Borderline fixed point

Nonlinear terms can change a star into a spiral) Here s another example that shows that borderline fixed points are sensitive to nonlinear terms. Consider the system in polar coordinates given by r = -r, 3 = 1/ln r. 

A < 0. The parabola - 4A = 0 is the borderline between nodes and spirals star nodes and degenerate nodes live on this parabola. The stability of the nodes and spirals is determined by t. When t < 0, both eigenvalues have negative real parts, so the fixed point is stable. Unstable spirals and nodes have t > 0. Neutrally stable centers live on the borderline t = 0, where the eigenvalues are purely imaginary. 

Figure 5.2.8 shows that saddle points, nodes, and spirals are the major types of fixed points they occur in large open regions of the (A, t) plane. Centers, stars, degenerate nodes, and non-isolated fixed points are borderline cases that occur along curves in the (A,t) plane. Of these borderline cases, centers are by far the most important. They occur very commonly in frictionless mechanical systems where energy is conserved. 

Is it really safe to neglect the quadratic terms in (1) In other words, does the linearized system give a qualitatively correct picture of the phase portrait near (x, y ) The answer is yes, as long as the fixed point for the linearized system is not one of the borderline cases discussed in Section 5.2. In other words, if the linearized system predicts a saddle, node, or a spiral, then the fixed point realty is a saddle, node, or spiral for the original nonlinear system. See Andronov et al. (1973) for a proof of this result, and Example 6.3.1 for a concrete illustration. 

The borderline cases (centers, degenerate nodes, stars, or non-isolated fixed points) are much more delicate. They can be altered by small nonlinear terms, as weTl see in Example 6.3.2 and in Exercise 6.3.11. 

Now because stable nodes and saddle points are not borderline cases, we can be certain that the fixed points for the full nonlinear system have been predicted correctly. 

The thermal decomposition of organic compounds can also be employed to generate small carbon clusters or atoms. The borderline with chemical vapor deposition (CVD) as presented in the next section is not really fix. In both cases, the method is based on the thermal decomposition of organic precursors. Processes both with and without catalyst have been reported. Contrary to the chemical vapor deposition, however, the catalyst (if applied) is not coated onto a substrate, but the substance or a precursor is added directly to the starting material ( floating catalyst ). The resulting mixture is then introduced into the reactor either in solid or in liquid state by a gas stream. From this point of view the HiPCo-process could also be considered a pyrolytic preparation of SWNT, but due to its importance it is usually regarded as autonomous method. 

Fig. 6.5 Phase diagrams of side-chain association. The binodal (solid line), the spinodals (borderline of the gray areas), microphase separation transition Une (broken line), critical solution points (white circles), and Lifshitz points (black circles) are shown. The homogeneous mixture region, microphase region, and the macroscopicaUy unstable region are indicated by H, M, and U, respectively. Parameters are fixed at wa = 1000, f = 200, = 10, A.q = 1.0, and tjfi = 1.0. The...

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