Degenerate node

When there s only one eigendirection, the fixed point is a degenerate node. A... 

A good way to think about the degenerate node is to imagine that it has been created by deforming an ordinary node. The ordinary node... 

Another way to get intuition about this case is to realize that the degenerate node is on the borderline between a spiral and a node. The trajectories are trying to wind around in a spiral, but they don t quite make it. ... 

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. 

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. 

In Exercise 6.8.1, you are asked to show that spirals, centers, degenerate nodes and stars all have / = +1. Thus, a saddle point is truly a different animal from all the other familiar types of isolated fixed points. 

Show that each of the following fixed points has an index equal to -bl. a) stable spiral b) unstable spiral c) center d) star e) degenerate node... 

With c — c, the singularity at (1,0) becomes a degenerate node. Figure 4b, with equal exponents A = = — 1. The eigenvector... 

Fig. 4. Evolution of the eigenvectors of the singularity at (1,0) with wave speed for quadratic autocatalysis (a) c < c, the singularity at (1,0) is a focus with complex eigenvalues and is approached by a spiraling trajectory that crosses out of the allowed region (b) c = c, (1,0) is a degenerate node and can be approached by an allowable connection from (0,0) (c) c > c, (1,0) is a regular node with distinct eigenvectors, an allowed trajectory approaches along...

Figure 15.3 The six benzene tt molecular orbitals. The bonding orbitals >p2 and t 3 have the same energy and are said to be degenerate, as are the antibonding orbitals tf/4 and 5. The orbitals and 4 have no tt electron density on two carbons because of a node passing through these atoms.

First-degree AV nodal blockade occurs due to inhibition of conduction within the upper portion of the node.15 Mobitz type I second-degree AV nodal blockade occurs as a result of inhibition of conduction further down within the node.12,15 Mobitz type II second-degree AV nodal blockade is caused by inhibition of conduction within or below the level of the bundle of His.12,15 Third-degree AV nodal blockade maybe a result of inhibition of conduction either within the AV node or within the bundle of His or the His-Purkinje system.12,15 AV block may occur as a result of age-related AV node degeneration. 

Idiopathic degeneration of the atrioventricular node Myocardial ischemia or infarction Neurocardiac syncope Carotid-sinus hypersensitivity... 

Long-term management of patients with AV nodal blockade due to idiopathic degeneration of the AV node requires implantation of a permanent pacemaker.12... 

After intratracheal instillation of nickel chloride or nickel sulphate in rats, a modest inflammatory response with increased number of macrophages and polynuclear leucocytes was obtained, together with increased activities of lactate dehydrogenase and -glucuronidase in bronchoalveolar fluid [351]. More severe lesions were characterized by type II cell hyperplasia with epithelialization of alveoli, and in some animals, fibroplasia of the pulmonary interstitium. By inhalation in rats, the nickel salts produced chronic inflammation and degeneration of the bronchiolar epithelium [352, 353]. There was also atrophy of the olfactory epithelium and hyperplasia of the bronchial and mediastinal lymph nodes. Nickel sulphate also produced a low incidence of emphysema and fibrosis [353]. 

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