Theory INDEX

In Section 6.3 we learned how to linearize a system about a fixed point. Linearization is a prime example of a local method it gives us a detailed microscopic view of the trajectories near a fixed point, but it can t tell us what happens to the trajectories after they leave that tiny neighborhood. Furthermore, if the vector field starts with quadratic or higher-order terms, the linearization tells us nothing. 

In this section we discuss index theory, a method that provides global information about the phase portrait. It enables us to answer such questions as Must a closed trajectory always encircle a fixed point If so, what types of fixed points are permitted What types of fixed points can coalesce in bifurcations The method also yields information about the trajectories near higher-order fixed points. Finally, we can sometimes use index arguments to rule out the possibility of closed orbits in certain parts of the phase plane. 

The index of a closed curve C is an integer that measures the winding of the vector field on C. The index also provides information about any fixed points that might happen to lie inside the curve, as we ll see. 

As X moves counterclockwise around C, the angle 0 changes continuously since the vector field is smooth. Also, when x returns to its starting place, 0 returns to its original direction. Hence, over one circuit, 0 has changed by an integer multiple of 2 r. Let [0] be the net change in 0 over one circuit. Then the index of the closed curve C with respect to the vector field f is defined as 

is the net number of counterclockwise revolutions made by the vector field as x moves once counterclockwise around C. 

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Pure, low temperature organic Hquid viscosities can be estimated by a group contribution method (7) and a method combining aspects of group contribution and coimectivity indexes theories (222). Caution is recommended in the use of these methods because the calculated absolute errors are as high as 100% for individual species in a 150-compound, 10-family test set (223). A new method based on a second-order fit of Benson-type groups with numerous steric correctors is suggested as an alternative. Lower errors are claimed for the same test set. 

Ty). Numerous authors have discussed certain special aspects of refractive index theory in simple dense fluids, macromolecular fluids and in fluids near the critical point. Also, various experimental studies of jRm have been carried out for simple fluids and water. " Frequency-dependent polarizabilities of atoms and simple molecules have been calculated. ... 

Todeschini, R., Consonni, V. and Maiocchi, A. (1998). The K Correlation Index Theory Development and its Applications in Chemometrics. Chemom.lntell.Lab.Syst., 46,13-29. 

Matter and antimatter ) There s an intriguing analogy between bifurcations of fixed points and collisions of particles and anti-particles. Let s explore this in the context of index theory. For example, a two-dimensional version of the saddle-node bifurcation is given by x = a + x, y = -y, where a is a parameter. 

Suppose we have a strong suspicion, based on numerical evidence or otherwise, that a particular system has no periodic solutions. How could we prove this In the last chapter we mentioned one method, based on index theory (see Examples... 

For / > 1, librations are impossible because any libration must encircle a fixed point, by index theory—but there are no fixed points when / > 1- Hence we only need to consider rotations. 

Method 4. Index theory approach.. This method is based on the Poincare-Hopf index theorem found in differential topology, see, e.g., Gillemin and Pollack (1974). Similarly to the univalence mapping approach, it requires a certain sign from the Hessian, but this requirement need hold only at the equilibrium point. 

Observe that the condition (—1) JT is trivially satisfied if is negative definite which is implied by the condition (2.2) of contraction mapping, i.e., this method is also somewhat weaker than the contraction mapping argument. Moreover, the index theory condition need only hold at the equilibrium. This... 

Conley, C., and Zehnder, E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Comm. Pure Appl. Math. 87 (1984), No. 2, 207-255. 

With this background in this review we will now talk about the basic theory of the reactivity index including definitions of local and global softness along with relative nucleophilicity and electrophilicity. We will as well cover the role of response function in deriving the excited state reactivity index theory, followed... 

In this review we have revisited the reactivity index theory from HSAB principle within the domain of DFT. We have presented an overview of the reactivity index theory from concept to industrial application. We have demonstrated that a theory within the DFT domain based on the theory of electronegativity and explored in the realm of electron affinity and ionization potential is capable to deliver a simple... 

Chatterjee A (2011) Excited state reactivity index theory application for small moieties, hit J Quantum Chem 111 3821... 

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