Resonant Normal Form Research Articles

Normal forms and integrability of ODE systems

We consider a special case of the Euler-Poisson system describing the motion of a rigid body with a fixed point. This is an autonomous sixth-order ODE system with one parameter. Among the stationary points of the system, we select two one-parameter families with resonance (0, 0, ?, ??, 2?, ?2?) of eigenvalues of the matrix of the linear part. At these stationary points, we compute the resonant normal form of the system using a program based on the MATHEMATICA package. Our results show that, if there exists an additional first integral of the system, then its normal form is degenerate. Therefore, we assume that the integrability of the ODE system can be established based on its normal form.

On Metastability in FPU

We present an analytical study of the Fermi–Pasta–Ulam (FPU) α–model with periodic boundary conditions. We analyze the dynamics corresponding to initial data with one low frequency Fourier mode excited. We show that, correspondingly, a pair of KdV equations constitute the resonant normal form of the system. We also use such a normal form in order to prove the existence of a metastability phenomenon. More precisely, we show that the time average of the modal energy spectrum rapidly attains a well defined distribution corresponding to a packet of low frequencies modes. Subsequently, the distribution remains unchanged up to the time scales of validity of our approximation. The phenomenon is controlled by the specific energy.

The defocusing mechanism of isochronous resonances

We present a numerical study concerning the defocusing mechanism of isochronous resonance island chains in the presence of two permanent robust tori. The process is initialized and concluded through bifurcations of fixed points located on the robust tori. Our approach is based on a Hamiltonian system derived from the resonant normal form. Choosing a convenient parameter in this system, we are able to depict a comprehensive analysis of the dynamics of the problem.

Application of the resonant normal form to high-order nonlinear ODEs using Mathematica

This paper describes an example of a local analysis of a dynamical system in normal form method by using the Mathematica system. We approximated families of local conditionally periodic solutions of the double pendulum system.

PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA

AbstractThis paper concerns perturbations of smooth vector fields on $\mathbb{T}^n$ (constant if $n\geq3$) with zeroth-order $C^\infty$ and Gevrey $G^\sigma$, $\sigma\geq1$, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the $C^\infty$ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in $C^\infty$ and Gevrey $G^\sigma$. Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the ‘size’ of different exceptional sets, including some inhomogeneous examples.AMS 2000 Mathematics subject classification: Primary 37C15; 11J13. Secondary 58J40; 11J20; 35H05

RESONANT NORMAL FORMS AS CONSTRAINED LINEAR SYSTEMS

We show that a nonlinear dynamical system in Poincaré–Dulac normal form (in ℝn) can be seen as a constrained linear system; the constraints are given by the resonance conditions satisfied by the spectrum of (the linear part of) the system and identify a naturally invariant manifold for the flow of the "parent" linear system. The parent system is finite dimensional if the spectrum satisfies only a finite number of resonance conditions, as implied e.g. by the Poincaré condition. In this case our result can be used to integrate resonant normal forms, and sheds light on the geometry behind the classical integration method of Horn, Lyapounov and Dulac.

Analysis of a model for resonant extraction of intense beams by normal forms and frequency map

A simple 1D model is proposed to explore the resonant extraction of intense beams from a synchrotron as performed in the SIS synchrotron in GSI (Darmstadt). The model Hamiltonian consists of a constant focusing, a thin sextupole, and a smooth space charge field. Hyperbolic normal forms are used to estimate the extraction times and the emittance of the extracted beam; the quality of the reconstruction is tested in absence of space charge. The effect of space charge on the dynamical behavior of the beam near the $1/3$ betatron resonance is numerically investigated using the frequency map analysis and qualitatively explained with perturbation theory. A polynomial approximation to the one turn map is obtained by replacing the exact space charge force with a sequence of polynomial kicks, and the resonant normal forms reproduce quite accurately the nonlinear tunes and the fixed points position. At low order an analytical estimate of the area of the stable region is proposed to recover the self-consistency of the model.

The nonlinear Schrödinger equation as a resonant normal form

Averaging theory is used to study the dynamics of dispersive equationstaking the nonlinear Klein Gordon equation on the line as a model problem:For approximatively monochromatic initial data of amplitude $\epsilon$, we showthat the corresponding solution consists of two non interacting wave packets,each one being described by a nonlinear Schrödinger equation. Such solutionsare also proved to be stable over times of order $1/ \epsilon^2$. We think that this approachputs into a new light the problem of obtaining modulations equationsfor general dispersive equations. The proof of our results requires a new useof normal forms as a tool for constructing approximate solutions.

Normal form and exponential stability for some nonlinear string equations

We study small amplitude solutions of the nonlinear wave equation¶¶\(u_{tt}-u_{xx}=\psi(u) \qquad u(0,t)=0=u(\pi,t)\)(0.1)¶¶ with an analytic nonlinearity of the type \(\psi(u)=\pm u^{2k-1} + {\cal O}(u^{2k})$, $k\geq2\). For this system we introduce a new method to compute the resonant normal form obtaining a simple formula for it. Then we specialize to the case k = 2 and find all periodic solutions of the averaged system and also of what we call the “simplified system”, namely a Hamiltonian system that we will prove to approximate (0.1) up to an error exponentially small with the energy of the initial datum. Furthermore, for one of such orbits we prove a strong (Nekhoroshev type) stability property with respect to the complete dynamics.

Evaluation of nonlinear resonances in 4D symplectic mappings

We discuss the evaluation of the stability, position and width of resonances in four-dimensional symplectic mappings. The approach presented in this paper is based on the computation of the resonant perturbative series carried out by the program ARES. This piece of code allows to perform the evaluation of the perturbative series and of the interpolating Hamiltonian. Given a symplectic map in the neighbourhood of an elliptic fixed point, the new code NERO performs the analysis of the orbits of the interpolating Hamiltonian both in the nonresonant and the resonant case. For each resonant normal form, the interpolating Hamiltonian is computed and the position and the stability of the resonant orbits and the width of the islands are evaluated; this analysis is carried out through the direct inspection of the coefficients of the interpolating Hamiltonian. All the computations are carried out at an arbitrary order; the first significant perturbative order is taken as a first guess, and a Newton method is used to evaluate the higher orders effect.

Numeric Deduction in Symbolic Computation. Application to Normalizing Transformations

Algorithms of numeric (in exact arithmetic) deduction of analytical expressions, proposed and described by Shevchenko and Vasiliev (1993), are developed and implemented in a computer algebra code. This code is built as a superstructure for the computer algebra package by Shevchenko and Sokolsky (1993a) for normalization of Hamiltonian systems of ordinary differential equations, in order that high complexity problems of normalization could be solved. As an example, a resonant normal form of a Hamiltonian describing the hyperboloidal precession of a dynamically symmetric satellite is derived by means of the numeric deduction technique. The technique provides a considerable economy, about 30 times in this particular application, in computer memory consumption. It is naturally parallelizable. Thus the economy of memory consumption is convertible into a gain in computation speed.

Stability and Geometry of Third-Order Resonances in four-Dimensional Symplectic Mappings

We analyze four-dimensional symplectic mappings in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a third-order resonance. Using the perturbative tools of resonant normal forms, the geometry of the orbits and the existence of elliptic or hyperbolic one-dimensional tori (fixed lines) is worked out. This allows one to give an analytical estimate of the stability domain when the resonance is unstable. A comparison with numerical results for the four-dimensional Henon mapping is given.

Local analysis of formal stability and existence of fixed points in 4d symplectic mappings

Symplectic mappings in a four-dimensional phase space are analysed; in the neighbourhood of an elliptic fixed point whose eigenvalues are close to satisfy a double resonance condition, the perturbative tools of resonant normal forms are outlined. The analysis of the interpolating Hamiltonian of the normal form allows one to determine the existence and the stability of the fixed points which are due to the double resonance: only some combinations of stabilities are allowed. Due to the asymptotic character of the series, the results are formal, but supported by numerical computations.

Time scale to ergodicity in the Fermi-Pasta-Ulam system.

We study the approach to near-equipartition in the N-dimensional Fermi-Pasta-Ulam Hamiltonian with quartic (hard spring) nonlinearity. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy which is independent of N, one can reach equipartition if one waits for a time proportional to N(2). Below this critical energy the time to equipartition is exponentially long. We develop a theory to determine the time evolution and the excitation of the nonlinear modes based on a resonant normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for equipartition, the time scale to equipartition, and the form of the nonlinear modes below equipartition, in qualitative agreement with the numerical results. (c) 1995 American Institute of Physics.

Analysis of resonant structures of four-dimensional symplectic mappings, using normal forms.

The geometry of the resonant orbits of symplectic four-dimensional mappings, in the neighborhood of an elliptic fixed point, is analyzed in the framework of a perturbative approach based on resonant normal forms. The analysis of the truncated interpolating Hamiltonian allows one to determine the classification, the location, and the stability of the integrable resonant structures in phase space.

Classical and quantal aspects of resonant normal forms

The Birkhoff-Gustavson normal form is employed to study separately chaos and resonances in a system with two degrees of freedom. In the integrable regime, tunnelling effects are appreciable when the nearest level spacings show oscillations. Tunnelling among states in the libration and rotation tori regions is also observed. The regularity of avoided crossings due to tunnelling indicates a collective effect and is associated with an isolated resonance. The spectral fluctuations also show a strong level correlation. The Husimi distribution, on the other hand, is insensitive to avoided crossings. An integrable approximation to the overlap of resonances is obtained and a theoretical description is given for an isolated cubic resonance plus a complex orbit. In the non-integrable regime chaos is stronger after overlapping and preferentially at low energies.

Resonant normal forms, interpolating Hamiltonians and stability analysis of area preserving maps

The geometrical and dynamical properties of area preserving maps in the neighborhood of an elliptic fixed point are analyzed in the framework of resonant normal forms. The interpolating flow is not obtained from a map tangent to the identity, but from the normal form of the given map and a time independent interpolating Hamiltonian H is introduced. On this Hamiltonian the local stability properties of the fixed point and the geometric structure of the orbits are transparent. Numerical agreement between the level lines of H and the orbits of the map suggests that the perturbative expansion of H is asymptotic. This is confirmed by a rigorous error analysis, based on majorant series: the error for the normal form expansion grows as n! while the truncation error for H also has a factorial growth and in a disc of radius r can be made exponentially small with 1/ r. The boundary of the global stability domain is considered; for the quadratic map the identification with the inner envelope of the homoclinic tangle of the hyperbolic fixed point is strongly suggested by numerical evidence.

New periodic orbits and basins for families of resonant normal forms

A special class of periodic orbits of parameter dependent families of truncated resonant normal forms is constructed. Their existence is shown for arbitrarily large periods. Explicit analytical criteria are derived for constructing the complete basin of attraction of an invariant circle for normal forms for which the eigenvalue of the linearized mapping is not a multiple root of unity. Persistence of these basins is shown for small perturbations of the parameters. Hence, the existence of bounded orbits for all times established, generically.