Global phase portrait

The system has the spatially homogeneous stationary point (x, y) of a stable focus type. At the point k02 the first catastrophe takes place the imaginary part of the eigenvalues vanishes, both the eigenvalues in this sensitive state being equal, Ax = A2. Such a catastrophe is not related to a loss of stability by the point (x, y) only a global phase portrait changes... 

Fig. 10.6.7. This picture illustrates how the separatrices of a saddle point organize the global phase portrait.

Let us now consider the behavior of the system when the Kerr coupling constant is switched on (e12 / 0). For brevity and clarity, we restrict our discussion to the question of how the attractors in Fig. 20 change when both oscillators interact with each other. To answer this question, let us have a look at the joint auto-nomized spectrum of Lyapunov exponents for the two oscillators A,j, A,2, L3, A-4, L5 versus the interaction parameter 0 < ( 2 < 0.7. The spectrum is seen in Fig. 32 and describes the dynamical properties of our oscillators in a global sense. The dynamics of individual oscillators can be glimpsed at the appropriate phase portraits. Let us now fix our attention on a detailed analysis of Fig. 32. For the limit value ei2 = 0, the dynamics of the uncoupled oscillators has already been presented in Fig. 20. In the case of very weak interaction 0 < C 2 < 0.0005, the system of coupled oscillators manifests chaotic behavior. For C 2 = 0.0005 we obtain the spectrum 0.06,0.00, —0.21, 0.54, 0.89. It is interesting to... 

Although the system (4.2) looks similar to the equations for the chemostat in Chapter 1, the analysis is more difficult because the system is no longer competitive. Stephanopoulos and Lapidus used a very clever index argument to generate phase portraits. However, such arguments are only local [HWW] determined the global asymptotic behavior. 

In this section we discuss index theory, a method that provides global information about the phase portrait. It enables us to answer such questions as Must a closed trajectory always encircle a fixed point If so, what types of fixed points are permitted What types of fixed points can coalesce in bifurcations The method also yields information about the trajectories near higher-order fixed points. Finally, we can sometimes use index arguments to rule out the possibility of closed orbits in certain parts of the phase plane. 

As we can see, in all the three cases a qualitative change in the phase portrait takes place. However, there is a fundamental difference between cases (IA), (IB) and (II) in cases (IA), (IB) the change in the phase portrait nearby a stationary point occurs while in case (II) the neighbourhood of the stationary point occurs while in case (II) the neighbourhood of the stationary point does not vary, but a global character of the phase portrait is altered. In other words, the catastrophe appearing in case (II) is finer and more difficult to examine. 

When 0stationary state (1, 1) is a stable node. When a = otj equation (6.71) has two equal real roots, kr = k2 < 0. The value kt is determined from the requirement of vanishing of the discriminant A of the quadratic equation (6.71). This is a sensitive state when a increasing exceeds the value ax, a change in the phase portrait of the system (6.67) takes place the stationary state is, for < a < a0 = (y — 1) a stable focus (the real part of kx >2 is negative). In accordance with what we established in Chapter 5, the catastrophe does not alter a phase portrait nearby a stationary point but is of a global character. 

Note that the second sensitive state is of most interest the requirement v = 0 implies the lack of diffusion, while the condition a = 0 signifies the lack of a chemical reaction. The sensitive states 1, 3 correspond to catastrophes of the change in a phase portrait in the neighbourhood of a given singular point while the sensitive state 2 represents the catastrophe of a global change of a phase portrait. 

Although many of the measurements of growth, like the ones I have already cited, might present a portrait of a company that has enjoyed a seamless path of expansion, the actual experience from innovative start-up to global brand has been less direct. I recognize four phases in our quarter-century history early start-up, explosive growth, slow growth, and transition transformation. 

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