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Eventually-fixed point

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... [Pg.171]

Lemma 2 follows immediately from the additivity of rule R90, since any configuration is a superposition of configurations emerging from the initial states with a single nonzero site. The proof of Lemma 3 is left as an exercise for the reader. Note, however, that the first part of Lemma 3 shows that when N = 2 all configurations must eventually evolve to a fixed point consisting of all zero sites, since... [Pg.240]

For small enough values of p so that pf p) < p for all 0 < p < 1, p = 0 will be the only fixed point. As p increases, there will eventually be some density p for which pf p ) > p in this case, we can expect there to be nonzero fixed point densities as well. Qualitatively, the mean-field-predicted behaviors will depend on the shape of the iterative map. If / has a concave downward profile, for example (i.e. if/" < 0 everywhere), then, as p decreases, Poo decreases continuously to zero at some critical value of p = Pc- Note also that the iterative map /jet for the deterministic rule associated with its minimally diluted probabilistic counterpart is given by /jet = //p-... [Pg.356]

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. [Pg.330]

When the initial system is sufficiently far from 0, the system approaches Si or S2 Both correspond to homochiral states with 0i OO = 1. When the initial configuration is close to the racemic state or the diagonal line r = s, the system approaches a racemic fixed point U4 at first. But, while the recycling process returns the chiral enantiomers back to achiral substrate, the majority enantiomer increases its population at the cost of the minority one along the flow curve diverging from U4 to Si or to S2, and eventually the whole system becomes homochiral. [Pg.114]

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. [Pg.505]

We can now see that the rate of relaxation of a perturbation depends on the distance of the perturbed solution from the true solution (or fixed point), and on the magnitude of the Jacobian. If the fixed point is stationary, then the perturbed solution tends exactly to the fixed point. This case is illustrated via the reaction T with kf= ki, = 10 s (Fig. 4.9) which has a simple steady-state corresponding to an equilibrium with X=Y = 1. Solutions with initial conditions away from the steady-state eventually relax to the steady-state. [Pg.376]

Figure 2.1.1 shows that a particle starting at Xg = r/4 moves to the right faster and faster until it crosses x = 7t/2 (where sinx reaches its maximum). Then the particle starts slowing down and eventually approaches the stable fixed point x = rr from the left. Thus, the qualitative form of the solution is as shown in Figure 2.1.2. [Pg.17]

When p = 0, all trajectories flow toward a stable fixed point at 0 = 0 (Figure 4.5.1a). Thus the firefly eventually entrains with zero phase difference in the case Q = a>. In other words, the firefly and the stimulus flash simultaneously if the firefly is driven at its natural frequency. [Pg.105]

If we continue to increase p, the stable and unstable fixed points eventually coalesce in a saddle-node bifurcation at // = 1. For p > 1 both fixed points have disappeared and now phase-locking is lost the phase difference 0 increases indefinitely, corresponding to phase drift (Figure 4.5.1c). (Of course, once 0 reaches 2jt the oscillators are in phase again.) Notice that the phases don t separate at a uniform rate, in qualitative agreement with the experiments of Hanson (1978) 0 increases most slowly when it passes under the minimum of the sine wave in Figure 4.5.1 c, at 0 = r/2, and most rapidly when it passes under the maximum at 0 = -kI2. ... [Pg.105]

What is the fate of such a bounded trajectory If there are fixed points inside C, then of course the trajectory might eventually approach one of them. But what if there aren t any fixed points Your intuition may tell you that the trajectory can t... [Pg.149]

In this section we ll follow in Lorenz s footsteps. He took the analysis as far as possible using standard techniques, but at a certain stage he found himself confronted with what seemed like a paradox. One by one he had eliminated all the known possibilities for the long-term behavior of his system he showed that in a certain range of parameters, there could be no stable fixed points and no stable limit cycles, yet he also proved that all trajectories remain confined to a bounded region and are eventually attracted to a set of zero volume. What could that set be And how do the trajectories move on it As we ll see in the next section, that set is the strange attractor, and the motion on it is chaotic. [Pg.311]

Hence, if we start with a enormous solid blob of initial conditions, it eventually shrinks to a limiting set of zero volume, like a balloon with the air being sucked out of it. All trajectories starting in the blob end up somewhere in this limiting set later we ll see it consists of fixed points, limit cycles, or for some parameter values, a strange attractor. [Pg.313]

At first the trajectory seems to be tracing out a strange attractor, but eventually it stays on the right and spirals down toward the stable fixed point (Recall that both C and C are still stable at r = 21. jThetimeseriesof y vs. t shows the same result an initially erratic solution ultimately damps down to equilibrium (Figure 9.5.3). [Pg.332]

Figure 9.15 Evolution of the polyad phase sphere from the local mode to the normal mode limit as the strength of the 1 1 coupling term (antithetical to the local mode limit) is increased from 0 (part a) to oo (part f). As the coupling term, <5 in Eq. (9.4.174) increases from 0, first in part (c) one trajectory (level 4, at highest E) falls through the unstable fixed point into the normal mode region (antisymmetric stretch) eventually, in part (e), the resonance zone fills the entire phase sphere finally, in part (f), the normal mode limit is reached (from Xiao and Kellman, 1989). Figure 9.15 Evolution of the polyad phase sphere from the local mode to the normal mode limit as the strength of the 1 1 coupling term (antithetical to the local mode limit) is increased from 0 (part a) to oo (part f). As the coupling term, <5 in Eq. (9.4.174) increases from 0, first in part (c) one trajectory (level 4, at highest E) falls through the unstable fixed point into the normal mode region (antisymmetric stretch) eventually, in part (e), the resonance zone fills the entire phase sphere finally, in part (f), the normal mode limit is reached (from Xiao and Kellman, 1989).
Figure 16 Sketches of numerically generated separatrices on the Poincare map. After extracting the manodromy matrix of a hyperbolic fixed point (p2> the asymptotic eigenvectors W+ and W can be obtained. (A) If no other fix points are nearby in the chaotic sea, the separatrix branch formed by repeated mappings in positive time of points initially on will eventually meet the branch formed by repeated mappings in negative time of points initially on W at a single point hj (called a homoclinic point). The closed curve so generated is the separatrix 5. (B) If a second fixed point (p2> 2)2 associated with a different periodic orbit is nearby, a separatrix 5 may be formed by the intersection of branches arising from the two orbits at two points hi and (called heteroclinic points). Figure 16 Sketches of numerically generated separatrices on the Poincare map. After extracting the manodromy matrix of a hyperbolic fixed point (p2> the asymptotic eigenvectors W+ and W can be obtained. (A) If no other fix points are nearby in the chaotic sea, the separatrix branch formed by repeated mappings in positive time of points initially on will eventually meet the branch formed by repeated mappings in negative time of points initially on W at a single point hj (called a homoclinic point). The closed curve so generated is the separatrix 5. (B) If a second fixed point (p2> 2)2 associated with a different periodic orbit is nearby, a separatrix 5 may be formed by the intersection of branches arising from the two orbits at two points hi and (called heteroclinic points).
Plate-Column Capacity The maximum allowable capacity of a plate for handling gas and liquid flow is of primaiy importance because it fixes the minimum possible diameter of the column. For a constant hquid rate, increasing the gas rate results eventually in excessive entrainment and flooding. At the flood point it is difficult to obtain net downward flow of hquid, and any liquid fed to the column is carried out with the overheaa gas. Furthermore, the column inven-toiy of hquid increases, pressure drop across the column becomes quite large, and control becomes difficult. Rational design caUs for operation at a safe margin below this maximum aUowable condition. [Pg.1371]

Extensive experiments were in fact needed before optimal test and acquisition conditions were eventually set (for details, see ). In any fixed strain and frequency conditions, data acquisition is made in order to record 10,240 points at the rate of 512 pt/s. Twenty cycles are consequently recorded at each strain step, with the immediate requirement that the instrument is set in order to apply a sufficient number of cycles (for instance, 40 cycles at 1.0 Hz, 20 cycles at 0.5 Hz the stability condition with the RPA) for the steady harmonic regime to be reached. Data acquisition is activated as the set strain is reached and stable. [Pg.825]


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See also in sourсe #XX -- [ Pg.4 , Pg.6 ]




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