Cusp point

This discussion will be limited to functions of one variable that can be plotted in 2-space over the interval considered and that constitute the upper boundar y of a well-defined area. The functions selected for illustration are simple and well-behaved, they are smooth, single valued, and have no discontinuities. When discontinuities or singularities do occur (for example the cusp point of the Is hydrogen orbital at the nucleus), we shall integrate up to the singularity but not include it. 

Figure 10. Divisions of the Em map for the quadratic Hamiltonian in Eq. (39). Va, Vb, and Vx are the energies of the two isomers and that of the saddle point between them. The solid lines are relative equilibria, and the cusp points, a , indicate points at which the secondary minimum and the saddle point in Fig. 9 merge at a point of inflection.

These equations represent a parametric curve with parameter y. From a set of values of the parameter y it is possible to draw a curve in the xq — yo plane, so we obtain a bifurcation curve as a function of parameter y. This curve with a cusp point can be considered as the border that dividing the plane xq — yo into domains with one and three equilibrium states respectively. 

Exercise 5. Taking into account exercise 4 prove that the parametric equations of curve with cusp point are the following ... 

The cusp curve begins at the cusp point which has coordinates... 

Fig. 7.5. The 0ad 1 parameter plane showing the hysteresis line and the isola cusp described by eqns (7.34)—(7.36). The plane is divided into five regions (see inset for details in vicinity of cusp point) corresponding to the qualitative forms in Fig. 7.4. The numerical values are appropriate to the exponential approximation, y = 0, but the qualitative form of the diagram holds for ally < J.

The important feature of these equations is that the winged cusp point exists for physically acceptable values of the various quantities (x, tres, 0ad, and tn > 0, 9C > — y 1) provided y stays within the above range. The system can be unfolded from the singular point by varying 0ad, tN, and 9C, and in this way all seven of the stationary-state patterns shown in Fig. 7.8. 

Three different patterns of multistability appear possible from Fig. 12.5. For a cut across the diagram such as that represented by (b), the rate curve shows an isola as r passes through the region of multistability. If p just exceeds the value corresponding to the cusp point C, then a horizontal cut across the diagram such as (c) intersects the boundary four times the stationary-state reaction rate locus has four turning points and gives a mushroom pattern. Finally, for 0.020133 < p < 0.021442, the traverse (d) shows... 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

In figure 2b, there are clearly folds in the left-hand side of the 3/2 and 2/1 resonance horns. This phenomenon had not (when we observed it) been seen in other forced oscillators such as the Brusselator model (Kai Tomita 1979) and the non-isothermal cstr (Kevrekidis et al. 1986), although it may have been missed in previous numerical studies that did not use arc-length continuation. It is however also to be found in unpublished work of Marek s group. The cusp points at M and L are quite different from the apparent cusp ... 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

The cusp points on the Wulff plot generally correspond to low-index planes. This is because the surface tension of a solid is primarily determined by the strength of the bonding of the individual surface atoms. Atoms in the low-index planes form the greatest number of bonds and hence have less energy than atoms in less densely packed planes. Thus, the equilibrium shape generally consists almost entirely of low-index planes. 

The only limitations of this method that are not also present in the classical method relate to the shape of the fiber. Although the above quantitative theory considers a cylindrical fiber, the method should work equally well (on a go-no go basis) for small diameter fibers of any cross section that do not have cusps pointed outward. Referring to Figure 2, irregular cross sections such as A, B, and C are satisfactory. A fiber of cross section D, however, must sink at contact angles much greater than... 

To summarize the results so far, we plot the bifurcation curves h = +hfr) in the (r,h) plane (Figure 3.6.2). Note that the two bifurcation curves meet tangentially at (r, /i) = (0,0) such a point is called a cusp point. VJe also label the regions that correspond to different numbers of fixed points. Saddle-node bifurcations occur all along the boundary of the regions, except at the cusp point, where we have a codimension-2 bifurcation. (This fancy terminology essentially means that we have had to tune two parameters, h and r, to achieve this type of bifurcation. Un-... 

It is easy to see that a single Gaussian doesn t represent an s-type atomic orbital as well as the STO because the Gaussian is rounded at r = 0 where the atomic orbital has a cusp point. A linear combination of many Gaussians can be made to approach the atomic orbital as closely as we please near (but not precisely at) the nucleus depending on software efficiency, the power of the computer, and the amount of CPU time we wish to expend. This idea led to programs in which 3 Gaussians... 

Figure 12 Sampling efficiency defined as the ratio of converged eigenvalues divided by NK, for a fixed number of points (Ng = 64) as a function of the grid spacing Aq. The optimal sampling spacing is marked. At Aqapl the sampling efficiency approaches tt/4 at the cusp point Aq. ...

Fig. 11.1 Two-dimensional bifurcation diagram calculated by continuation from the Citri-Epstein mechanism for the chlorite-iodide system. Plot of 2D space of constraints is chosen to be the ratio of input concentrations, [CIO ]o/[I ]o> versus the logarithm of the reciprocal resideuce time, logfco-Notation SSI, SS2, region of steady states with high [1 ] and low [I ], respectively Osc, region of periodic oscillations Exc, region of excitahihty snl, sn2, curves of saddle-node bifurcations of steady states hp, curve of Hopf bifiucatious sup, ciuve of saddle-node bifurcations of periodic orbits swt, swallow tail (a small area of tristability) C, cusp point TB, Takens-Bogdanov point (terminus of hp on snl). (From [5].)...

In figure 7.9 the loci of codimension-1 points are depicted in the f q space. As the heat of reaction is increased, the cusp point H crosses the /-boundary at a codimension-3 point and moves into the unfeasible region. Although the area between the loci of... 

It is seen from Figure 3.97 that the maximum A (minimum B) of the critical line coincides with the maximum (minimum) of the upper (lower) cusp point line. Thi.s relation is valid generally. The extremum condition of the cusp point line requires dx, be equal to zero along the cusp line. Calculations involving Fxjiiations 14, 15, and 31 lead to an equation symmetrical with Equation 31... 

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