Floquet analysis

The discussion following Eqn. (5.1.8) imply a single Hopf bifurcation when Reynolds number increases beyond Rccr It is interesting to note that Landau (1944) talked about further instabilities following the nonlinear saturation of the primary instability mode. This is akin to Floquet analysis of the resulting time periodic system (Bender Orszag (1978)). The possibility of multiple bifurcation was also mentioned in Drazin Reid (1981) who stated that in more complete models of hydrodynamic stability we shall see that there may he further bifurcations from the solution A = 0, e.g. where the next least stable mode of the basic flow becomes unstable, and from the solution A = Ae- To the knowledge of the present authors, no theoretical analysis exist that showed multiple bifurcation before for this flow. Here,... 

Using Rydberg atoms and microwave fields it has been possible to observe virtually all one electron strong field phenomena. The attraction of these experiments is that they can be more controlled than most laser experiments, with the result that more quantitative information can be extracted. The insights gained from these experiments can be profitably transferred to optical experiments. To demonstrate the latter point we demonstrate that apparently non-resonant microwave ionization, in fact, occurs by resonant transitions through intermediate states. These experiments demonstrated clearly the power of Floquet analysis of such processes, and the ideas were subsequently applied to the analogous problem of laser multiphoton ionization. 

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

However, integrabihty imposes a criterion for obtaining DBs analytically. DBs are obtained analytically for integrable systems, while for non-integrable systems it is obtained by various numerical methods viz. spectral collocation method, finite-difference method, finite element method, Floquet analysis, etc. As evident from many numerical experiments, DBs mobility is achieved by an appropriate perturbation [42]. From the practical application perspective, dissipative DBs are more relevant than their Hamiltonian counterparts. The latter with the character of an attractor for different initial conditions in the corresponding basin of attraction may appear whenever power balance, instead of energy conservation, governs the nonlinear lattice dynamics. The attractor character for dissipative DBs allows for the existence of quasi-periodic and even chaotic DBs [54, 55]. 

The specific models we will analyse in this section are an isothermal autocatalytic scheme due to Hudson and Rossler (1984), a non-isothermal CSTR in which two exothermic reactions are taking place, and, briefly, an extension of the model of chapter 2, in which autocatalysis and temperature effects contribute together. In the first of these, chaotic behaviour has been designed in much the same way that oscillations were obtained from multiplicity with the heterogeneous catalysis model of 12.5.2. In the second, the analysis is firmly based on the critical Floquet multiplier as described above, and complex periodic and aperiodic responses are observed about a unique (and unstable) stationary state. The third scheme has coexisting multiple stationary states and higher-order periodicities. 

We now turn to an example where full use has been made of the bifurcation analysis based on Floquet multipliers, as described in 5.4.3. 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

Since in the Floquet representation the Hamiltonian K defined on the enlarged Hilbert space is time-independent, the analysis of the effect of perturbations (like, e.g., transition probabilities) can be done by stationary perturbation theory, instead of the usual time-dependent one. Here we will present a formulation of stationary perturbation theory based on the iteration of unitary transformations (called contact transformations or KAM transformations) constructed such that the form of the Hamiltonian gets simplified. It is referred to as the KAM technique. The results are not very different from the ones of Rayleigh-Schrodinger perturbation theory, but conceptually and in terms of speed of convergence they have some advantages. 

The preceding analysis is well adapted when one considers slowly varying laser parameters. One can study the dressed Schrodinger equation invoking adiabatic principles by analyzing the Floquet Hamiltonian as a function of the slow parameters. 

If we apply this analysis locally in the Floquet spectrum, it provides the matching between the adiabatic evolution far from any avoided crossings and a local adiabatic or diabatic behavior around them. 

Lim, Y., Floquet, R, Joulia, X. and Kim, S. (1999). Multiobjective optimization in terms of economics and potential environment impact for process design and analysis in a chemical process simulator, Industrial Engineering Chemistry Research 38, pp. 4729-4741. 

Spectral analysis of the linearized semiflow along a periodic solution is called Floquet theory. The eigenvalues p of the linearized period map are called Floquet multipliers. A Floquet exponent is a complex / such that exp(/3r) is a Floquet multiplier of the system, where t denotes the minimal period. A periodic solution is hyperbolic if, and only if, it possesses only the trivial Floquet multiplier p = 1 on the unit circle, and this multiplier has algebraic multiplicity one. Otherwise it is called non-hyperbolic. In ODEs hyperbolic periodic solutions possess stable and unstable manifolds, similarly to the case of hyperbolic equilibria. Non-hyperbolic periodic solutions possess center manifolds. 

DS operations form groups and, consequently, DS analysis of Floquet states is analogous to symmetry analysis of stationary states. In particular, one can label QEs and their corresponding Floquet states with appropriate quantum numbers, determine symmetry properties of... 

A Lyapunov exponent is a generalized measure trf the growth or decay of small perturbations away from a particular dynamical state. For perturbations around a fixed point or steady state, the Lyapunov exponents are identical to the stability eigenvalues of the Jacobian matrix discussed in an earlier section. For a limit cycle, the Lyapunov exponents are called Floquet exponents and are determined by carrying out a stability analysis in which perturbations are applied to the asymptotic, periodic state that characterizes the limit cycle. For chaotic states, at least one of the Lyapunov exponents will mm out to be positive. Algorithms for the calculation of Lyapunov exponents are discussed in a later section in conjunction with the analysis of experimental data. These algorithms can be used for simulations that yield possibly chaotic results as well as for the analysis of experimental data. 

This is not necessarily the way we actually calculate the element currents. In fact this was done by direct calculations of the currents in the finite array in question by using the SPLAT program discussed in Chapter 3. Typical examples have already been presented in Figs. 1.3b and 1.3c. Clearly the currents in Fig. 1.3c are seen to be highly erratic. To find out what current components actually are contained in such a distribution, we simply ran a Fourier analysis and obtained the current spectrum shown in Fig. 4.5f. While the current spike at Tex = 0.707 is easy to associate with the Floquet currents obtained for an infinite array exposed to a plane wave incident at 45°, the two other spikes at Tex = 1.25 remained somewhat of a mystery until the explanation in Section 4.6 above was introduced. 

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