Internal bifurcation

Consider next a Banach space B of dynamical systems X of the Morse- Smale class in a compact region G. Let dB denote the boundary of B, Any system Xq G dB is structurally unstable. We will assume then that a system Xo G dB is a boundary system of the Morse-Smale class, if in any of its neighborhoods there are systems with infinitely many periodic orbits (basically, this means the presence of transverse homoclinic trajectories). The other systems on dB correspond to internal bifurcations within the Morse-Smale class. 

The primary scope of this book will focus on the analysis of the internal bifurcations within the class of systems with simple dynamics, such as Morse-Smale systems. Furthermore, we will restrict our study mostly to bifurcations of codimension-one. The reason for this restriction is that some bifurcations of higher codimension turn out to be boundary bifurcations in many cases, such as when the normal forms for the equilibrium states are three-dimensional. Nevertheless, we will examine some codimension-two cases which are concerned with equilibrium states and the loss of stability of periodic orbits. Meanwhile, let us start our next section with a discussion of some questions related to structurally unstable heteroclinic connections. 

Note that only cases (1) and (2) correspond to structurally stable systems the other cases are non-rough. In essence, a bifurcation of a homoclinic-8 with a negative saddle value is an internal bifurcation in the Morse-Smale class. 

CEA involves exposure of the carotid bifurcation in the neck to a point along the internal carotid artery (ICA) beyond which the atherosclerotic plaque terminates. 

Touho H, Morisako T, Hashimoto Y, Karasawa J. Embolectomy for acute embolic occlusion of the internal carotid artery bifurcation. Surg Neurol 1999 51 313-320. 

Dendrimers have a star-like centre (functionality e.g. 4) in contrast to a star however, the ends of the polymer chains emerging from the centre again carry multifunctional centres that allow for a bifurcation into a new generation of chains. Multiple repetition of this sequence describes dendrimers of increasing generation number g. The dynamics of such objects has been addressed by Chen and Cai [305] using a semi-analytical treatment. They treat diffusion coefficients, intrinsic viscosities and the spectrum of internal modes. However, no expression for S(Q,t) was given, therefore, up to now the analysis of NSE data has stayed on a more elementary level. 

Sanchez, A. L., A. Linan, and F. A. Williams. 1992. A bifurcation analysis of high-temperature ignition of H2-O2 diffusion flames. 24th Symposium (International) on Combustion Proceedings. Pittsburgh, PA The Combustion Institute. 1529-37. 

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

However, many equations that we shall encounter in this book will have more than one solution for a given set of the parameters, indicating that the modeled chemical/biological process can actually be in one of several steady states, depending on its recent history or start-up and on the internal dynamics of the system as described by differential equations. Such behavior signifies bifurcation and this is where the fun begins. 

Most tissue histamine is sequestered and bound in granules (vesicles) in mast cells or basophils the histamine content of many tissues is directly related to their mast cell content. The bound form of histamine is biologically inactive, but many stimuli, as noted below, can trigger the release of mast cell histamine, allowing the free amine to exert its actions on surrounding tissues. Mast cells are especially rich at sites of potential tissue injury—nose, mouth, and feet internal body surfaces and blood vessels, particularly at pressure points and bifurcations. 

Let B be a CBS with at least two cells. An edge of B is called internal if it is the intersection of two cells, and external otherwise. Cell c is an end cell (bifurcation cell)... 

Localization of this steady state as a point of intercept for the null dines x = 0 and y = 0 as a function of the k x value is shown in Fig. 16. At low k x this point is localized sufficiently close to the region of probable initial conditions (at k x = 0 it becomes a boundary steady state). It is the proximity of the initial conditions to the steady state outside the reaction polyhedron that accounts for the slow transition regime. Note that, besides two real-valued steady states, the system also has two complex-valued steady states. At bifurcation values of the parameters, the latter become real and appear in the reaction simplex as an unrough internal steady state. The proximity of complex-valued roots of the system to the reaction simplex also accounts for the generation of slow relaxations. 

Carotid revascularization, initially by CEA, was introduced in early 1950s as a method to prevent stroke due to atherosclerosis of the carotid bifurcation and internal carotid artery (ICA). At least four prospective randomized trials have demonstrated... 

The above scenario is typical of nonlinear dynamical systems when the amplitude of the internally generated oscillations becomes sufficiently large. In the bifurcation diagram of Fig. 12.5 this occurs when the slope of the feedback characteristics exceeds a critical value. However, similar scenarios can be produced through variation of other parameters such as, for instance, the damping of the arteriolar oscillator. 

R. Seydel (Ed.) Proc. of Bifurcation and Chaos Analysis, Algorithms, Applications . International Series of Numerical Mathematics, Vol. 97, pp. 231-236. Basel Birkhauser Verlag. 

The carotid bifurcation can be seen in 80% of the adult population at the level of cervical vertebrae 3-5. Distal to the physiologic dilatation of the carotid bulb the internal carotid artery (ICA) proceeds dor-sally to the external carotid artery (ECA) into the petrous bone. Entering the bony carotid canal, it describes a sharp bend and thus causes turbulent flow patterns with somewhat typically symmetrical signal loss on TOF-MRA, which should not be misinterpreted as real stenoses (Fig. 5.2). 

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