves. We derive weakly nonlinear approximations of the first type and investigate the associated small-divisor problem. We find that waves of the second type exhibit striking nonlinear behaviour, e.g. the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.

Time-periodic solutions of Hamiltonian PDEs using pseudoholomorphic curves

We extend the pseudoholomorphic curve methods from Floer theory to infinite-dimensional phase spaces and use our results to prove the existence of a forced time-periodic solution to a general Hamiltonian PDE with regularizing nonlinearity. In particular, when the nonlinearity is sufficiently regularizing, bounded and time-periodic, we prove an infinite-dimensional version of Gromov-Floer compactness by using ideas from the theory of Diophantine approximations to overcome the small divisor problem. Furthermore, in the case when the infinite-dimensional phase space is a product of a finite-dimensional closed symplectic manifold with linear symplectic Hilbert space, we prove a cup-length estimate for the number of periodic solutions.

Optimal linearization of vector fields on the torus in non-analytic Gevrey classes

We study linear and non-linear small divisors problems in analytic and non-analytic regularity. We observe that the Bruno arithmetic condition, which is usually attached to non-linear analytic problems, can also be characterized as the optimal condition to solve the linear problem in some fixed non-quasi-analytic class. Based on this observation, it is natural to conjecture that the optimal arithmetic condition for the linear problem is also optimal for non-linear small divisors problems in any reasonable non-quasi-analytic classes. Our main result proves this conjecture in a representative non-linear problem, which is the linearization of vector fields on the torus, in the most natural non-quasi-analytic class, which is the Gevrey class. The proof follows Moser’s argument of approximation by analytic functions, and uses works of Popov, Rüssmann and Pöschel in an essential way.

Diophantine approximation as Cosmic Censor for Kerr\u2013AdS black holes

The purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant Lambda <0 and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations psi of Kerr–AdS solving Box _gpsi -frac{2}{3}Lambda psi =0 with reflecting boundary conditions imposed at infinity. Understanding the behavior of psi at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass {mathfrak {m}} = M sqrt{-Lambda } and angular momentum {mathfrak {a}} = a sqrt{-Lambda } satisfy a certain non-Diophantine condition, then perturbations psi arising from generic smooth initial data blow up |psi |rightarrow +infty at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for Lambda ge 0—in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters {mathfrak {m}}, {mathfrak {a}} for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking–Reall bound. On the other hand, we conjecture that for a set of parameters {mathfrak {m}}, {mathfrak {a}} which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon |psi |le C. This suggests that the validity of the C^0-formulation of Strong Cosmic Censorship for Lambda <0 may change in a spectacular way according to the notion of genericity imposed.

Hamiltonian Floer Theory for Nonlinear Schrödinger Equations and the Small Divisor Problem

Abstract We prove the existence of infinitely many time-periodic solutions of nonlinear Schrödinger equations using pseudo-holomorphic curve methods from Hamiltonian Floer theory. For the generalization of the Gromov–Floer compactness theorem to infinite dimensions, we show how to solve the arising small divisor problem by combining elliptic methods with results from the theory of diophantine approximations.

Response solutions for strongly dissipative quasi-periodically forced systems with arbitrary nonlinearities and frequencies

We consider quasi-periodically systems in the presence of dissipation and study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term. When the dissipation is large enough and a suitable function involving the forcing has a simple zero, response solutions are known to exist without assuming any non-resonance condition on the frequency vector. We analyse the case of non-simple zeroes and, in order to deal with the small divisors, we confine ourselves to two-dimensional frequency vectors, so as to use the properties of continued fractions. We show that, if the order of the zero is odd (if it is even, in general no response solution exists), a response solution still exists provided the inverse of the parameter measuring the dissipation belongs to a set given by the union of infinite intervals depending on the convergents of the ratio of the two components of the frequency vector. The intervals may be disjoint and as a consequence we obtain the existence of response solutions in a set with “holes”. If we want the set to be connected we have to require some non-resonance condition on the frequency: in fact, we need a condition weaker than the Bryuno condition usually considered in small divisors problems.

Existence of Bifurcating Quasipatterns in Steady Bénard–Rayleigh Convection

Extending the results obtained in the sixties for bifurcating periodic patterns, the existence of bifurcating quasipatterns in the steady Benard–Rayleigh convection problem is proved. These are two-dimensional patterns, quasiperiodic in any horizontal direction, invariant under horizontal rotations of angle $${\pi/q}$$ . There is a small divisor problem for $${q \geqq 4}$$ . Using the results of Berti–Bolle–Procesi in 2010, we adapt it to a Navier–Stokes system ruling the Benard–Rayleigh convection problem. Our solution is approximated by the truncated power series which was formally obtained by Iooss in 2009, but which is divergent in general (Gevrey series). First, we formulate the problem in introducing a suitable parameter, able to move the spectrum of the linearized operator, as a whole, as for the Swift–Hohenberg PDE model. For using the Nash–Moser process, we are faced with the problem of inverting a linear operator which is the differential at a non zero point. There are two new difficulties: (i) First, the extra dimension leading to a more complicated spectrum of the linear operator. This first difficulty leads to use specific projections for reducing the spectrum of the studied operator, which we want to invert, to a finite set very close to 0. (ii) The second difficulty is the fact that the linearization L(N) at a non-zero point leads to a non-selfadjoint operator, contrary to what occurs in previous works. This is more serious, and leads to use the spectrum of L(N)L(N)* which depends mainly quadratically on the main parameter. A careful study of the “bad set”of parameters, with an assumption on the convexity of the eigenvalues of this operator, allows us to obtain a good estimate, as it is necessary for using the results of Berti et al. for solving ”the range equation”. We again use separation properties of the Fourier spectrum (see the Bourgain and Craig results) for obtaining an estimate in high Sobolev norms. It then remains to solve the one-dimensional “bifurcation equation. For any $${q \geqq 4}$$ , and provided that a weak transversality conjecture is realized, we prove the existence of a bifurcating convective quasipattern of order 2q, above the critical Rayleigh number.

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.

Finite amplitude steady-state wave groups with multiple near resonances in deep water

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

Multifractal analysis of the Brjuno function

The Brjuno function B is a 1-periodic, nowhere locally bounded function, introduced by Yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1. We show that \(T^p_\alpha \) regularity, introduced by Calderon and Zygmund, is the only one which is relevant in order to unfold the pointwise regularity properties of B; we determine its \(T^p_\alpha \) regularity at every point and show that it is directly related to the irrationality exponent \(\tau (x)\): its p-exponent at x is exactly \(1/\tau (x)\). This new example of multifractal function puts into light a new link between dynamical systems and fractal geometry. Finally we also determine the Holder exponent of a primitive of B.

Generalised continuation by means of right limits

Several theories have been proposed to generalise the concept of analytic continuation to holomorphic functions of the disc for which the circle is a natural boundary. Elaborating on Breuer-Simon’s work on right limits of power series, Baladi-Marmi-Sauzin recently introduced the notion of renascent right limit and rrl-continuation. We discuss a few examples and consider particularly the classical example of Poincare simple pole series in this light. These functions are represented in the disc as series of infinitely many simple poles located on the circle; they appear, for instance, in small divisor problems in dynamics. We prove that any such function admits a unique rrl-continuation, which coincides with the function obtained outside the disc by summing the simple pole expansion. We also discuss the relation with monogenic regularity in the sense of Borel.

Asymptotic analysis of the parametric instability of nonlinear hyperbolic equations

This paper is concerned with parametric resonance under nonlinear periodic perturbations of differential equations which are abstract analogues of hyperbolic systems. A modification of the Krylov-Bogolyubov averaging method capable of circumventing the well-known small divisor problem is applied to reduce the description of solutions of perturbed equations at resonance to the study of autonomous dynamical systems in finite-dimensional spaces. Bibliography: 28 titles.

Proof of Quasipatterns for the Swift–Hohenberg Equation

This paper establishes the existence of quasipatterns solutions of the Swift–Hohenberg PDE. In a former approach we avoided the use of Nash–Moser scheme, but our proof contains a gap. The present proof of existence is based on the works by Berti et al related to the Nash–Moser scheme. For solving the small divisor problem, we need to introduce a new free parameter related to the freedom in the choice of parameterization of the bifurcating solution. Thanks to a transversality condition, the result gives only a bifurcating set, located in a small hornlike region centered on a curve, with the origin at the bifurcation point.

Standing Waves in Near-Parallel Vortex Filaments

A model derived in (Klein et al., J Fluid Mech 288:201–248, 1995) for n near-parallel vortex filaments in a three dimensional fluid region takes into consideration the pairwise interaction between the filaments along with an approximation for motion by self-induction. The same system of equations appears in descriptions of the fine structure of vortex filaments in the Gross–Pitaevski model of Bose–Einstein condensates. In this paper we construct families of standing waves for this model, in the form of n co-rotating near-parallel vortex filaments that are situated in a central configuration. This result applies to any pair of vortex filaments with the same circulation, corresponding to the case n = 2. The model equations can be formulated as a system of Hamiltonian PDEs, and the construction of standing waves is a small divisor problem. The methods are a combination of the analysis of infinite dimensional Hamiltonian dynamical systems and linear theory related to Anderson localization. The main technique of the construction is the Nash–Moser method applied to a Lyapunov–Schmidt reduction, giving rise to a bifurcation equation over a Cantor set of parameters.

Localization in the Ground State of an Interacting Quasi-Periodic Fermionic Chain

We consider a many body fermionic system with an incommensurate external potential and a short range interaction in one dimension. We prove that, for certain densities and weak interactions, the zero temperature thermodynamical correlations are exponentially decaying for large distances, a property indicating persistence of localization in the interacting ground state. The analysis is based on Renormalization Group, and convergence of the renormalized expansion is achieved using fermionic cancellations and controlling the small divisor problem assuming a Diophantine condition for the frequency.

Small divisors problem in dynamical systems and analytic invariant curves for an iterative equation related to a rational difference equation

We study the existence of analytic invariant curves of the iterative equation f(f(x)) = ax/(f(x)x + b), which arises from second-order rational difference equations. By reducing the equation with the Schroder transformation to an auxiliary equation we discuss not only the constant α at resonance, that is, at a root of the unity, but also those α near resonance under the Bryuno condition.

Amplification of pulses in nonlinear geometric optics

In this companion paper to our study of amplification of wavetrains J.-F. Coulombel, O. Guès and M. Williams, Semilinear geometric optics with boundary amplification, Anal. PDE7(3) (2014) 551–625, we study weakly stable semilinear hyperbolic boundary value problems with pulse data. Here weak stability means that exponentially growing modes are absent, but the so-called uniform Lopatinskii condition fails at some boundary frequency in the hyperbolic region. As a consequence of this degeneracy there is again an amplification phenomenon: outgoing pulses of amplitude O(ε2) and wavelength ε give rise to reflected pulses of amplitude O(ε), so the overall solution has amplitude O(ε). Moreover, the reflecting pulses emanate from a radiating pulse that propagates in the boundary along a characteristic of the Lopatinskii determinant. In the case of N × N systems considered here, a single outgoing pulse produces on reflection a family of incoming pulses traveling at different group velocities. Unlike wavetrains, pulses do not interact to produce resonances that affect the leading order profiles. However, pulse interactions do affect lower-order profiles and so these interactions have to be estimated carefully in the error analysis. Whereas the error analysis in the wavetrain case dealt with small divisor problems by approximating periodic profiles by trigonometric polynomials (which amounts to using a high frequency cutoff), in the pulse case we approximate decaying profiles with nonzero moments by profiles with zero moments (a low frequency cutoff). Unlike the wavetrain case, we are now able to obtain a rate of convergence in the limit describing convergence of approximate to exact solutions.

Multiscale Time Averaging, Reloaded

We develop a rigorously controlled multi-time scale averaging technique; the averaging is done on a finite time interval, properly chosen, and then, via iterations and normal form transformations, the time intervals are scaled to arbitrary order. Here, we consider as an example the problem of a finite dimensional conservative dynamical system, which is quasi periodic and dominated by slow frequencies, leading to small divisor problems in perturbative schemes. Our estimates hold for arbitrary long time intervals similar to the Nekhoroshev-type results.