Fixed points attracting

Should the solid undergo a two-particle or a classical crossover before the single-particle one, nesting at Q0 is not relevant. Below the two-particle crossover, the RPA pole defines the attractive fixed point. Equation (32) can be simplified to the bare essential elements... 

A pattern emerges which explains the period doubling cascade seen in Fig. 1.8. At r = rk,k = 1,2,..., the 2 attracting fixed points of become unstable and has to be considered. Of the fixed... 

Fig. 2.3 shows a qualitative sketch of Fig. 1.8. Solid lines indicate attracting fixed points and cycles, dashed lines indicate repelling fixed points and cycles.

We say that x = 0 is an attracting fixed point in Figures 5.1.5a-c all trajectories that start near x approach it as z —> oo. That is, x(z) —> x as z oo. in fact x attracts all trajectories in the phase plane, so it could be called globally attracting. 

Our example also illustrates some general mathematical concepts. Given an attracting fixed point x, we define its basin of attraction to be the set of initial conditions Xfl such that x(r) —> x as t —> w. For instance, the basin of attraction for the node at (3,0) consists of all the points lying below the stable manifold of the saddle. This basin is shown as the shaded region in Figure 6.4.8. 

Show that a conservative system cannot have any attracting fixed points. Solution Suppose x were an attracting fixed point. Then all points in its basin... 

If attracting fixed points can t occur, then what kind of fixed points can occur One generally finds saddles and centers, as in the next example. 

To show that the system is not conservative, it suffices to show that it has an attracting fixed point. (Recall that a conservative system can never have an attracting fixed point—see Example 6.5.1.)... 

Solution No. Trajectories are repelled to infinity, and never return. So infinity acts like an attracting fixed point. Chaotic behavior should be aperiodic, and that excludes fixed points as well as periodic behavior. ... 

Since the sequence xn must be bounded, the parameter b has to satisfy the condition 0 < 4b < 1. Hence, when be[0, 1/4], the sequence xn approaches zero, i.e. the attracting fixed point x (1) irrespective of the xt value (xt must be different from x <2)). 

In the N-chain model this attractive fixed point is... 

The common boundary of basins of attraction of these two fixed points is a 1- dimensional line, which is also an invariant submanifold for the renormalization flows. This line has one attractive fixed point ... 

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