Showing papers on "Floquet theory published in 1977"

Intense Field Multiphoton Ionization via Complex Dressed States: Application to the H Atom

Abstract: Extension of Floquet theory to include continuum as well as bound atomic states yields a practical technique for computation of multiphoton ionization rates in the region where rms field strengths approach the strength of the internal atomic fields.

On the Instability of an Internal Gravity Wave

Abstract: The linear stability of an internal gravity wave of arbitrary amplitude in an unbounded stratified inviscid Boussinesq fluid is considered mathematically. The instability is shown to be governed by a Floquet system and treated by a generalization of the method of normal modes. Some properties of the Floquet system, and in particular those of its parametric instability, are analysed. The parametric instability is related to the theory of resonant wave interactions; and the surface of marginal stability in the control space of the amplitude and wavenumbers is shown to be describable by the catastrophe theory of Thom. Finally some results of numerical calculations of the marginal surface are shown. The main physical conclusion is to confirm that the internal gravity wave is unstable always, even when its amplitude is small and so its local Richardson number is large everywhere for all time. It is suggested, by various illustrations and arguments, that the methods developed in this paper are applicable to the instability of many symmetric nonlinear waves.

Harmonic balance and the Hopf bifurcation

Abstract: A form of the Hopf bifurcation theorem specially suited to systems in block-diagram form has been developed, which allows one to deal with high or infinite order linear elements solely in terms of their transfer functions. The results are proved by the method of harmonic balance, and, for general non-linear systems, lead to criteria for the existence and stability of bifurcated orbits generalizing those derived by various authors for systems of ordinary differential equations. In the particular case of control loops with a single non-linearity, a simple addition to the Nyquist diagram of the loop determines the amplitude and frequency of bifurcated orbits, and whether they occur when the equilibrium is stable or unstable. The analysis is independent of the central manifold theorem, and of Floquet theory.

Scattering by periodic metal surfaces with sinusoidal height profiles--A theoretical approach

Abstract: A theory of scattering by periodic metal surfaces is presented that utilizes the physical optics approximation to determine the current distribution in the metal surface to first order, but modifies this approximate distribution by multiplication with a Fourier series whose fundamental period is that of the surface profile (Floquet's theorem). The coefficients of the Fourier series are determined from the condition that the field radiated by the current distribution into the lower (shielded) half-space must cancel the primary plane wave in this space range. The theory reduces the scatter problem to the familiar task of solving a linear system. For certain basic types of surface profiles, including the sinusoidal profile considered here, the coefficients of the linear system are obtained as closed form expressions in well-known functions (Bessel functions for sinusoidal profiles and exponential functions for piecewise linear profiles). The theory is thus amenable to efficient computer evaluation. Comparison of numerical results based on this theory with data obtained by recent numerical schemes shows that for depths of surface grooves less than a wavelength and for unrestricted groove widths, reliable and comparable, if not more accurate, data is obtained, in many cases at considerably cheaper computational cost.

Entrainment of a Limit Cycle by a Periodic External Excitation

Abstract: Entrainment of a limit cycle by a periodic external excitation 1s investigated with the Prigogine-Lefever·Nicolis model for chemical reaction. In the neighbourhood of the marginal point a quasi-harmonic theory is developed and the theoretical prediction is compared with the numerical computation. The stability of the entrained oscillation is examined in particular. The instabilities are classified into two types, i.e., hard- and soft-mode instabilities, by the use of the Floquet exponents which are, calculated by a non-perturbational method. The hard·mode instability corresponds to the limit of entrainment and the amplitude is subject to a modulation beyond the threshold. The frequency of modulation is estimated from the Floquet exponent, which is compared with the results of numerical computation. The soft. mode instability corresponds to a jump phenomenon in the amplitude. The smaller amplitude branch is liable to a modulation due to a superimposed hard-mode instability. As a whole, a reasonable agreement is obtained between numerical and theoretical results.

Higher‐order Bragg coupling in periodic media with gain or loss

Abstract: We consider wave propagation in longitudinally periodic media with gain or loss. An extended coupled‐waves (ECW) approach and Floquet approach are used to study index and gain (or loss) coupling in periodic media at the first few Bragg resonances by the use of Brillouin diagrams. Phase speeding or slowing that is dependent upon the type of coupling occurs in periodic media. Even and odd resonances display different dispersion characteristics. Several examples of coupling due to multiharmonic periodicities are also covered. Coupling or feedback strength can be continuously varied by changing the relative phase of multiharmonic periodicities.

Third order linear differential equations with periodic coefficients

Abstract: The third order linear differential equation Ly = y"' + p2(t)y" + p1(t)y' + po(t)y = 0 is studied, where p,(t) are continuous real-valued and periodic of period o > 0. Various criteria are obtained which guarantee "partial" asymptotic stability or instability by means of effective bounds on the Floquet characteristic multipliers of Ly = 0.

Linear periodic systems

Abstract: The purpose of this chapter is to develop the theory of a linear periodic RFDE(f) that is analogous to the Floquet theory for ordinary differential equations. It is shown by example that a complete Floquet theory will not exist. However, it is possible to define characteristic multipliers and exploit the compactness of the solution operator to show that a Floquet representation exists on the generalized eigenspace of a characteristic multiplier. The decomposition of the space C by a characteristic multiplier is also applied to the variation-of-constants formula.

Selected topics in transmission-line theory for EMP internal interaction problems. Final report

Abstract: This report discusses a number of modifications of a single-wire transmission line model for use in EMP internal interaction studies. The importance of including the effects of local line perturbations (such as a cable clamp) is discussed in terms of the reflection coefficient presented to waves on the transmission line, and general curves for determining when to neglect such effects are given. Next, the behavior of a finite-length transmission line with periodic loading is investigated and results are compared with those of an infinite line using Floquet's theorem. Finally, the bulk current response of a general multi-conductor transmission is related to the current on a single-wire line and a prescription for determining an equivalent single-wire characteristic impedance and a load impedance from the multiconductor quantities is described.