Middle bifurcation

The precise nature of bifurcation at IV4 = N r (likewise, for other Ni, i = 1,2,3). It is expected that at JV4 = Nf the lower and middle limiting current branches fuse and annihilate each other, leaving an inflection point at the voltage-current curve for N4 < N". ... 

The Hopf bifurcation analysis proceeds as described previously, the required condition being that the trace of the Jacobian matrix corresponding to eqns (12.45) and (12.46) should become equal to zero for some stationary-state concentration given by the lower root from (12.51). (The solution with the upper root corresponds to the middle branch of stationary states for... 

FIGURE 8 Detailed bifurcation diagram for variations in the reactant partial pressures (a, and a2) when y, = 0.001 and y2 = 0.002. The middle diagram is the superposition of the four surrounding bifurcation curves. The points K, L, M and N correspond to double zero eigenvalues and the points G and H are metacritical Hopf points. 

An example is shown in figure 3 for section AA near the bottom of the 2/1 resonance horn of figure 2. As the frequency is increased from left to right, the torus becomes phase locked as a pair of period 2 saddle nodes develop on it. The saddle nodes then separate with the saddles alternating with the node and the invariant circle is now composed of the unstable manifolds of the saddles whereas the stable manifolds of the saddles come from the unstable period 1 focus in the middle of the circle and from infinity. As the frequency is increased further, the saddles rotate around the circle and recombine with their neighbouring nodes in another saddle-node bifurcation. 

In a sense, the numerical results, i. e., the high level of instability in computing the middle solution to our BVP near the bifurcation points and elsewhere mimics the instability... 

If we increase / and 7 to 1.2 and 25, respectively, we find a much larger interval of values with multiple pellet efficiencies. The curve shows a nascent 5-fold bifurcation kink in the lower part of the middle branch and it requires us to solve nearly 2.7 times as many BVPs in 285 seconds. 

These indicate the limit of our successful numerical BVP integrations near the bifurcation points. In between the x and o marks on the middle branch, the curve is drawn using interpolation of our successful BVP solutions data, while in between two adjacent x or two adjacent o marks, the curve is drawn by extrapolating nearby computed function data. This is done automatically by MATLAB s plot commands. 

The bifurcation diagram represents an imperfect pitchfork diagram, where the middle steady state persists over the entire range of Kc, even for negative values of Kc, i.e., even for positive feed back control, which would destabilize the system. 

Changing Kc increases the slope of the heat removal line because its slope is 1 + K If a bifurcation diagram is drawn for this nonadiabatic case with Kc as the bifurcation parameter and the jacket cooling temperature is the temperature of the middle steady state ym, we obtain a pitchfork type bifurcation diagram as shown in Figure 9 (A-2). 

Fig. 68. Typical dispersion relations, displaying the growth rate of perturbations, A(n), vs. their wave number, n, of an S-NDR system for three different homogeneous steady states [33]. The lowest curve depicts the case of a stable homogeneous state, the middle one is close to a Turing-type bifurcation in which a stationary structure with the integer wave number closest to the maximum of the curve is born. The up-most curve shows a situation for which the homogeneous state is unstable with respect to perturbations lying within the wavelength range n, for which X(n) > 0.

Figure 1. Possible forms of transformation of an unstable bifurcation diagram (middle column) into either one of two possible stable forms (left or right column) at the Hysteresis (a), Isola (b, c) and Double limit varieties (d, e).

The middle cerebral artery (MCA) is divided into four segments. The horizontal Ml segment reaches laterally to the bifurcation or trifurcation, the insular segments within the Sylvian fissure are named... 

The internal carotid artery starts as the carotid sinus at the bifurcation of the common carotid artery at the level of the thyroid cartilage. It runs up the neck, without any branches, to the base of the skull where it passes through the foramen lacerum to enter the carotid canal of the petrous bone. It then runs through the cavernous sinus in an S-shaped curve (the carotid siphon) pierces the dura and exits just medial to the anterior clinoid process. It then bifurcates into the anterior cerebral artery and the larger middle cerebral artery. 

Schmahmann JD (2003). Vascular syndromes of the thalamus. Stroke 34 2264-2278 Schulz UG, Rothwell PM (2001). Major variation in carotid bifurcation anatomy a possible risk factor for plaque development Stroke 32 2522-2529 Scott BL, Jankovic J (1996). Delayed-onset progressive movement disorders after static brain lesions. Neurology 46 68-74 Wardlaw JM, Merrick MV, Ferrington CM et al. (1996). Comparison of a simple isotope method of predicting likely middle cerebral artery occlusion with transcranial Doppler ultrasound in acute ischaemic stroke. Cerebrovascular Diseases 6 32-39 Wardlaw JM, Lewsi SC, Dennis MS etal. (1999). Is it reasonable to assume a particular embolic source from the type of stroke Cerebrovascular Diseases 9(Supp 1) 14... 

Thus, we speculate that, in the processes where qualitatively different flows emerge, the system would experience the situation where normal hyperbolicity breaks down. This speculation is based on the argument that, in order for the flows along the normal directions of NHIMs to bifurcate, one of the Lyapunov exponents of the normal directions must change its sign from plus to minus or from minus to plus. In the middle of these changes, normal hyperbolicity breaks down. 

The middle cerebral artery, which originates at the division of the internal carotid artery, passes through the lateral sulcus (Sylvian fissure) en route to the lateral convexity of the cerebral hemisphere, to which it supplies blood. The middle cerebral artery travels along the surface of the insular cortex, over the inner surface of the frontal, temporal, and parietal lobes, and appears on the lateral convexity. The posterior cerebral arteries originate at the bifurcation of the basilar artery, and each one passes around the lateral margin of the midbrain. 

The study of cerebrovascular disease has advanced markedly in recent years with advances in non-invasive imaging methods such as MR angiography and CT angiography as well as an improved understanding of the immune system in the pathogenesis of atherosclerosis. Atherosclerotic cerebrovascular disease is a common cause of strokes and shows a predilection for sites such as the bifurcation of the common carotid artery into the internal and external carotid arteries and the aortic arch and the major intracranial arteries such as the basilar artery and the middle cerebral arteries. Occlusive atherosclerotic vascular disease of these large extracranial arteries is responsible for as many as 20-30% of ischemic strokes and intracranial steno-occlusive disease causes around 5-10% of ischemic strokes. 

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