Showing papers on "Floquet theory published in 1998"

Theory and simulation of rotational shear stabilization of turbulence

Abstract: Numerical simulations of ion temperature gradient (ITG) mode transport with gyrofluid flux tube codes first lead to the rule that the turbulence is quenched when the critical E×B rotational shear rate γE−crit exceeds the maximum of ballooning mode growth rates γ0 without E×B shear [Waltz, Kerbel, and Milovich, Phys. Plasmas 1, 2229 (1994)]. The present work revisits the flux tube simulations reformulated in terms of Floquet ballooning modes which convect in the ballooning mode angle. This new formulation avoids linearly unstable “box modes” from discretizing in the ballooning angle and illustrates the true nonlinear nature of the stabilization in toroidal geometry. The linear eigenmodes can be linearly stable at small E×B shear rates, yet Floquet mode convective amplification allows turbulence to persist unless the critical shear rate is exceeded. The flux tube simulations and the γE−crit≈γ0 quench rule are valid only at vanishing relative gyroradius. Modifications and limits of validity on the quench rul...

Elliptic algebro-geometric solutions of the KdV and AKNS hierarchies - an analytic approach

Abstract: We provide an overview of elliptic algebro-geometric solutions of the KdV and AKNS hierarchies, with special emphasis on Floquet theoretic and spectral theoretic methods. Our treatment includes an effective characterization of all stationary elliptic KdV and AKNS solutions based on a theory developed by Hermite and Picard.

Periodic Orbits, Lyapunov Vectors, and Singular Vectors in the Lorenz System

Abstract: Some theoretical issues related to the problem of quantifying local predictability of atmospheric flow and the generation of perturbations for ensemble forecasts are investigated in the Lorenz system. A periodic orbit analysis and the study of the properties of the associated tangent linear equations are performed. In this study a set of vectors are found that satisfy Oseledec theorem and reduce to Floquet eigenvectors in the particular case of a periodic orbit. These vectors, called Lyapunov vectors, can be considered the generalization to aperiodic orbits of the normal modes of the instability problem and are not necessarily mutually orthogonal. The relation between singular vectors and Lyapunov vectors is clarified, and transient or asymptotic error growth properties are investigated. The mechanism responsible for super-Lyapunov growth is shown to be related to the nonorthogonality of Lyapunov vectors. The leading Lyapunov vectors, as defined here, as well as the asymptotic final singular vectors, are tangent to the attractor, while the leading initial singular vectors, in general, point away from it. Perturbations that are on the attractor and maximize growth should be considered in meteorological applications such as ensemble forecasting and adaptive observations. These perturbations can be found in the subspace of the leading Lyapunov vectors.

Beam Dynamics: A New Attitude and Framework

Abstract: A pictorial view in one degree of freedom from the Hamiltonian to the map classification of one-turn maps from linear to nonlinear maps vector fields and canonical transformations the ring floquet rings a theoretical construct power series and analytic/symbolic calculations examples of the analytical normalization the layout in the laboratory frame symplectic integration "small" rings - using the correct Hamiltonian fringe effects in ring dynamics large ring approximations and the rest inclusion of radiation.

Phase noise in oscillators: DAEs and colored noise sources

Abstract: Oscillators are key components of electronic systems. Undesired perturbations, i.e. noise, in practical electronic systems adversely affect the spectral and timing properties of oscillators resulting in phase noise, which is a key performance limiting factor, being a major contributor to bit-error-rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled data systems. We first present a theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of differential algebraic equations (DAEs), which extends our recent results on perturbation analysis of autonomous ordinary differential equations (ODEs). In developing the above theory, we rely on novel results we establish for linear periodically time varying (LPTV) systems: Floquet theory for DAEs. We then use this nonlinear perturbation analysis to derive the stochastic characterization, including the resulting oscillator spectrum, of phase noise in oscillators due to colored (e.g., 1/f noise), as opposed to white noise sources. The case of white noise sources has already been treated by us in a recent publication (A. Demir et al., 1998). The results of the theory developed in this work enabled us to implement a rigorous and effective analysis and design tool in a circuit simulator for low phase noise oscillator design.

Fast Fourier nonlinear vibration analysis

Abstract: We present an implementation of the multi-harmonic balance method (MHB) where intensive use of the Fast Fourier Transform algorithm (FFT) is made at all stages of calculations. The MHB method is not modified in essence, but computations are organized to obtain a very attractive method that can be applied systematically on general nonlinear vibration problems. The resulting nonlinear algebraic problem is solved by a particular implementation of a continuation method. Nonlinear vibration results are analyzed a posteriori by a Floquet method to determine their stability. The technique is applied on a series of problems of different nature, demonstrating the robustness and flexibility of the approach.

Efficient analysis of periodic structures

Abstract: An algorithm which allows the analysis of optical periodic structures with a very large number of periods with minimum numerical problems is presented, For this purpose the stable impedance transfer is combined with Floquet's theorem. Numerical results for a Bragg grating with up to 20000 periods are presented featuring the very moderate numerical effort.

Effects of the Coriolis force on the stability of Stuart vortices

Abstract: A detailed investigation of the effects of the Coriolis force on the three-dimensional linear instabilities of Stuart vortices is proposed This exact inviscid solution describes an array of co-rotating vortices embedded in a shear flow When the axis of rotation is perpendicular to the plane of the basic flow, the stability analysis consists of an eigenvalue problem for non-parallel versions of the coupled Orr–Sommerfeld and Squire equations, which is solved numerically by a spectral method The Coriolis force acts on instabilities as a ‘tuner’, when compared to the non-rotating case A weak anticyclonic rotation is destabilizing: three-dimensional Floquet modes are promoted, and at large spanwise wavenumber their behaviour is predicted by a ‘pressureless’ analysis This latter analysis, which has been extensively discussed for simple flows in a recent paper (Leblanc & Cambon 1997) is shown to be relevant to the present study The basic mechanism of short-wave breakdown is a competition between instabilities generated by the elliptical cores of the vortices and by the hyperbolic stagnation points in the braids, in accordance with predictions from the ‘geometrical optics’ stability theory On the other hand, cyclonic or stronger anticyclonic rotation kills three-dimensional instabilities by a cut-off in the spanwise wavenumber Under rapid rotation, the Stuart vortices are stabilized, whereas inertial waves propagate

Nonlinear beat Cepheid models

Abstract: The numerical hydrodynamic modeling of beat Cepheid behavior has been a long-standing quest in which purely radiative models have failed miserably. We find that beat pulsations occur naturally when turbulent convection is accounted for in our hydrodynamics codes. The development of a relaxation code and of a Floquet stability analysis greatly facilitates the search for and analysis of beat Cepheid models. The conditions for the occurrence of beat behavior can be understood easily and at a fundamental level with the help of amplitude equations. Here a discriminant arises whose sign decides whether single-mode or double-mode pulsations can occur in a model, and this depends only on the values of the nonlinear coupling coefficients between the fundamental and the first overtone modes. For radiative models is always found to be negative, but with sufficiently strong turbulent convection its sign reverses.

High-frequency electromagnetic modes in a dielectric-ring loaded beam tunnel

Abstract: A numerical code (PERIODIC-HYBRID) is developed for the calculation of the dispersion relation, the electromagnetic field components, and the quality factor of a dielectrically loaded, periodically corrugated gyrotron beam tunnel. The Floquet theorem is used to express the fields in the vacuum region, and an eigenfunction expansion is employed in each dielectric indentation. The boundary conditions imposed at the interface lead to a linear system of equations, which is appropriately truncated for the numerical implementation. The code has been benchmarked, and the results have been interpreted as the product of coupling between cavity and waveguide modes.

A physical interpretation of the Floquet description of magic angle spinning nuclear magnetic resonance spectroscopy

Abstract: A physical interpretation of the Floquet description for magic angle spinning (MAS) nuclear magnetic resonance (NMR) is proposed. The effect of the spatial rotation on the spin system in sample spinning is analysed and described in terms of orbital angular momentum operators. The analogy between rotations in real space and in spin space is emphasized. The transformation properties of the irreducible tensors in real space are used to construct a Floquet Hamiltonian for MAS NMR, that is time independent and comprises one term associated with pure sample rotation. The remaining terms are associated with the spin system, and consist of spin-phonon type Floquet operators generating simultaneous transitions between rotational states and spin states. Finally, two different definitions for the Floquet density operator are compared.

Nonlinear Beat Cepheid Models

Abstract: The numerical hydrodynamic modelling of beat Cepheid behavior has been a longstanding quest in which purely radiative models have failed miserably. We find that beat pulsations occur naturally when turbulent convection is accounted for in our hydrodynamics codes. The development of a relaxation code and of a Floquet stability analysis greatly facilitates the search for and analysis of beat Cepheid models. The conditions for the occurrence of beat behavior can be understood easily and at a fundamental level with the help of amplitude equations. Here a discriminant D arises whose sign decides whether single mode or double mode pulsations can occur in a model, and this D depends only on the values of the nonlinear coupling coefficients between the fundamental and the first overtone modes. For radiative models D is always found to be negative, but with sufficiently strong turbulent convection its sign reverses.

Coherent and incoherent chaotic tunneling near singlet-doublet crossings

Abstract: In the spectrum of systems showing chaos-assisted tunneling, three-state crossings are formed when a chaotic singlet intersects a tunnel doublet. We study the dissipative quantum dynamics in the vicinity of such crossings. A harmonically driven double well coupled to a bath serves as a model. Markov and rotating-wave approximations are introduced with respect to the Floquet spectrum of the time-dependent central system. The resulting master equation is integrated numerically. We find various types of transient tunneling, determined by the relation of the level width to the inherent energy scales of the crossing. The decay of coherent tunneling can be significantly retarded or accelerated. Modifications of the quantum asymptotic state by the crossing are also studied. The comparison with a simple three-state model shows that in contrast to the undamped case, the participation of states outside the crossing cannot be neglected in the presence of dissipation. # S1063-651X! 98" 05212-X$

An Implicit Floquet Analysis for Rotorcraft Stability Evaluation

Abstract: Floquet theory has been extensively used for assessing the stability characteristics of periodic systems. In classical application of the theory, the transition matrix of the system is explicitly computed flrst, then its eigenvalues are evaluated. Stability of the system depends on the dominant eigenvalue: if this eigenvalue is larger than unity, the system is unstable. The proposed implicit Floquet analysis extracts the dominant eigenvalues of the transition matrix using the Arnoldi algorithm, without the explicit computation of this matrix. As a result, the proposed method yields stability information at a far lower computational cost than that of classical Floquet analysis, and is ideally suited for stability computations of systems involving a large number of degrees of freedom. Examples of application of the proposed methodology are presented that demonstrate its accuracy and computational e‐ciency.

Parametric Instability of a Beam Due to Axial Excitations and to Boundary Conditions

Abstract: In this paper the stability of the lateral dynamic behavior of a pinned-pinned, clamped-pinned and clamped-clamped beam under axial periodic force or torque is studied. The time-varying parameter equations are derived using the Rayleigh-Ritz method. The stability analysis of the solution is based on Floquet's theory and investigated in detail. The Rayleigh-Rirz results are compared to those of a finite element modal reduction. It is shown that the lateral instabilities of the beam depend on the forcing frequency, the type of excitation and the boundary conditions. Several experimental tests enable the validation of the numerical results.

Asymptotic boundary condition for corrugated surfaces, and its application to scattering from circular cylinders with dielectric filled corrugations

Abstract: A simple asymptotic solution for periodic corrugated surfaces when the corrugation period approaches zero is presented. The authors enforce an asymptotic boundary condition between the region inside the corrugations and the outer region, after a simple expansion of the fields in the corrugated region. This expansion is obtained by considering the corrugated region as a homogeneous region inside which the fields have no derivatives in the direction normal to the walls of the corrugations. The asymptotic corrugation boundary condition (ACBC) replaces the use of Floquet mode expansions (FME) of the field in the outer region, and it overcomes the known limitations of the surface impedance approach. The authors show how to apply the method to calculate the plane wave scattering from a circular cylinder with dielectric-filled corrugations. The series solution is obtained and verified with a solution based on Floquet mode expansions. Excellent agreement is obtained between the two solutions after introducing the ratio of the corrugation width (w) and the period (p) into the ACBC and with λ/p is about 10 or more.

Propagation of longitudinal leaky surface waves under periodic metal grating structure on lithium tetraborate

Abstract: The dispersion properties of longitudinal leaky surface waves propagating under the periodic Al strip grating on lithium tetraborate (Li/sub 2/B/sub 4/O/sub 7/; LBO) are described theoretically and experimentally for applications of the mode to high frequency SAW devices. A theoretical method developed here is based on Floquet's theorem using space harmonics as an orthogonal function set and real boundary integral equations derived from the method of weighted residuals for a period of each region, i.e., substrate, metal, and free space. The boundary integral equations are solved by using the Galerkin procedure. The periodic strip gratings with both single-electrodes and double-electrodes are investigated, considering the convergency of the numerical computation for the number of the space harmonics. As a result, the propagation loss for shorted gratings was found to be relatively low in the thickness range of the Al strip below about 1% for the single-electrodes and 2% for the double-electrodes, although it greatly increases for a thickness over 2% for the single-electrodes and 3% for the double-electrodes.

Dynamics of a Rotating System With a Nonlinear Eddy-Current Damper

Abstract: We investigate the nonlinear dynamics and stability of a rotating system with an electromagnetic noncontact eddy-current damper. The damper is modeled by a thin nonmagnetic disk that is translating and rotating with a shaft in an air gap of a direct current electromagnet. The damper dissipates energy of the rotating system lateral vibration through induced eddy-currents. The dynamical system also includes a cubic restoring force representing nonlinear behavior of rubber o-rings supporting the shaft. The equilibrium state of the balanced rotating system with an eddy-current damper becomes unstable via a Hopf bifurcation and exact solutions for the limit cycle radius and frequency of the self-excited oscillation are obtained analytically. Forced vibration induced by the rotating system mass imbalance is also investigated analytically and numerically. System response includes periodic and quasiperiodic solutions. Stability of the periodic solutions obtained from the balanced self-excited motion and the imbalance forced response is analyzed by use of Floquet theory. This analysis enables an explanation of the nonlinear dynamics and stability phenomena documented for rotating systems controlled by electromagnetic eddy-current dampers.

Stable and efficient numerical method for solving the Schrödinger equation to determine the response of tunneling electrons to a laser pulse

Abstract: The form Ψ(x, t)=[F0(x)+F1(x, t)e−iωt+F−1(x, t)eiωt]e−iEt/ℏ is used for the wave function in the transient solutions. This expression is similar to the three dominant terms in the steady-state solution from the Floquet theory, except that now F1 and F−1 depend on t as well as x. The function F0 is the static solution, and separate partial differential equations are given for F1 and F−1. Polynomial extrapolation is used to satisfy boundary conditions at the ends of the grid. The numerical solutions are shown to converge and to be numerically stable even for simulated times exceeding 2000 cycles of the radiation field. The examples show delays corresponding to the semiclassical tunneling transit time, the classical time for traversing the inverted barrier. A resonance is seen when electrons promoted above the barrier by absorbing quanta from the radiation field have the closed line integral of momentum between the turning points equal to an integral multiple of Planck's constant. A second resonance occurs when the period of oscillation for the radiation equals the semiclassical tunneling transit time for electrons that absorb one photon from the radiation but are still below the barrier. This resonance decays at a rate corresponding to the tunneling dwell time, and, thus, it is not present in the steady state. These observations suggest a semiclassical picture of the tunneling process. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 70: 703–710, 1998

Symbolic Computation of Local Stability and Bifurcation Surfaces for Nonlinear Time-Periodic Systems

Abstract: A new technique is presented for symbolic computation of local stability boundaries and bifurcation surfaces for nonlinear multidimensional time-periodic dynamical systems as an explicit function of the system parameters. This is made possible by the recent development of a symbolic computational algorithm for approximating the parameter-dependent fundamental solution matrix of linear time-periodic systems. By evaluating this matrix at the end of the principal period, the parameter-dependent Floquet Transition Matrix (FTM), or the linear part of the Poincare map, is obtained. The subsequent use of well-known criteria for the local stability and bifurcation conditions of equilibria and periodic solutions enables one to obtain the equations for the bifurcation surfaces in the parameter space as polynomials of the system parameters. Further, the method may be used in conjunction with a series expansion to obtain perturbation-like expressions for the bifurcation boundaries. Because this method is not based on expansion in terms of a small parameter, it can be successfully applied to periodic systems whose internal excitation is strong. Also, the proposed method appears to be more efficient in terms of cpu time than the truncated point mapping method. Two illustrative example problems, viz., a parametrically excited simple pendulum and a double inverted pendulum subjected to a periodic follower force, are included.

Dynamics of Time-Varying Discrete-Time Linear Systems: Spectral Theory and the Projected System

Abstract: We study structural properties of linear time-varying discrete-time systems. At first an associated system on projective space is introduced as a basic tool to understand the linear dynamics. We study controllability properties of this system and characterize in particular the control sets and their cores. Sufficient conditions for an upper bound on the number of control sets with nonempty interior are given. Furthermore, exponential growth rates of the linear system are studied. Using finite-time controllability properties in the cores of control sets the Floquet spectrum of the linear system may be described. In particular, the closure of the Floquet spectrum is contained in the Lyapunov spectrum.

Dynamical quenching of field-induced dissociation of H2+ in intense infrared lasers

Abstract: The dynamics of dissociation of the hydrogen molecular ion H2+ in an intense infrared (IR) field is studied by a series of wave packet simulations. In these simulations, the molecular ion is assumed to be instantly prepared at the initial time by a sudden ionization of the ground-state H2 parent molecule, and a variety of frequency and intensity conditions of the laser field are considered. A new stabilization mechanism, called dynamical dissociation quenching, is found operative in the IR spectral range. In a time-resolved picture, this effect is shown to arise when a proper synchronization between the molecular motions and the laser field oscillations is ensured. In the Floquet, dressed molecule picture, the effect is related to interferences between the Floquet resonances that are excited initially by the nonadiabatic, sudden preparation of the ion. The Floquet analysis of the wave packets in this low frequency regime reveals important intersystem couplings between Floquet blocks, reflecting the highly...

Inflationary reheating classes via spectral methods

Abstract: Inflationary reheating is almost completely controlled by the Floquet indices, $\mu_k$. Using spectral theory we demonstrate that the stability bands (where $\mu_k = 0$) of the Mathieu and Lam\'e equations are destroyed even in Minkowski spacetime, leaving a fractal Cantor set or a measure zero set of stable modes in the cases where the inflaton evolves in an almost-periodic or stochastic manner respectively. These two types of potential model the expected multi-field and quantum backreaction effects during reheating.