Higher-order fixed point

Question 4.6 (Hitchin). Gonsider a hnite group action on a K3 surface which preserves a hyper-Kahler structure. (Such actions were classified by Mukai [58].) It naturally induces the action on the Hilbert scheme of points on the K3 surface. Its fixed point component is a compact hyper-Kahler manifold as in 4.2. Is the component a new hyper-Kahler manifold The known compact irreducible hyper-Kahler manifolds are equivalent to the Hilbert scheme of points on a /F3 surface, or the higher order Kummar variety (denoted by Kr in [6]) modulo deformation and birational modification, (cf. [57, p.l68j)... 

Fixed points for the calibration of the optical pyrometers are the silver, gold, and copper points and higher temperature secondary fixed points such as those given in Table 2. Calibration at other temperatures can be accomplished by use of a rotating sector or a filter of accurately known transmission factor between the fixed-point source and the pyrometer, in order to simulate a source of lower temperature in accordance with the Planck equation. Such sectors or filters are used also to permit the optical pyrometer to be used for the measurement of temperatures much above 2000 K. 

Note that, for those terms with oti = pj, the problem of eliminating higher-order terms is the same as that in the Birkhoff normal form for stable fixed points. Then, it is possible to eliminate higher-order terms as long as the set of frequencies (d n = 2,..., N) satisfies the nonresonance condition... 

Higher-order and non-isolated fixed points at least one eigenvalue is zero. 

These ideas also generalize neatly to higher-order systems. A fixed point of an th-order system is hyperbolic if all the eigenvalues of the linearization lie off the imaginary axis, i.e., Re(Aj iO for / = ,. . ., . The important Hartman-Grobman theorem states that the local phase portrait near a hyperbolic fixed point is topologically equivalent to the phase portrait of the linearization in particular, the stability type of the fixed point is faithfully captured by the linearization. Here topologically equivalent means that there i s a homeomorphism (a continuous deformation with a continuous inverse) that maps one local phase portrait onto the other, such that trajectories map onto trajectories and the sense of time (the direction of the arrows) is preserved. 

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. 

Nasty fixed point) The system x = xy- x y + y, y = y + x - xy has a nasty higher-order fixed point at the origin. Using polar coordinates or otherwise, sketch the phase portrait. 

After dividing through by x and neglecting higher-order terms, get ii+2- x /(> 0. Hence there is a pair of fixed points with x = -yj 6 ii + 2) for H slightly greater than —2. Thus a supercritical pitchfork bifurcation occurs at... 

When we have too few points to justify linearizing the function between adjacent points (as the trapezoidal integration does) we can use an algorithm based on a higher-order polynomial, which thereby can more faithfully represent the curvature of the function between adjacent measurement points. The Newton-Cotes method does just that for equidistant points, and is a moving polynomial method with fixed coefficients, just as the Savitzky-Golay method used for smoothing and differentation discussed in sections 8.5 and 8.8. For example, the formula for the area under the curve between x, and xn, is... 

This point was examined at some length [18] for systems of various organic liquids vs. water. The discussion above of the orientation of group dipoles—i.e., of quadrupole and higher order pole effects—constitutes an extension of the earlier discussion [18]. To complete this theoretical treatment, it will be necessary to examine in detail the group-dipole interactions in fixed orientations, in a model for the system where nearest-neighbor interactions are considered separately from the attractions of further-removed molecules. 

The situation changes dramatically when higher order terms are included in the initial polynomial approximation for the propagator amplitude [35]. Beginning with the initial, two-loop approximation that is quadratic in a, one finds that the dynamical system d(t a), a t a) exhibits a nontrivial, stable fixed point or attractor (t). This indicates that the corresponding approximation to the photon propagator amplitude is an asymptotically self-similar fractal object. 

There are two other fixed points (0.254037, 0.022159, 0.07098) and (0.2000, 0.0666, 0.0666) on the line separating the basins of attraction of the fixed points corresponding to the collapsed and the swollen phases. These are purely repulsive, and cannot be reached starting with our choice of initial condition. These correspond to higher order multicritical points( tetra critical). 

We now consider the above derivation from a slightly different point of view, which is extremely helpful in organizing the calculations of the higher order energies. To evaluate the summation over y, for fixed i, in Eq. (6.50)... 

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