Oscillatory orbitals

Figure C3.6.7 Cubic (jir = 0) and linear (r = 0) nullclines for tire FitzHugh-Nagumo equation, (a) The excitable domain showing trajectories resulting from sub- and super-tlireshold excitations, (b) The oscillatory domain showing limit cycle orbits small inner limit cycle close to Hopf point large outer limit cycle far from Hopf point.

Figure 5. Dimensionless widths (left) and its oscillatory part (right) corresponding to the scar function spectral peaks for periodic orbit number 3 in Fig. 2.

The maximum Lyapunov exponent (MLE) was computed to provide major evidence of nonperiodic oscillatory behavior. MLE is one of the most important features in nonlinear science to distinguish chaotic from non chaotic behavior. Essentially MLE measures the distance between attractor orbits... 

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

The link between the oscillatory behaviour of the structural energy curves and the moments of the local density of states can be made explicit by writing the bond order of a given bond as a many-atom expansion about that bond (Pettifor (1989)). Considering for simplicity the case of s orbitals, on a lattice where all sites are equivalent, the bond order can be expressed exactly (Aoki (1993)) as... 

The interpretation of a spectrum from a dynamical point of view can also be applied to a spectrum containing a broad feature associated with direct and/or indirect dissociation reactions. From such spectra dynamics of a dissociating molecule can also be extracted via the Fourier transform of a spectrum. An application of the Fourier transform to the Hartley band of ozone by Johnson and Kinsey [3] demonstrated that a small oscillatory modulation built on a broad absorption feature contains information of the classical trajectories of the vibrational motion on PES, so-called unstable periodic orbits, at the transition state of a unimolecular dissociation. 

FIGURE 2 The birth and growth of limit cycle oscillations in the I - a, jS, Tr space for a system with non-zero e and k displaying a mushroom stationary-state pattern. Oscillatory behaviour originates from a supercritical Hopf bifurcation along the upper branch and terminates via homoclinic orbit formation. 

The aim of this article is to show that the new quasi-two-dimensional organic conductor p -(BEDO-TTF)5[CsHg(SCN)4]2 [hereafter called (BEDO)CsHg] (BEDO-TTF - bis-(ethylenedioxy)tetrathiafulvalene) which contains closed and open orbits displays rather complicated oscillatory spectra associated with magnetic breakdown (MB) and quantum interference (QI) effects. Tight binding band structure calculations for this compound are proposed to characterise its Fermi surface. The aim of the article includes also an investigation of the optical conductivity anisotropy with polarized infrared reflectance spectra. 

One finds that the oscillatory structure of dti (r) observed in Fig. 5.1 is completely due to the density correction Av r, dens), as can be gleaned from Fig. 5.3 . As expected, the functional correction Avx(r,fctl) is only relevant for the high density small-r regime, i.e. the innermost orbitals. In this regime, however, the Av r) from the LDA is rather different from the exact OPM result. In order to demonstrate that both components of Avj (r) contribute to the error of the LDA it is again advantageous to consider Hg Similar to the situation for AE the exact Avx r,fctl) vanishes. As is clear from Fig. 5.4 the LDA does not reproduce the exact Av ir). In particular, Av r,fctl) does not vanish but is of the same order of magnitude as the exact Avjr, dens). 

---