Fast periodic orbit

Thus, a trajectory of the system (12.4.7) for small e behaves in the following way during some finite time it comes to a small neighborhood of one of the invariant manifolds Mgq or Mpo, so that its a -coordinate is nearly fixed. Then, it drifts along the invariant manifold so that it corresponds to a slow change of x. As for the initial system (12.4.6) one observes a jump of the 2/-variables to the invariant manifold followed by a finite speed motion in the ar-variable. In addition, if this is a manifold of a fast periodic orbit, then we have a fast circular motion of the y-variables on the manifold, as depicted in Fig. 12.4.7. 

To find the evolution of x along the cylinder Mpo we must find the equation y = t/po(r x) of the corresponding fast periodic orbits, then substitute y = ypo te x) into the right-hand side of the first equation of (12.4.6) and average it over one period. The result of the averaging gives a first order approximation for the x-motion along a stable branch of Mpo namely ... 

In fact, the triggering from one stable branch to another is the most typical phenomenon in singularly perturbed systems, so one may encounter for our blue sky catastrophe every time when jumps between the branches of fast periodic orbits and fast equilibrium states are observed. 

Strongly open potentials that have no energy minimum but instead a saddle point, a maximum, or possibly no equilibrium point. In such potentials, the dissociation process is direct and very fast so that the lifetimes are very short. If there is no equilibrium point, it is possible that no resonance exists. We may expect sparse spectra that are regular, or irregular depending on the spectrum of interfering periodic orbits. 

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

Lega, E. and Froeschle, C. (2001). On the relationship between fast Lyapunov indicator and periodic orbits for symplectic mappings. Celest. Mech. and Dynamical Astronomy, 15 1-19. 

Classical mechanically, we know today, that periodic orbits govern the flow of trajectories in a collinear collision, In a sense, one may say that classical collinear collisions are well understood. However the world is quantal and it is of greater interest to also try and understand the quantum mechanics of collinear collisions. In the past decade various numerical techniques have been devised which enable a relatively fast and cheap evaluation of exact quantal collinear reaction probabilities. Here too, a study of classical periodic orbits has provided insight into the quantum mechanics. In section III we show how periodic orbits may be used as an analytic tool for understanding quantal phenomena such as Feshbach resonances and tunneling. ... 

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

That may seem a silly question to ask at a silicon conference But silicon is of exceptional structural interest. A primary reason is silicon s position in the Periodic Table. In its normal compounds it is 4-coordinate, and so has no vacant s or p orbitals, which would lead to very fast reactions. Compounds of lower coordination number can be made, but they must be stabilized by bulky groups. Silicon also has... 

As you should recall from general chemistry, a favorable equilibrium constant is not sufficient to ensure that a reaction will occur. In addition, the rate of the reaction must be fast enough that the reaction occurs in a reasonable period of time. The reaction rate depends on a number of factors. First, the reactants, in this case the acid and the base, must collide. In this collision the molecules must be oriented properly so that the orbitals that will form the new bond can begin to overlap. The orientation required for the orbitals of the reactants is called the stereoelectronic requirement of the reaction. (,Stereo means dealing with the three dimensions of space.) In the acid-base reaction, the collision must occur so that the atomic orbital of the base that is occupied by the unshared pair of electrons can begin to overlap with the is orbital of the acidic hydrogen. 

However, a special relativistic, or kinematical, correction, is necessary it is the Thomas precession. The electron orbiting around the nucleus with speed v (where v is a reasonably large fraction of the speed of light c) causes the period of one full rotation around the nucleus to be T in the fast-moving electron rest frame, but a longer time T (time dilatation) in the stationary rest frame of nucleus [see Eq. (2.13.11)] ... 

The second effect, the rapid or fast oscillations, also briefly mentioned in Sect. 2.2.3, shows many similarities to the usual SdH effect and was, therefore, in the beginning interpreted as being due to closed orbits in the DW state. However, as pointed out already, the resistance oscillations, although periodic in 1/5, show a temperature and magnetic field dependence which is not understandable within the usual Lifshitz-Kosevich theory (3.6). 

Recently Tambe et al. (284) extended this model and included two different types of adsorption sites for A and B, while permitting the conversion of sites from one type to the other. The authors used the same coverage dependency and the same parameters as Pikios and Luss (283). Introducing the possibility of adsorption on different sites generated a qualitatively new dynamic behavior for the system characterized by a finite amplitude/ infinite period bifurcation that yielded a homoclinic orbit. This new feature was observed when the equilibration between the two types of sites was slow compared to the other reactions. However, if equilibration is fast and the equilibrium constant is assumed to be one, this model is equivalent to the one discussed by Pikos and Luss (283). 

Kudin, K., Scuseria, G. (1998). A fast multipole algorithm for the efficient treatment of the Coulomb problem in electronic structure calculations of periodic systems with Gaussian orbitals. Chem. Phys. Lett. 289, 611-616. 

Kudin, K., Scuseria, G. (2000). Linear-scaling density-functional theory with Gaussian orbitals and periodic boundary conditions efficient evaluation of energy and forces via the fast multipole method. Phys. Rev. B61,16443. 

Funcdonal Theory with Gaussian Orbitals and Periodic Boundary Condidons Efficient Evaluation of Energy and Forces via the Fast Multipole Method. 

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