Birkhoff Normal Form Research Articles

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part I. Finite-dimensional discretization

We consider discretized Hamiltonian PDEs associated with a Hamiltonian function that can be split into a linear unbounded operator and a regular nonlinear part. We consider splitting methods associated with this decomposition. Using a finite-dimensional Birkhoff normal form result, we show the almost preservation of the actions of the numerical solution associated with the splitting method over arbitrary long time and for asymptotically large level of space approximation, provided the Sobolev norm of the initial data is small enough. This result holds under generic non-resonance conditions on the frequencies of the linear operator and on the step size. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.

Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part II. Abstract splitting

We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size, and under a condition of zero momentum on the nonlinearity. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for rounded numerical schemes avoiding at each step possible high frequency energy drift. We apply these results to nonlinear Schrödinger equations as well as the nonlinear wave equation.

Quantum Birkhoff normal forms

We consider quantum analogues of several theorems on Birkhoff normal forms and prove a quantum analogue of the theorem on reducing a Hamiltonian to the normal form and the analogue of the theorem on reducing a Hamiltonian to the real normal form. We obtain the normal form explicitly in the nonresonant case. We consider the uniqueness problems of the normal form and of the normalizing transformation in the nonresonant and resonant cases.

Stability of some solutions for elliptic equations on a cylindrical domain

We study analytically a class of solutions for the elliptic equation where α >0 and ε is a small parameter. This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every α >0, it contains solutions that are defined for large values of time and they are very close (of order O ( ε )) to a linear torus for long times (of order O ( ε −1 )). The proof uses the fact that the equation leaves invariant a smooth centre manifold and, for the restriction of the system to the centre manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

Stability of some waves in the dissipative Boussinesq system

In this paper we study analytically a class of waves in the variant of the classical dissipative Boussinesq system given by ∂ t u = − ∂ x v − α ∂ x x x v + β ∂ x x t u − ϵ ∂ x ( u v ) , ∂ t v = − ∂ x u + c ∂ x x x u + β ∂ x x t v − ϵ v ∂ x v , where β , c > 0 , ϵ is a small parameter and α ∈ ( 0 , 1 ) . This equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every β , c and almost every α ≤ 1 , it contains solutions that are defined for large values of time and they are very close (of order O ( ϵ ) ) to a linear torus for long times (of order O ( ϵ − 1 ) ). The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem, the Birkhoff normal form and Neckhoroshev-type estimates.

Normal Forms for Semilinear Quantum Harmonic Oscillators

We consider the semilinear harmonic oscillator $$i\psi_t=(-\Delta +{|x|}^{2} +M)\psi +\partial_2 g(\psi,\bar \psi), \quad x\in \mathbb{R}^{d},\, t\in \mathbb{R},$$where M is a Hermite multiplier and g a smooth function globally of order 3 at least. We prove that such a Hamiltonian equation admits, in a neighborhood of the origin, a Birkhoff normal form at any order and that, under generic conditions on M related to the non resonance of the linear part, this normal form is integrable when d = 1 and gives rise to simple (in particular bounded) dynamics when d ≥ 2. As a consequence we prove the almost global existence for solutions of the above equation with small Cauchy data. Furthermore we control the high Sobolev norms of these solutions.

The stability of the equilibrium of a nonlinear planar system and application to the relativistic oscillator

In this paper we present a sufficient condition for the stability of the equilibrium of a nonlinear planar system. The proof is based on the computation of the corresponding Birkhoff normal forms. The result does not involve small parameters. Applications to the relativistic oscillator are also given.

Nonlinear dynamical analysis for displaced orbits above a planet

Nonlinear dynamical analysis and the control problem for a displaced orbit above a planet are discussed. It is indicated that there are two equilibria for the system, one hyperbolic (saddle) and one elliptic (center), except for the degenerate hzmax, a saddle-node bifurcation point. Motions near the equilibria for the nonresonance case are investigated by means of the Birkhoff normal form and dynamical system techniques. The Kolmogorov–Arnold–Moser (KAM) torus filled with quasiperiodic trajectories is measured in the τ 1 and τ 2 directions, and a rough algorithm for calculating τ 1 and τ 2 is proposed. A general iterative algorithm to generate periodic Lyapunov orbits is also presented. Transitions in the neck region are demonstrated, respectively, in the nonresonance, resonance, and degradation cases. One of the important contributions of the paper is to derive necessary and sufficiency conditions for stability of the motion near the equilibria. Another contribution is to demonstrate numerically that the critical KAM torus of nontransition is filled with the (1,1)-homoclinic orbits of the Lyapunov orbit.

An Inverse Problem for Trapping Point Resonances

We consider semi-classical Schrodinger operator P(h) = − h2Δ + V(x) in \({{\mathbb R}^n}\) such that the analytic potential V has a non-degenerate critical point x0 = 0 with critical value E0 and we can define resonances in some fixed neighborhood of E0 when h > 0 is small enough. If the eigenvalues of the Hessian are \({\mathbb Z}\) -independent the resonances in hδ-neighborhood of E0 (δ > 0) can be calculated explicitly as the eigenvalues of the semi-classical Birkhoff normal form. Assuming that potential is symmetric with respect to reflections about the coordinate axes we show that the classical Birkhoff normal form determines the Taylor series of the potential at x0. As a consequence, the resonances in a hδ-neighborhood of E0 determine the first N terms in the Taylor series of V at x0. The proof uses the recent inverse spectral results of V. Guillemin and A. Uribe.

Nekhoroshev stability atL4orL5in the elliptic-restricted three-body problem - application to Trojan asteroids

The problem of analytical determination of the stability domain around the points L4 or L5 of the Lagrangian equilateral configuration of the three-body problem has served in the literature as a basic celestial mechanical model probing the predictive power of the so-called Nekhoroshev theory of exponential stability in non-linear Hamiltonian dynamical systems. All analytical investigations in this framework have so far been based on the circular restricted three-body problem (CRTBP). In this work, we extend the analytical estimates of Nekhoroshev stability in the case of the planar elliptic-restricted three-body problem (ERTBP). To this end, we introduce an explicit symplectic mapping model for the planar ERTBP, obtained via Hadjidemetriou's method, which generalizes the family of mappings discussed in earlier papers. The mapping is based on an expansion of the disturbing function up to a sufficiently high order in the eccentricities and the variations of the semimajor axis, and it is given as a series around a period-one fixed point of the system. Within the domain of the mapping's convergence, we then compute a Birkhoff normal form for 4D symplectic mappings as well as the associated approximate integrals of motion which can be expressed in terms of proper elements. The variations of the integrals predicted by the remainder of the normal form series provide a lower bound for the domain of Nekhoroshev stability for a time at least equal to the age of the Solar System. In the case of Jupiter's Trojans, the domain of the mapping's convergence lies entirely within the region of librational motion, in which the longitude of the perihelion of the asteroid librates around a fixed point value of ϖ. For most asteroids outside this domain macroscopic chaotic diffusion cannot be ruled out. The present analysis provides a physically relevant estimate of the domain of Nekhoroshev stability for proper librations (Dp < 100), and marginal for proper eccentricities (ep < 0.01). The formalism is developed in a general way allowing for applications in both, our Solar System and extrasolar system dynamics.

Analysis of a renormalization group method and normal form theory for perturbed ordinary differential equations

For singular perturbation problems, the renormalization group (RG) method of Chen, Goldenfeld, and Oono [Phys. Rev. E. 49 (1994) 4502–4511] has been shown to be an effective general approach for deriving reduced or amplitude equations that govern the long time dynamics of the system. It has been applied to a variety of problems traditionally analyzed using disparate methods, including the method of multiple scales, boundary layer theory, the WKBJ method, the Poincaré–Lindstedt method, the method of averaging, and others. In this article, we show how the RG method may be used to generate normal forms for large classes of ordinary differential equations. First, we apply the RG method to systems with autonomous perturbations, and we show that the reduced or amplitude equations generated by the RG method are equivalent to the classical Poincaré–Birkhoff normal forms for these systems up to and including terms of O ( ϵ 2 ) , where ϵ is the perturbation parameter. This analysis establishes our approach and generalizes to higher order. Second, we apply the RG method to systems with nonautonomous perturbations, and we show that the reduced or amplitude equations so generated constitute time-asymptotic normal forms, which are based on KBM averages. Moreover, for both classes of problems, we show that the main coordinate changes are equivalent, up to translations between the spaces in which they are defined. In this manner, our results show that the RG method offers a new approach for deriving normal forms for nonautonomous systems, and it offers advantages since one can typically more readily identify resonant terms from naive perturbation expansions than from the nonautonomous vector fields themselves. Finally, we establish how well the solution to the RG equations approximates the solution of the original equations on time scales of O ( 1 / ϵ ) .

Rotation numbers and Lyapunov stability of elliptic periodic solutions

Using the relation between the Hill's equations and the Ermakov-Pinney equations established by Zhang [27], we will give some interesting lower bounds of rotation numbers of Hill's equations. Based on the Birkhoff normal forms and the Moser twist theorem, we will prove that two classes of nonlinear, scalar, time-periodic, Newtonian equations will have twist periodic solutions, one class being regular and another class being singular.

Symbolic computation of the Birkhoff normal form in the problem of stability of the triangular libration points

The problem of stability of the triangular libration points in the planar circular restricted three-body problem is considered. A software package, intended for normalization of autonomous Hamiltonian systems by means of computer algebra, is designed so that normalization problems of high analytical complexity could be solved. It is used to obtain the Birkhoff normal form of the Hamiltonian in the given problem. The normalization is carried out up to the 6th order of expansion of the Hamiltonian in the coordinates and momenta. Analytical expressions for the coefficients of the normal form of the 6th order are derived. Though intermediary expressions occupy gigabytes of the computer memory, the obtained coefficients of the normal form are compact enough for presentation in typographic format. The analogue of the Deprit formula for the stability criterion is derived in the 6th order of normalization. The obtained floating-point numerical values for the normal form coefficients and the stability criterion confirm the results by Markeev (1969) and Coppola and Rand (1989), while the obtained analytical and exact numeric expressions confirm the results by Meyer and Schmidt (1986) and Schmidt (1989). The given computational problem is solved without constructing a specialized algebraic processor, i.e., the designed computer algebra package has a broad field of applicability.

Results on Normal Forms for FPU Chains

In this paper we prove, among other results, that near the equilibirum position, any periodic FPU chain with an odd number N of particles admits a Birkhoff normal form up to order 4, whereas any periodic FPU chain with N even admits a resonant normal form up to order 4. This resonant normal form of order 4 turns out to be completely integrable. Further, for N odd, we obtain an explicit formula of the Hessian of its Hamiltonian at the fixed point.

Scattering Poles near the Real Axis for Two Strictly Convex Obstacles

To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator $M$ along the trapped ray between the two obstacles. Using this method Ikawa and G{\'e}rard established the existence of parallel rows of poles in a strip $Im z\leq c$ as $Re z$ tends to infinity. Assuming that the boundaries are analytic and the eigenvalues of Poincar{\'e} map are non-resonant we use the Birkhoff normal form for $M$ to improve this result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.

Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds

AbstractThis paper is devoted to the proof of almost global existence results for Klein‐Gordon equations on Zoll manifolds (e.g., spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the Laplacian perturbed by a potential on Zoll manifolds. © 2007 Wiley Periodicals, Inc.

Existence and stability of periodic solutions for second-order semilinear differential equations with a singular nonlinearity

It is proved that a periodically forced second-order equation with a singular nonlinearity in the origin with linear growth in infinity possesses a -periodic stable solution for high values of the mean value of the forcing term. The method of proof combines a rescaling argument with the analysis of the first twist coefficient of the Birkhoff normal form for the Poincaré map.

“Bottom of the well” semi-classical trace invariants

Let $\hat H$ be an $\h$-admissible pseudodifferential operator whose principal symbol, $H$, has a unique non-degenerate global minimum. We give a simple proof that the semi-classical asymptotics of the eigenvalues of $\hat H$ corresponding to the ``bottom of the well determine the Birkhoff normal form of $H$ at the minimum. We treat both the resonant and the non-resonant cases.

Some inverse spectral results for semi-classical Schrödinger operators

We show that the Birkhoff normal form of a classical Hamiltonian $H(x,\xi) = \norm{\xi}^2+V(x)$ at a non-degenerate minimum $x_0$ of the potential determines the Taylor series of the potential at $x_0$, provided the eigenvalues of the Hessian are linearly independent over $\bbQ$ and $V$ satisfies a symmetry condition near $x_0$. As a consequence, if $x_0$ is the unique global minimum of $V$, the low-lying eigenvalues of the semi-classical Schr\odinger operator, $-\h^2\Delta + V(x)$, determine the Taylor series of the potential at $x_0$.

Birkhoff normal form for partial differential equations with tame modulus

We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrodinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the d-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions