Stable fixed point

One fixed point, stable or unstable, in the stroboscopic map corresponds to a periodic solution of period t. If there are two distinct points, each mapping into the other, we have a periodic solution of twice the period. Figure 30, drawn with a continuous fill-in, illustrates the way in which different kinds of periodic solution can coexist. The invariant circle (actually ovoid with a pointed... 

Unstable Fixed Point -> Stable Limit Cycle... 

Thus, the saddle-node bifurcation leads to the appearance of a pair of fixed points, stable and unstable, whereas the period-doubling leads to the appearance of an unstable orbit of period two. An unstable fixed point vanishes at y = 0 (corresponding to a single-circuit homoclinic loop) when 0 = 0 and 0 = 7T. An unstable orbit of period two approaches y = 0 (a double homoclinic loop) when... 

Figure C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Figure C3.6.7(a) shows tire u= 0 and i )= 0 nullclines of tliis system along witli trajectories corresponding to sub-and super-tlireshold excitations. The trajectory arising from tire sub-tlireshold perturbation quickly relaxes back to tire stable fixed point. Three stages can be identified in tire trajectory resulting from tire super-tlireshold perturbation an excited stage where tire phase point quickly evolves far from tire fixed point, a refractory stage where tire system relaxes back to tire stable state and is not susceptible to additional perturbation and tire resting state where tire system again resides at tire stable fixed point.

At the critical value a = oi = 1, however, becomes unstable and the a-dependent fixed point becomes stable. This exchange of stability between two fixed points of a map is known as a transcritical bifurcation. By using the same linear-stability analysis as above, we see that remains stable if — 1 < a(l — Xjjj) < 1, or for all a such that 1 < a < 3. Something more interesting happens at a — 3. 

Fig. 4.4 A schematic representation of the pitchfork bifurcation a stable fixed point bifurcates into a period-2 limit cycle plus an unstable fixed point.

We see that the phase-plane is broken up into a sequence of fixed points and a series of both open and closed constant-energy curves. The origin (0= =0) and its periodic equivalents (0 27rn, = 0), are stable fixed points (or elliptic... 

The possible asymptotic, or equilibrium, values of the density Poo = hmt >oo p(t) are obtained by solving for the stable fixed points of the equation p = f p). Recall that a given solution p is stable if f p) < 1. 

Table 7.3 lists the four rules in this minimally-diluted rule-family, along with their corresponding iterative maps. Notice that since rules R, R2 and R3 do not have a linear term, / (p = 0) = 0 and mean-field-theory predicts a first-order phase transition. By first order we mean that the phase transition is discontinuous there is an abrupt, discontinuous change at a well defined critical probability Pc, at which the system suddenly goes from having poo = 0 as the only stable fixed point to having an asymptotic density Poo 7 0 as the only stable fixed point (see below). 

Figure 7.8 shows a plot of the iterative map /2(p) for rule R2 as a function of p for four different values of p p = 1 (top curve), p > Pc, P = Pc and p < Pc, where Pc 0.5347. Notice that all four curves have zero first and second derivatives at the origin. This ensures the existence of some critical value Pc such that for all p < Pc, p t + 1) < p t) and thus that limt->oo p t) = 0. In fact, for all 0 < p < Pc the origin is the only stable fixed point. At p = Pc, another stable fixed point ps 0.373 appears via a tangent bifurcation. For values of p greater than Pc, /2 undergoes a... 

Now let us consider the stability of the two systems around fixed points of /(cr), and therefore around homogeneous solutions of the CML. From chapter 4 we recall that (7 = 0 is a stable fixed point for a < 1 and cr = 1 - 1/a is a stable fixed point for 1 < a < 3. Let us see whether our diffusive coupling leads to any instability. 

Landau proposed in 1944 that turbulence arises essentially through the emergence of an ever increasing number of quasi-periodic motions resulting from successive bifurcations of the fluid system [landau44]. For small TZ, the fluid motion is, as we have seen, laminar, corresponding to a stable fixed point in phase space. As Ti is... 

Two alternatives present themselves in formulating algorithms for the tracking of segments of stable and unstable manifolds. The first involves observing the initial value problem for an appropriately chosen familv of initial conditions, henceforth referred to as simulation of invariant manifolds. A second generation of algorithms for the computation of invariant manifolds involves numerical fixed point techniques. 

The geometric version of TST laid out in Section II is centered around the NHIM that defines the dividing surface and its stable and unstable manifolds that act as separatrices. The NHIMs at different energies are in turn organized by the saddle point. It forms a fixed point of the dynamics—that is it is itself an invariant object—and it provides the Archimedean point in which the geometric phase-space structure is anchored. 

Increasing a leads to the effective double-well potential shown earlier with two elliptic (stable) and one hyperbolic (unstable) fixed points. The elliptic fixed points become unstable for parameter values below... 

Figure 3. Classical phase portraits (upper panel), residual quantum wavefunctions (middle panel), and ionization probability versus time (in units of the period T) (bottom panel). The parameters are (A) F = 5.0, iv = 0.52 (B) F = 20, iv = 1.04 and (C) F = 10 and u> = 2.0. Note that the peak structure of the final wavefunction reflects both stable and unstable classical fixed points. For case C, the peaks are beginning to coalesce reflecting the approach of the single-well effective potentiai (see text). 

Figure Jh Homoclinic tangle associated with the fixed point at (—a, 0) for case A. Near the fixed point, the solid line gives the unstable direction while the dashed line is the stable direction. The size of Planck s constant h is shown to illustrate that several states can be supported by the single structure. An estimate of the number of states is given by the number of h boxes needed to cover the structure.

Remark 1. The concept for local stable and unstable manifolds becomes clear when one represents the stable and unstable manifolds of the hyperbolic fixed point (periodic orbit) locally. For details see (Wiggins, 1989) or (Wiggins, 1988). 

In order to identify the periodic orbits (POs) of the problem, we need to extract the periodic points (or fixed points) from the Poincare map. Adopting the energy F = 0.65 eV, Fig. 31 displays the periodic points associated with some representative POs of the mapped two-state system. The properties of the orbits are collected in Table VI. The orbits are labeled by a Roman numeral that indicates how often trajectory intersects the surfaces of section during a cycle of the periodic orbit. For example, the two orbits that intersect only a single time are labeled la and lb and are referred to as orbits of period 1. The corresponding periodic points are located on the p = 0 axis at x = 3.330 and x = —2.725, respectively. Generally speaking, most of the short POs are stable and located in... 

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