Repelling fixed point

X2 is attracting whereas is repelling. This accounts for the sharp corner in the graph of Xoo r) at r — tq. For r > ri = 3, neither of the two fixed points of fr is stable. In order to understand the period doubling at r = ri, it is necessary to consider the fixed points (and their stability properties) of the second iterate of fr. The second iterate f x) is a quartic polynomial in x. Therefore, the fixed point equation X xf ) has four solutions given by... 

For r > 1 the fixed point x = 0 loses its stability. This means that starting at xq = e with e arbitrarily small and positive, the iterates of e will tend away from 0. Thus, for r > 1 we call x = 0 a repelling fixed point, or a repeller. 

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.

The second fixed point yields = 2 — r. Thus, X2 is repelling for... 

The fixed point becomes neutral again at r = 3, and repelling for r > 3. Therefore, the first bifurcation of the logistic map happens at r = 3. 

Each of these systems has a fixed point x — 0 with f x ) = 0. However the stability is different in each case. Figure 2.4.1 shows that (a) is stable and (b) is unstable. Case (c) is a hybrid case we II call half-stable, since the fixed point is attracting from the left and repelling from the right. We therefore indicate this type of fixed point by a half-filled circle. Case (d) is a whole line of fixed points perturbations neither grow nor decay. 

Solution If the eigenvalues are complex, the fixed point is either a center (Figure 5.2.4a) or a spiral (Figure 5.2.4b). We ve already seen an example of a center in the simple harmonic oscillator of Section 5.1 the origin is surrounded by a family of closed orbits. Note that centers are neutrally stable, since nearby trajectories are neither attracted to nor repelled from the fixed point. A spiral would occur if the harmonic oscillator were lightly damped. Then the trajectory would just fail to... 

Can we conclude that there is a closed orbit inside the trapping region No There is a fixed point in the region (at the intersection of the nullclines), and so the conditions of the Poincard-Bendixson theorem are not satisfied. But if this fi xed point is a repeller, then we can prove the existence of a closed orbit by considering... 

The repeller drives all neighboring trajectories into the shaded region, and since this region is free of fixed points, the Poincare-Bendixson theorem applies. 

Now we find conditions under which the fixed point is a repeller. 

Solution By the argument above, it suffices to find conditions under which the fixed point is a repeller, i.e., an unstable node or spiral. In general, the Jacobian is... 

All that remains is to see under what conditions (if any) the fixed point is a repeller. The Jacobian at (x, y ) is... 

Show that it is impossible for the Lorenz system to have either repelling fixed points or repelling closed orbits. (By repelling, we mean that all trajectories starting near the fixed point or closed orbit are driven away from it.)... 

Solution Repellers are incompatible with volume contraction because they are sources of volume, in the following sense. Suppose we encase a repeller with a closed surface of initial conditions nearby in phase space. (Specifically, pick a small sphere around a fixed point, or a thin tube around a closed orbit.) A short time later, the surface will have expanded as the corresponding trajectories are driven away. Thus the volume inside the surface would increase. This contradicts the fact that all volumes contract. ... 

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. ... 

Area contraction is the analog of the volume contraction that we found for the Lorenz equations in Section 9.2. As in that case, it yields several conclusions. For instance, the attractor A for the baker s map must have zero area. Also, the baker s map cannot have any repelling fixed points, since such points would expand area elements in their neighborhood. 

In the region Di there are two fixed points, one of which is a saddle, and the other one is stable for (o,6) Df, and repelling when (a, 6) G Di. Transition from Di to D2 is accompanied with the period-doubling bifurcations of the fixed point, correspondingly, stable on the route Df and repelling on... 

Insect preferences for certain types of food can be considered from a chemical ecological point of view as follows presence of attractant, fixing factor oviposition-stimulant, and feeding stimulant absence of repellent, oviposition deterrent, feeding deterrent, nutritional defect, and growth-deterrent. Conversely, the opposite is true for certain food types undisturbed by insects. 

Coulomb charged the pith balls in the apparatus with electrostatic charges. The first charged ball was fixed in place the second was attached to a horizontal bar suspended by a fiber or wire. When the two balls had like charges, they repelled one another. The force of that repulsion was measured by the distance between the two balls, which was the point where the tension in the twisting fiber equaled the force of repulsion. Using this difficult, sensitive instrument, Coulomb came up with the law now named for him. 

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